LMFDB - Embedded newform 4304.2.a.k.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4304 = 2^{4} \cdot 269$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{15} - 3 x^{14} - 25 x^{13} + 73 x^{12} + 236 x^{11} - 652 x^{10} - 1091 x^{9} + 2672 x^{8} + \cdots + 272$ x^15 - 3x^14 - 25x^13 + 73x^12 + 236x^11 - 652x^10 - 1091x^9 + 2672x^8 + 2772x^7 - 5331x^6 - 3794x^5 + 5012x^4 + 2263x^3 - 2106x^2 - 412x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-2.86111$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.86111 q^{3} -1.55421 q^{5} -1.29443 q^{7} +5.18596 q^{9} +2.30728 q^{11} -6.92228 q^{13} -4.44678 q^{15} +1.59232 q^{17} -6.13155 q^{19} -3.70352 q^{21} -0.902947 q^{23} -2.58442 q^{25} +6.25426 q^{27} -7.64018 q^{29} +8.63662 q^{31} +6.60140 q^{33} +2.01183 q^{35} +5.09826 q^{37} -19.8054 q^{39} -5.81559 q^{41} +7.88367 q^{43} -8.06009 q^{45} -9.56763 q^{47} -5.32444 q^{49} +4.55580 q^{51} +2.12772 q^{53} -3.58602 q^{55} -17.5431 q^{57} +2.37427 q^{59} -0.479381 q^{61} -6.71287 q^{63} +10.7587 q^{65} -2.84815 q^{67} -2.58343 q^{69} -16.4286 q^{71} -5.70632 q^{73} -7.39430 q^{75} -2.98663 q^{77} -13.0537 q^{79} +2.33627 q^{81} -0.978767 q^{83} -2.47481 q^{85} -21.8594 q^{87} +1.73484 q^{89} +8.96043 q^{91} +24.7103 q^{93} +9.52975 q^{95} +6.73326 q^{97} +11.9655 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$15 q - 3 q^{3} + 5 q^{5} - 13 q^{7} + 14 q^{9} - 9 q^{11} - 2 q^{13} - 19 q^{15} + 6 q^{17} - 6 q^{19} + q^{21} - 25 q^{23} + 8 q^{25} - 15 q^{27} + 4 q^{29} - 19 q^{31} + 11 q^{33} - 16 q^{35} + 3 q^{37}+ \cdots - 16 q^{99}+O(q^{100})$ 15 * q - 3 * q^3 + 5 * q^5 - 13 * q^7 + 14 * q^9 - 9 * q^11 - 2 * q^13 - 19 * q^15 + 6 * q^17 - 6 * q^19 + q^21 - 25 * q^23 + 8 * q^25 - 15 * q^27 + 4 * q^29 - 19 * q^31 + 11 * q^33 - 16 * q^35 + 3 * q^37 - 38 * q^39 - 2 * q^41 - 8 * q^43 - 32 * q^47 - 2 * q^49 - 15 * q^51 + 16 * q^53 - 27 * q^55 - 2 * q^57 - 17 * q^59 - 25 * q^61 - 33 * q^63 + 14 * q^65 - 20 * q^67 - 7 * q^69 - 47 * q^71 - 12 * q^73 - 7 * q^75 + 18 * q^77 - 64 * q^79 + 15 * q^81 - 29 * q^83 - 11 * q^85 - 19 * q^87 + 6 * q^89 - 13 * q^91 + 19 * q^93 - 50 * q^95 - 10 * q^97 - 16 * q^99

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	2.86111	1.65186	0.825932	−	0.563770i	$-0.190650\pi$
$3$	2.86111	1.65186	0.825932	+	0.563770i	$0.190650\pi$
$4$	0	0
$5$	−1.55421	−0.695066	−0.347533	−	0.937668i	$-0.612981\pi$
$5$	−1.55421	−0.695066	−0.347533	+	0.937668i	$0.612981\pi$
$6$	0	0
$7$	−1.29443	−0.489250	−0.244625	−	0.969618i	$-0.578665\pi$
$7$	−1.29443	−0.489250	−0.244625	+	0.969618i	$0.578665\pi$
$8$	0	0
$9$	5.18596	1.72865
$10$	0	0
$11$	2.30728	0.695672	0.347836	−	0.937555i	$-0.386917\pi$
$11$	2.30728	0.695672	0.347836	+	0.937555i	$0.386917\pi$
$12$	0	0
$13$	−6.92228	−1.91989	−0.959947	−	0.280181i	$-0.909605\pi$
$13$	−6.92228	−1.91989	−0.959947	+	0.280181i	$0.909605\pi$
$14$	0	0
$15$	−4.44678	−1.14815
$16$	0	0
$17$	1.59232	0.386194	0.193097	−	0.981180i	$-0.438147\pi$
$17$	1.59232	0.386194	0.193097	+	0.981180i	$0.438147\pi$
$18$	0	0
$19$	−6.13155	−1.40667	−0.703337	−	0.710856i	$-0.748308\pi$
$19$	−6.13155	−1.40667	−0.703337	+	0.710856i	$0.748308\pi$
$20$	0	0
$21$	−3.70352	−0.808174
$22$	0	0
$23$	−0.902947	−0.188277	−0.0941387	−	0.995559i	$-0.530010\pi$
$23$	−0.902947	−0.188277	−0.0941387	+	0.995559i	$0.530010\pi$
$24$	0	0
$25$	−2.58442	−0.516883
$26$	0	0
$27$	6.25426	1.20363
$28$	0	0
$29$	−7.64018	−1.41875	−0.709373	−	0.704834i	$-0.751022\pi$
$29$	−7.64018	−1.41875	−0.709373	+	0.704834i	$0.751022\pi$
$30$	0	0
$31$	8.63662	1.55118	0.775591	−	0.631235i	$-0.217452\pi$
$31$	8.63662	1.55118	0.775591	+	0.631235i	$0.217452\pi$
$32$	0	0
$33$	6.60140	1.14916
$34$	0	0
$35$	2.01183	0.340061
$36$	0	0
$37$	5.09826	0.838149	0.419075	−	0.907952i	$-0.362355\pi$
$37$	5.09826	0.838149	0.419075	+	0.907952i	$0.362355\pi$
$38$	0	0
$39$	−19.8054	−3.17140
$40$	0	0
$41$	−5.81559	−0.908243	−0.454121	−	0.890940i	$-0.650047\pi$
$41$	−5.81559	−0.908243	−0.454121	+	0.890940i	$0.650047\pi$
$42$	0	0
$43$	7.88367	1.20225	0.601124	−	0.799155i	$-0.294720\pi$
$43$	7.88367	1.20225	0.601124	+	0.799155i	$0.294720\pi$
$44$	0	0
$45$	−8.06009	−1.20153
$46$	0	0
$47$	−9.56763	−1.39558	−0.697791	−	0.716301i	$-0.745834\pi$
$47$	−9.56763	−1.39558	−0.697791	+	0.716301i	$0.745834\pi$
$48$	0	0
$49$	−5.32444	−0.760635
$50$	0	0
$51$	4.55580	0.637940
$52$	0	0
$53$	2.12772	0.292265	0.146133	−	0.989265i	$-0.453317\pi$
$53$	2.12772	0.292265	0.146133	+	0.989265i	$0.453317\pi$
$54$	0	0
$55$	−3.58602	−0.483538
$56$	0	0
$57$	−17.5431	−2.32363
$58$	0	0
$59$	2.37427	0.309104	0.154552	−	0.987985i	$-0.450607\pi$
$59$	2.37427	0.309104	0.154552	+	0.987985i	$0.450607\pi$
$60$	0	0
$61$	−0.479381	−0.0613785	−0.0306893	−	0.999529i	$-0.509770\pi$
$61$	−0.479381	−0.0613785	−0.0306893	+	0.999529i	$0.509770\pi$
$62$	0	0
$63$	−6.71287	−0.845742
$64$	0	0
$65$	10.7587	1.33445
$66$	0	0
$67$	−2.84815	−0.347957	−0.173979	−	0.984749i	$-0.555662\pi$
$67$	−2.84815	−0.347957	−0.173979	+	0.984749i	$0.555662\pi$
$68$	0	0
$69$	−2.58343	−0.311008
$70$	0	0
$71$	−16.4286	−1.94971	−0.974856	−	0.222837i	$-0.928468\pi$
$71$	−16.4286	−1.94971	−0.974856	+	0.222837i	$0.928468\pi$
$72$	0	0
$73$	−5.70632	−0.667875	−0.333937	−	0.942595i	$-0.608377\pi$
$73$	−5.70632	−0.667875	−0.333937	+	0.942595i	$0.608377\pi$
$74$	0	0
$75$	−7.39430	−0.853820
$76$	0	0
$77$	−2.98663	−0.340358
$78$	0	0
$79$	−13.0537	−1.46866	−0.734328	−	0.678795i	$-0.762503\pi$
$79$	−13.0537	−1.46866	−0.734328	+	0.678795i	$0.762503\pi$
$80$	0	0
$81$	2.33627	0.259585
$82$	0	0
$83$	−0.978767	−0.107434	−0.0537168	−	0.998556i	$-0.517107\pi$
$83$	−0.978767	−0.107434	−0.0537168	+	0.998556i	$0.517107\pi$
$84$	0	0
$85$	−2.47481	−0.268430
$86$	0	0
$87$	−21.8594	−2.34357
$88$	0	0
$89$	1.73484	0.183892	0.0919462	−	0.995764i	$-0.470691\pi$
$89$	1.73484	0.183892	0.0919462	+	0.995764i	$0.470691\pi$
$90$	0	0
$91$	8.96043	0.939308
$92$	0	0
$93$	24.7103	2.56234
$94$	0	0
$95$	9.52975	0.977732
$96$	0	0
$97$	6.73326	0.683659	0.341830	−	0.939762i	$-0.388953\pi$
$97$	6.73326	0.683659	0.341830	+	0.939762i	$0.388953\pi$
$98$	0	0
$99$	11.9655	1.20258
$100$	0	0
$101$	−9.58202	−0.953447	−0.476723	−	0.879053i	$-0.658176\pi$
$101$	−9.58202	−0.953447	−0.476723	+	0.879053i	$0.658176\pi$
$102$	0	0
$103$	1.44410	0.142291	0.0711455	−	0.997466i	$-0.477335\pi$
$103$	1.44410	0.142291	0.0711455	+	0.997466i	$0.477335\pi$
$104$	0	0
$105$	5.75606	0.561734
$106$	0	0
$107$	−12.7616	−1.23372	−0.616858	−	0.787075i	$-0.711595\pi$
$107$	−12.7616	−1.23372	−0.616858	+	0.787075i	$0.711595\pi$
$108$	0	0
$109$	3.73136	0.357399	0.178700	−	0.983904i	$-0.442811\pi$
$109$	3.73136	0.357399	0.178700	+	0.983904i	$0.442811\pi$
$110$	0	0
$111$	14.5867	1.38451
$112$	0	0
$113$	6.42695	0.604597	0.302298	−	0.953213i	$-0.402246\pi$
$113$	6.42695	0.604597	0.302298	+	0.953213i	$0.402246\pi$
$114$	0	0
$115$	1.40337	0.130865
$116$	0	0
$117$	−35.8986	−3.31883
$118$	0	0
$119$	−2.06115	−0.188945
$120$	0	0
$121$	−5.67644	−0.516040
$122$	0	0
$123$	−16.6391	−1.50029
$124$	0	0
$125$	11.7878	1.05433
$126$	0	0
$127$	−13.2716	−1.17766	−0.588831	−	0.808256i	$-0.700412\pi$
$127$	−13.2716	−1.17766	−0.588831	+	0.808256i	$0.700412\pi$
$128$	0	0
$129$	22.5561	1.98595
$130$	0	0
$131$	10.3728	0.906277	0.453138	−	0.891440i	$-0.350304\pi$
$131$	10.3728	0.906277	0.453138	+	0.891440i	$0.350304\pi$
$132$	0	0
$133$	7.93689	0.688215
$134$	0	0
$135$	−9.72046	−0.836604
$136$	0	0
$137$	14.9538	1.27759	0.638796	−	0.769377i	$-0.279433\pi$
$137$	14.9538	1.27759	0.638796	+	0.769377i	$0.279433\pi$
$138$	0	0
$139$	12.0388	1.02111	0.510557	−	0.859844i	$-0.329439\pi$
$139$	12.0388	1.02111	0.510557	+	0.859844i	$0.329439\pi$
$140$	0	0
$141$	−27.3740	−2.30531
$142$	0	0
$143$	−15.9717	−1.33562
$144$	0	0
$145$	11.8745	0.986122
$146$	0	0
$147$	−15.2338	−1.25646
$148$	0	0
$149$	21.4746	1.75927	0.879635	−	0.475650i	$-0.157787\pi$
$149$	21.4746	1.75927	0.879635	+	0.475650i	$0.157787\pi$
$150$	0	0
$151$	17.6466	1.43606	0.718030	−	0.696012i	$-0.245044\pi$
$151$	17.6466	1.43606	0.718030	+	0.696012i	$0.245044\pi$
$152$	0	0
$153$	8.25770	0.667595
$154$	0	0
$155$	−13.4232	−1.07817
$156$	0	0
$157$	12.3281	0.983891	0.491945	−	0.870626i	$-0.336286\pi$
$157$	12.3281	0.983891	0.491945	+	0.870626i	$0.336286\pi$
$158$	0	0
$159$	6.08765	0.482782
$160$	0	0
$161$	1.16880	0.0921147
$162$	0	0
$163$	−21.1082	−1.65332	−0.826660	−	0.562702i	$-0.809762\pi$
$163$	−21.1082	−1.65332	−0.826660	+	0.562702i	$0.809762\pi$
$164$	0	0
$165$	−10.2600	−0.798739
$166$	0	0
$167$	−14.1868	−1.09780	−0.548902	−	0.835886i	$-0.684954\pi$
$167$	−14.1868	−1.09780	−0.548902	+	0.835886i	$0.684954\pi$
$168$	0	0
$169$	34.9179	2.68599
$170$	0	0
$171$	−31.7980	−2.43165
$172$	0	0
$173$	14.6315	1.11241	0.556207	−	0.831044i	$-0.312256\pi$
$173$	14.6315	1.11241	0.556207	+	0.831044i	$0.312256\pi$
$174$	0	0
$175$	3.34535	0.252885
$176$	0	0
$177$	6.79306	0.510597
$178$	0	0
$179$	−0.881802	−0.0659090	−0.0329545	−	0.999457i	$-0.510492\pi$
$179$	−0.881802	−0.0659090	−0.0329545	+	0.999457i	$0.510492\pi$
$180$	0	0
$181$	8.21231	0.610416	0.305208	−	0.952286i	$-0.401274\pi$
$181$	8.21231	0.610416	0.305208	+	0.952286i	$0.401274\pi$
$182$	0	0
$183$	−1.37156	−0.101389
$184$	0	0
$185$	−7.92380	−0.582569
$186$	0	0
$187$	3.67393	0.268665
$188$	0	0
$189$	−8.09572	−0.588877
$190$	0	0
$191$	−21.4182	−1.54977	−0.774883	−	0.632105i	$-0.782191\pi$
$191$	−21.4182	−1.54977	−0.774883	+	0.632105i	$0.782191\pi$
$192$	0	0
$193$	−10.2805	−0.740007	−0.370003	−	0.929030i	$-0.620644\pi$
$193$	−10.2805	−0.740007	−0.370003	+	0.929030i	$0.620644\pi$
$194$	0	0
$195$	30.7819	2.20433
$196$	0	0
$197$	−26.6405	−1.89806	−0.949028	−	0.315191i	$-0.897931\pi$
$197$	−26.6405	−1.89806	−0.949028	+	0.315191i	$0.897931\pi$
$198$	0	0
$199$	−21.6989	−1.53820	−0.769098	−	0.639131i	$-0.779294\pi$
$199$	−21.6989	−1.53820	−0.769098	+	0.639131i	$0.779294\pi$
$200$	0	0
$201$	−8.14888	−0.574777
$202$	0	0
$203$	9.88970	0.694121
$204$	0	0
$205$	9.03868	0.631289
$206$	0	0
$207$	−4.68264	−0.325466
$208$	0	0
$209$	−14.1472	−0.978585
$210$	0	0
$211$	4.63038	0.318768	0.159384	−	0.987217i	$-0.449049\pi$
$211$	4.63038	0.318768	0.159384	+	0.987217i	$0.449049\pi$
$212$	0	0
$213$	−47.0039	−3.22066
$214$	0	0
$215$	−12.2529	−0.835642
$216$	0	0
$217$	−11.1795	−0.758916
$218$	0	0
$219$	−16.3264	−1.10324
$220$	0	0
$221$	−11.0225	−0.741452
$222$	0	0
$223$	8.44334	0.565408	0.282704	−	0.959207i	$-0.408769\pi$
$223$	8.44334	0.565408	0.282704	+	0.959207i	$0.408769\pi$
$224$	0	0
$225$	−13.4027	−0.893511
$226$	0	0
$227$	−15.3179	−1.01668	−0.508341	−	0.861156i	$-0.669741\pi$
$227$	−15.3179	−1.01668	−0.508341	+	0.861156i	$0.669741\pi$
$228$	0	0
$229$	4.52536	0.299044	0.149522	−	0.988758i	$-0.452226\pi$
$229$	4.52536	0.299044	0.149522	+	0.988758i	$0.452226\pi$
$230$	0	0
$231$	−8.54507	−0.562224
$232$	0	0
$233$	2.88395	0.188934	0.0944668	−	0.995528i	$-0.469885\pi$
$233$	2.88395	0.188934	0.0944668	+	0.995528i	$0.469885\pi$
$234$	0	0
$235$	14.8701	0.970022
$236$	0	0
$237$	−37.3481	−2.42602
$238$	0	0
$239$	11.0758	0.716432	0.358216	−	0.933639i	$-0.383385\pi$
$239$	11.0758	0.716432	0.358216	+	0.933639i	$0.383385\pi$
$240$	0	0
$241$	−15.6121	−1.00566	−0.502831	−	0.864385i	$-0.667708\pi$
$241$	−15.6121	−1.00566	−0.502831	+	0.864385i	$0.667708\pi$
$242$	0	0
$243$	−12.0785	−0.774834
$244$	0	0
$245$	8.27533	0.528691
$246$	0	0
$247$	42.4443	2.70067
$248$	0	0
$249$	−2.80036	−0.177466
$250$	0	0
$251$	21.6605	1.36720	0.683600	−	0.729857i	$-0.260413\pi$
$251$	21.6605	1.36720	0.683600	+	0.729857i	$0.260413\pi$
$252$	0	0
$253$	−2.08335	−0.130979
$254$	0	0
$255$	−7.08070	−0.443410
$256$	0	0
$257$	13.8992	0.867009	0.433504	−	0.901152i	$-0.357277\pi$
$257$	13.8992	0.867009	0.433504	+	0.901152i	$0.357277\pi$
$258$	0	0
$259$	−6.59936	−0.410064
$260$	0	0
$261$	−39.6216	−2.45252
$262$	0	0
$263$	5.07172	0.312736	0.156368	−	0.987699i	$-0.450021\pi$
$263$	5.07172	0.312736	0.156368	+	0.987699i	$0.450021\pi$
$264$	0	0
$265$	−3.30694	−0.203144
$266$	0	0
$267$	4.96356	0.303765
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	16.0387	0.974284	0.487142	−	0.873323i	$-0.338039\pi$
$271$	16.0387	0.974284	0.487142	+	0.873323i	$0.338039\pi$
$272$	0	0
$273$	25.6368	1.55161
$274$	0	0
$275$	−5.96298	−0.359581
$276$	0	0
$277$	3.53765	0.212557	0.106278	−	0.994336i	$-0.466107\pi$
$277$	3.53765	0.212557	0.106278	+	0.994336i	$0.466107\pi$
$278$	0	0
$279$	44.7891	2.68146
$280$	0	0
$281$	16.7978	1.00207	0.501036	−	0.865427i	$-0.332952\pi$
$281$	16.7978	1.00207	0.501036	+	0.865427i	$0.332952\pi$
$282$	0	0
$283$	1.12446	0.0668425	0.0334212	−	0.999441i	$-0.489360\pi$
$283$	1.12446	0.0668425	0.0334212	+	0.999441i	$0.489360\pi$
$284$	0	0
$285$	27.2657	1.61508
$286$	0	0
$287$	7.52790	0.444358
$288$	0	0
$289$	−14.4645	−0.850854
$290$	0	0
$291$	19.2646	1.12931
$292$	0	0
$293$	2.61290	0.152648	0.0763238	−	0.997083i	$-0.475682\pi$
$293$	2.61290	0.152648	0.0763238	+	0.997083i	$0.475682\pi$
$294$	0	0
$295$	−3.69013	−0.214848
$296$	0	0
$297$	14.4304	0.837334
$298$	0	0
$299$	6.25045	0.361473
$300$	0	0
$301$	−10.2049	−0.588200
$302$	0	0
$303$	−27.4152	−1.57496
$304$	0	0
$305$	0.745062	0.0426621
$306$	0	0
$307$	10.8096	0.616938	0.308469	−	0.951234i	$-0.400183\pi$
$307$	10.8096	0.616938	0.308469	+	0.951234i	$0.400183\pi$
$308$	0	0
$309$	4.13172	0.235045
$310$	0	0
$311$	−14.8093	−0.839756	−0.419878	−	0.907581i	$-0.637927\pi$
$311$	−14.8093	−0.839756	−0.419878	+	0.907581i	$0.637927\pi$
$312$	0	0
$313$	−18.2766	−1.03306	−0.516528	−	0.856270i	$-0.672776\pi$
$313$	−18.2766	−1.03306	−0.516528	+	0.856270i	$0.672776\pi$
$314$	0	0
$315$	10.4332	0.587847
$316$	0	0
$317$	0.414931	0.0233049	0.0116524	−	0.999932i	$-0.496291\pi$
$317$	0.414931	0.0233049	0.0116524	+	0.999932i	$0.496291\pi$
$318$	0	0
$319$	−17.6281	−0.986982
$320$	0	0
$321$	−36.5125	−2.03793
$322$	0	0
$323$	−9.76339	−0.543250
$324$	0	0
$325$	17.8900	0.992361
$326$	0	0
$327$	10.6758	0.590375
$328$	0	0
$329$	12.3847	0.682788
$330$	0	0
$331$	14.3106	0.786583	0.393292	−	0.919414i	$-0.371336\pi$
$331$	14.3106	0.786583	0.393292	+	0.919414i	$0.371336\pi$
$332$	0	0
$333$	26.4394	1.44887
$334$	0	0
$335$	4.42664	0.241853
$336$	0	0
$337$	−35.3270	−1.92438	−0.962191	−	0.272375i	$-0.912191\pi$
$337$	−35.3270	−1.92438	−0.962191	+	0.272375i	$0.912191\pi$
$338$	0	0
$339$	18.3882	0.998711
$340$	0	0
$341$	19.9271	1.07912
$342$	0	0
$343$	15.9532	0.861390
$344$	0	0
$345$	4.01521	0.216171
$346$	0	0
$347$	−26.6983	−1.43324	−0.716619	−	0.697465i	$-0.754311\pi$
$347$	−26.6983	−1.43324	−0.716619	+	0.697465i	$0.754311\pi$
$348$	0	0
$349$	−13.3352	−0.713818	−0.356909	−	0.934139i	$-0.616169\pi$
$349$	−13.3352	−0.713818	−0.356909	+	0.934139i	$0.616169\pi$
$350$	0	0
$351$	−43.2937	−2.31085
$352$	0	0
$353$	−17.4433	−0.928415	−0.464207	−	0.885727i	$-0.653661\pi$
$353$	−17.4433	−0.928415	−0.464207	+	0.885727i	$0.653661\pi$
$354$	0	0
$355$	25.5335	1.35518
$356$	0	0
$357$	−5.89718	−0.312112
$358$	0	0
$359$	33.0076	1.74207	0.871037	−	0.491217i	$-0.163448\pi$
$359$	33.0076	1.74207	0.871037	+	0.491217i	$0.163448\pi$
$360$	0	0
$361$	18.5960	0.978734
$362$	0	0
$363$	−16.2409	−0.852427
$364$	0	0
$365$	8.86885	0.464217
$366$	0	0
$367$	−3.72202	−0.194288	−0.0971439	−	0.995270i	$-0.530971\pi$
$367$	−3.72202	−0.194288	−0.0971439	+	0.995270i	$0.530971\pi$
$368$	0	0
$369$	−30.1594	−1.57004
$370$	0	0
$371$	−2.75419	−0.142991
$372$	0	0
$373$	19.7780	1.02407	0.512034	−	0.858965i	$-0.328892\pi$
$373$	19.7780	1.02407	0.512034	+	0.858965i	$0.328892\pi$
$374$	0	0
$375$	33.7262	1.74162
$376$	0	0
$377$	52.8874	2.72384
$378$	0	0
$379$	−1.19974	−0.0616264	−0.0308132	−	0.999525i	$-0.509810\pi$
$379$	−1.19974	−0.0616264	−0.0308132	+	0.999525i	$0.509810\pi$
$380$	0	0
$381$	−37.9714	−1.94534
$382$	0	0
$383$	8.55694	0.437239	0.218620	−	0.975810i	$-0.429845\pi$
$383$	8.55694	0.437239	0.218620	+	0.975810i	$0.429845\pi$
$384$	0	0
$385$	4.64186	0.236571
$386$	0	0
$387$	40.8844	2.07827
$388$	0	0
$389$	0.452394	0.0229373	0.0114686	−	0.999934i	$-0.496349\pi$
$389$	0.452394	0.0229373	0.0114686	+	0.999934i	$0.496349\pi$
$390$	0	0
$391$	−1.43778	−0.0727116
$392$	0	0
$393$	29.6778	1.49705
$394$	0	0
$395$	20.2883	1.02081
$396$	0	0
$397$	14.2666	0.716019	0.358010	−	0.933718i	$-0.383456\pi$
$397$	14.2666	0.716019	0.358010	+	0.933718i	$0.383456\pi$
$398$	0	0
$399$	22.7083	1.13684
$400$	0	0
$401$	−5.58254	−0.278779	−0.139389	−	0.990238i	$-0.544514\pi$
$401$	−5.58254	−0.278779	−0.139389	+	0.990238i	$0.544514\pi$
$402$	0	0
$403$	−59.7851	−2.97811
$404$	0	0
$405$	−3.63106	−0.180429
$406$	0	0
$407$	11.7631	0.583077
$408$	0	0
$409$	−36.4763	−1.80363	−0.901817	−	0.432117i	$-0.857767\pi$
$409$	−36.4763	−1.80363	−0.901817	+	0.432117i	$0.857767\pi$
$410$	0	0
$411$	42.7845	2.11041
$412$	0	0
$413$	−3.07334	−0.151229
$414$	0	0
$415$	1.52121	0.0746735
$416$	0	0
$417$	34.4442	1.68674
$418$	0	0
$419$	−12.8074	−0.625685	−0.312842	−	0.949805i	$-0.601281\pi$
$419$	−12.8074	−0.625685	−0.312842	+	0.949805i	$0.601281\pi$
$420$	0	0
$421$	23.5148	1.14604	0.573020	−	0.819541i	$-0.305772\pi$
$421$	23.5148	1.14604	0.573020	+	0.819541i	$0.305772\pi$
$422$	0	0
$423$	−49.6173	−2.41248
$424$	0	0
$425$	−4.11522	−0.199617
$426$	0	0
$427$	0.620527	0.0300294
$428$	0	0
$429$	−45.6967	−2.20626
$430$	0	0
$431$	−5.77235	−0.278044	−0.139022	−	0.990289i	$-0.544396\pi$
$431$	−5.77235	−0.278044	−0.139022	+	0.990289i	$0.544396\pi$
$432$	0	0
$433$	35.7621	1.71862	0.859309	−	0.511456i	$-0.170894\pi$
$433$	35.7621	1.71862	0.859309	+	0.511456i	$0.170894\pi$
$434$	0	0
$435$	33.9742	1.62894
$436$	0	0
$437$	5.53647	0.264845
$438$	0	0
$439$	−33.8958	−1.61776	−0.808880	−	0.587974i	$-0.799926\pi$
$439$	−33.8958	−1.61776	−0.808880	+	0.587974i	$0.799926\pi$
$440$	0	0
$441$	−27.6123	−1.31487
$442$	0	0
$443$	−9.29119	−0.441438	−0.220719	−	0.975337i	$-0.570840\pi$
$443$	−9.29119	−0.441438	−0.220719	+	0.975337i	$0.570840\pi$
$444$	0	0
$445$	−2.69631	−0.127817
$446$	0	0
$447$	61.4413	2.90607
$448$	0	0
$449$	30.4556	1.43729	0.718644	−	0.695379i	$-0.244763\pi$
$449$	30.4556	1.43729	0.718644	+	0.695379i	$0.244763\pi$
$450$	0	0
$451$	−13.4182	−0.631840
$452$	0	0
$453$	50.4889	2.37218
$454$	0	0
$455$	−13.9264	−0.652881
$456$	0	0
$457$	−15.6411	−0.731660	−0.365830	−	0.930682i	$-0.619215\pi$
$457$	−15.6411	−0.731660	−0.365830	+	0.930682i	$0.619215\pi$
$458$	0	0
$459$	9.95878	0.464836
$460$	0	0
$461$	28.5609	1.33022	0.665108	−	0.746747i	$-0.268386\pi$
$461$	28.5609	1.33022	0.665108	+	0.746747i	$0.268386\pi$
$462$	0	0
$463$	−4.83573	−0.224735	−0.112368	−	0.993667i	$-0.535843\pi$
$463$	−4.83573	−0.224735	−0.112368	+	0.993667i	$0.535843\pi$
$464$	0	0
$465$	−38.4052	−1.78100
$466$	0	0
$467$	0.657615	0.0304308	0.0152154	−	0.999884i	$-0.495157\pi$
$467$	0.657615	0.0304308	0.0152154	+	0.999884i	$0.495157\pi$
$468$	0	0
$469$	3.68674	0.170238
$470$	0	0
$471$	35.2721	1.62525
$472$	0	0
$473$	18.1899	0.836371
$474$	0	0
$475$	15.8465	0.727087
$476$	0	0
$477$	11.0343	0.505225
$478$	0	0
$479$	21.1910	0.968243	0.484122	−	0.875001i	$-0.339139\pi$
$479$	21.1910	0.968243	0.484122	+	0.875001i	$0.339139\pi$
$480$	0	0
$481$	−35.2916	−1.60916
$482$	0	0
$483$	3.34408	0.152161
$484$	0	0
$485$	−10.4649	−0.475188
$486$	0	0
$487$	−5.77562	−0.261718	−0.130859	−	0.991401i	$-0.541774\pi$
$487$	−5.77562	−0.261718	−0.130859	+	0.991401i	$0.541774\pi$
$488$	0	0
$489$	−60.3928	−2.73106
$490$	0	0
$491$	−20.7316	−0.935603	−0.467802	−	0.883834i	$-0.654954\pi$
$491$	−20.7316	−0.935603	−0.467802	+	0.883834i	$0.654954\pi$
$492$	0	0
$493$	−12.1656	−0.547911
$494$	0	0
$495$	−18.5969	−0.835869
$496$	0	0
$497$	21.2657	0.953896
$498$	0	0
$499$	−6.86605	−0.307366	−0.153683	−	0.988120i	$-0.549114\pi$
$499$	−6.86605	−0.307366	−0.153683	+	0.988120i	$0.549114\pi$
$500$	0	0
$501$	−40.5899	−1.81342
$502$	0	0
$503$	29.4097	1.31131	0.655656	−	0.755060i	$-0.272392\pi$
$503$	29.4097	1.31131	0.655656	+	0.755060i	$0.272392\pi$
$504$	0	0
$505$	14.8925	0.662708
$506$	0	0
$507$	99.9040	4.43689
$508$	0	0
$509$	−7.25878	−0.321740	−0.160870	−	0.986976i	$-0.551430\pi$
$509$	−7.25878	−0.321740	−0.160870	+	0.986976i	$0.551430\pi$
$510$	0	0
$511$	7.38645	0.326757
$512$	0	0
$513$	−38.3483	−1.69312
$514$	0	0
$515$	−2.24444	−0.0989017
$516$	0	0
$517$	−22.0752	−0.970868
$518$	0	0
$519$	41.8624	1.83756
$520$	0	0
$521$	12.6514	0.554269	0.277135	−	0.960831i	$-0.410615\pi$
$521$	12.6514	0.554269	0.277135	+	0.960831i	$0.410615\pi$
$522$	0	0
$523$	30.1359	1.31775	0.658876	−	0.752252i	$-0.271032\pi$
$523$	30.1359	1.31775	0.658876	+	0.752252i	$0.271032\pi$
$524$	0	0
$525$	9.57143	0.417731
$526$	0	0
$527$	13.7523	0.599058
$528$	0	0
$529$	−22.1847	−0.964552
$530$	0	0
$531$	12.3129	0.534333
$532$	0	0
$533$	40.2571	1.74373
$534$	0	0
$535$	19.8343	0.857513
$536$	0	0
$537$	−2.52293	−0.108873
$538$	0	0
$539$	−12.2850	−0.529153
$540$	0	0
$541$	29.5871	1.27205	0.636025	−	0.771668i	$-0.280577\pi$
$541$	29.5871	1.27205	0.636025	+	0.771668i	$0.280577\pi$
$542$	0	0
$543$	23.4963	1.00832
$544$	0	0
$545$	−5.79933	−0.248416
$546$	0	0
$547$	−44.7507	−1.91340	−0.956701	−	0.291072i	$-0.905988\pi$
$547$	−44.7507	−1.91340	−0.956701	+	0.291072i	$0.905988\pi$
$548$	0	0
$549$	−2.48605	−0.106102
$550$	0	0
$551$	46.8462	1.99571
$552$	0	0
$553$	16.8971	0.718540
$554$	0	0
$555$	−22.6709	−0.962325
$556$	0	0
$557$	29.8986	1.26684	0.633422	−	0.773806i	$-0.281650\pi$
$557$	29.8986	1.26684	0.633422	+	0.773806i	$0.281650\pi$
$558$	0	0
$559$	−54.5730	−2.30819
$560$	0	0
$561$	10.5115	0.443797
$562$	0	0
$563$	14.5073	0.611412	0.305706	−	0.952126i	$-0.401108\pi$
$563$	14.5073	0.611412	0.305706	+	0.952126i	$0.401108\pi$
$564$	0	0
$565$	−9.98887	−0.420235
$566$	0	0
$567$	−3.02414	−0.127002
$568$	0	0
$569$	13.5485	0.567984	0.283992	−	0.958827i	$-0.408341\pi$
$569$	13.5485	0.567984	0.283992	+	0.958827i	$0.408341\pi$
$570$	0	0
$571$	−9.23686	−0.386551	−0.193275	−	0.981145i	$-0.561911\pi$
$571$	−9.23686	−0.386551	−0.193275	+	0.981145i	$0.561911\pi$
$572$	0	0
$573$	−61.2798	−2.56000
$574$	0	0
$575$	2.33359	0.0973174
$576$	0	0
$577$	7.94141	0.330605	0.165303	−	0.986243i	$-0.447140\pi$
$577$	7.94141	0.330605	0.165303	+	0.986243i	$0.447140\pi$
$578$	0	0
$579$	−29.4137	−1.22239
$580$	0	0
$581$	1.26695	0.0525619
$582$	0	0
$583$	4.90926	0.203321
$584$	0	0
$585$	55.7942	2.30680
$586$	0	0
$587$	6.15312	0.253966	0.126983	−	0.991905i	$-0.459471\pi$
$587$	6.15312	0.253966	0.126983	+	0.991905i	$0.459471\pi$
$588$	0	0
$589$	−52.9559	−2.18201
$590$	0	0
$591$	−76.2214	−3.13533
$592$	0	0
$593$	5.02660	0.206418	0.103209	−	0.994660i	$-0.467089\pi$
$593$	5.02660	0.206418	0.103209	+	0.994660i	$0.467089\pi$
$594$	0	0
$595$	3.20347	0.131330
$596$	0	0
$597$	−62.0830	−2.54089
$598$	0	0
$599$	15.1656	0.619651	0.309826	−	0.950793i	$-0.399729\pi$
$599$	15.1656	0.619651	0.309826	+	0.950793i	$0.399729\pi$
$600$	0	0
$601$	−4.18136	−0.170561	−0.0852806	−	0.996357i	$-0.527179\pi$
$601$	−4.18136	−0.170561	−0.0852806	+	0.996357i	$0.527179\pi$
$602$	0	0
$603$	−14.7704	−0.601496
$604$	0	0
$605$	8.82241	0.358682
$606$	0	0
$607$	−39.7611	−1.61385	−0.806926	−	0.590653i	$-0.798870\pi$
$607$	−39.7611	−1.61385	−0.806926	+	0.590653i	$0.798870\pi$
$608$	0	0
$609$	28.2955	1.14659
$610$	0	0
$611$	66.2298	2.67937
$612$	0	0
$613$	−39.4842	−1.59475	−0.797376	−	0.603483i	$-0.793779\pi$
$613$	−39.4842	−1.59475	−0.797376	+	0.603483i	$0.793779\pi$
$614$	0	0
$615$	25.8607	1.04280
$616$	0	0
$617$	8.57042	0.345032	0.172516	−	0.985007i	$-0.444810\pi$
$617$	8.57042	0.345032	0.172516	+	0.985007i	$0.444810\pi$
$618$	0	0
$619$	−40.1520	−1.61385	−0.806923	−	0.590657i	$-0.798869\pi$
$619$	−40.1520	−1.61385	−0.806923	+	0.590657i	$0.798869\pi$
$620$	0	0
$621$	−5.64726	−0.226617
$622$	0	0
$623$	−2.24563	−0.0899693
$624$	0	0
$625$	−5.39871	−0.215949
$626$	0	0
$627$	−40.4768	−1.61649
$628$	0	0
$629$	8.11806	0.323688
$630$	0	0
$631$	−13.5787	−0.540559	−0.270279	−	0.962782i	$-0.587116\pi$
$631$	−13.5787	−0.540559	−0.270279	+	0.962782i	$0.587116\pi$
$632$	0	0
$633$	13.2480	0.526561
$634$	0	0
$635$	20.6269	0.818552
$636$	0	0
$637$	36.8573	1.46034
$638$	0	0
$639$	−85.1978	−3.37037
$640$	0	0
$641$	36.4879	1.44118	0.720592	−	0.693359i	$-0.243870\pi$
$641$	36.4879	1.44118	0.720592	+	0.693359i	$0.243870\pi$
$642$	0	0
$643$	40.3693	1.59201	0.796006	−	0.605289i	$-0.206943\pi$
$643$	40.3693	1.59201	0.796006	+	0.605289i	$0.206943\pi$
$644$	0	0
$645$	−35.0570	−1.38037
$646$	0	0
$647$	−34.4600	−1.35476	−0.677380	−	0.735633i	$-0.736885\pi$
$647$	−34.4600	−1.35476	−0.677380	+	0.735633i	$0.736885\pi$
$648$	0	0
$649$	5.47812	0.215035
$650$	0	0
$651$	−31.9859	−1.25363
$652$	0	0
$653$	22.0013	0.860977	0.430488	−	0.902596i	$-0.358341\pi$
$653$	22.0013	0.860977	0.430488	+	0.902596i	$0.358341\pi$
$654$	0	0
$655$	−16.1216	−0.629922
$656$	0	0
$657$	−29.5927	−1.15452
$658$	0	0
$659$	−38.9700	−1.51806	−0.759028	−	0.651058i	$-0.774325\pi$
$659$	−38.9700	−1.51806	−0.759028	+	0.651058i	$0.774325\pi$
$660$	0	0
$661$	−24.5979	−0.956746	−0.478373	−	0.878157i	$-0.658773\pi$
$661$	−24.5979	−0.956746	−0.478373	+	0.878157i	$0.658773\pi$
$662$	0	0
$663$	−31.5365	−1.22478
$664$	0	0
$665$	−12.3356	−0.478355
$666$	0	0
$667$	6.89867	0.267118
$668$	0	0
$669$	24.1573	0.933976
$670$	0	0
$671$	−1.10607	−0.0426993
$672$	0	0
$673$	28.2478	1.08887	0.544437	−	0.838802i	$-0.316743\pi$
$673$	28.2478	1.08887	0.544437	+	0.838802i	$0.316743\pi$
$674$	0	0
$675$	−16.1636	−0.622138
$676$	0	0
$677$	−20.1384	−0.773980	−0.386990	−	0.922084i	$-0.626485\pi$
$677$	−20.1384	−0.773980	−0.386990	+	0.922084i	$0.626485\pi$
$678$	0	0
$679$	−8.71576	−0.334480
$680$	0	0
$681$	−43.8261	−1.67942
$682$	0	0
$683$	14.3932	0.550742	0.275371	−	0.961338i	$-0.411199\pi$
$683$	14.3932	0.550742	0.275371	+	0.961338i	$0.411199\pi$
$684$	0	0
$685$	−23.2414	−0.888010
$686$	0	0
$687$	12.9476	0.493980
$688$	0	0
$689$	−14.7287	−0.561118
$690$	0	0
$691$	15.5381	0.591099	0.295549	−	0.955327i	$-0.404497\pi$
$691$	15.5381	0.591099	0.295549	+	0.955327i	$0.404497\pi$
$692$	0	0
$693$	−15.4885	−0.588360
$694$	0	0
$695$	−18.7108	−0.709742
$696$	0	0
$697$	−9.26028	−0.350758
$698$	0	0
$699$	8.25129	0.312092
$700$	0	0
$701$	−45.7146	−1.72662	−0.863308	−	0.504678i	$-0.831611\pi$
$701$	−45.7146	−1.72662	−0.863308	+	0.504678i	$0.831611\pi$
$702$	0	0
$703$	−31.2603	−1.17900
$704$	0	0
$705$	42.5451	1.60234
$706$	0	0
$707$	12.4033	0.466474
$708$	0	0
$709$	−11.8426	−0.444759	−0.222379	−	0.974960i	$-0.571382\pi$
$709$	−11.8426	−0.444759	−0.222379	+	0.974960i	$0.571382\pi$
$710$	0	0
$711$	−67.6959	−2.53880
$712$	0	0
$713$	−7.79841	−0.292053
$714$	0	0
$715$	24.8234	0.928342
$716$	0	0
$717$	31.6890	1.18345
$718$	0	0
$719$	44.7834	1.67014	0.835069	−	0.550144i	$-0.185427\pi$
$719$	44.7834	1.67014	0.835069	+	0.550144i	$0.185427\pi$
$720$	0	0
$721$	−1.86929	−0.0696159
$722$	0	0
$723$	−44.6678	−1.66121
$724$	0	0
$725$	19.7454	0.733326
$726$	0	0
$727$	30.8827	1.14537	0.572687	−	0.819774i	$-0.305901\pi$
$727$	30.8827	1.14537	0.572687	+	0.819774i	$0.305901\pi$
$728$	0	0
$729$	−41.5666	−1.53950
$730$	0	0
$731$	12.5533	0.464301
$732$	0	0
$733$	−39.4110	−1.45568	−0.727839	−	0.685748i	$-0.759475\pi$
$733$	−39.4110	−1.45568	−0.727839	+	0.685748i	$0.759475\pi$
$734$	0	0
$735$	23.6766	0.873326
$736$	0	0
$737$	−6.57149	−0.242064
$738$	0	0
$739$	−13.4352	−0.494221	−0.247110	−	0.968987i	$-0.579481\pi$
$739$	−13.4352	−0.494221	−0.247110	+	0.968987i	$0.579481\pi$
$740$	0	0
$741$	121.438	4.46113
$742$	0	0
$743$	−32.9131	−1.20746	−0.603732	−	0.797187i	$-0.706320\pi$
$743$	−32.9131	−1.20746	−0.603732	+	0.797187i	$0.706320\pi$
$744$	0	0
$745$	−33.3762	−1.22281
$746$	0	0
$747$	−5.07584	−0.185715
$748$	0	0
$749$	16.5191	0.603595
$750$	0	0
$751$	−49.7112	−1.81399	−0.906994	−	0.421144i	$-0.861629\pi$
$751$	−49.7112	−1.81399	−0.906994	+	0.421144i	$0.861629\pi$
$752$	0	0
$753$	61.9731	2.25843
$754$	0	0
$755$	−27.4266	−0.998157
$756$	0	0
$757$	39.0065	1.41772	0.708858	−	0.705351i	$-0.249211\pi$
$757$	39.0065	1.41772	0.708858	+	0.705351i	$0.249211\pi$
$758$	0	0
$759$	−5.96071	−0.216360
$760$	0	0
$761$	−17.3071	−0.627381	−0.313690	−	0.949525i	$-0.601565\pi$
$761$	−17.3071	−0.627381	−0.313690	+	0.949525i	$0.601565\pi$
$762$	0	0
$763$	−4.83000	−0.174858
$764$	0	0
$765$	−12.8342	−0.464023
$766$	0	0
$767$	−16.4354	−0.593447
$768$	0	0
$769$	2.67856	0.0965912	0.0482956	−	0.998833i	$-0.484621\pi$
$769$	2.67856	0.0965912	0.0482956	+	0.998833i	$0.484621\pi$
$770$	0	0
$771$	39.7672	1.43218
$772$	0	0
$773$	24.5568	0.883245	0.441623	−	0.897201i	$-0.354403\pi$
$773$	24.5568	0.883245	0.441623	+	0.897201i	$0.354403\pi$
$774$	0	0
$775$	−22.3206	−0.801780
$776$	0	0
$777$	−18.8815	−0.677370
$778$	0	0
$779$	35.6586	1.27760
$780$	0	0
$781$	−37.9054	−1.35636
$782$	0	0
$783$	−47.7837	−1.70765
$784$	0	0
$785$	−19.1605	−0.683869
$786$	0	0
$787$	−14.9291	−0.532166	−0.266083	−	0.963950i	$-0.585729\pi$
$787$	−14.9291	−0.532166	−0.266083	+	0.963950i	$0.585729\pi$
$788$	0	0
$789$	14.5108	0.516597
$790$	0	0
$791$	−8.31926	−0.295799
$792$	0	0
$793$	3.31841	0.117840
$794$	0	0
$795$	−9.46151	−0.335565
$796$	0	0
$797$	25.8652	0.916191	0.458096	−	0.888903i	$-0.348532\pi$
$797$	25.8652	0.916191	0.458096	+	0.888903i	$0.348532\pi$
$798$	0	0
$799$	−15.2347	−0.538966
$800$	0	0
$801$	8.99679	0.317886
$802$	0	0
$803$	−13.1661	−0.464622
$804$	0	0
$805$	−1.81657	−0.0640258
$806$	0	0
$807$	−2.86111	−0.100716
$808$	0	0
$809$	−43.7194	−1.53709	−0.768546	−	0.639794i	$-0.779020\pi$
$809$	−43.7194	−1.53709	−0.768546	+	0.639794i	$0.779020\pi$
$810$	0	0
$811$	40.5415	1.42360	0.711802	−	0.702381i	$-0.247880\pi$
$811$	40.5415	1.42360	0.711802	+	0.702381i	$0.247880\pi$
$812$	0	0
$813$	45.8886	1.60938
$814$	0	0
$815$	32.8066	1.14917
$816$	0	0
$817$	−48.3392	−1.69117
$818$	0	0
$819$	46.4684	1.62374
$820$	0	0
$821$	20.7203	0.723144	0.361572	−	0.932344i	$-0.382240\pi$
$821$	20.7203	0.723144	0.361572	+	0.932344i	$0.382240\pi$
$822$	0	0
$823$	4.26275	0.148590	0.0742950	−	0.997236i	$-0.476329\pi$
$823$	4.26275	0.148590	0.0742950	+	0.997236i	$0.476329\pi$
$824$	0	0
$825$	−17.0608	−0.593979
$826$	0	0
$827$	24.2342	0.842705	0.421353	−	0.906897i	$-0.361555\pi$
$827$	24.2342	0.842705	0.421353	+	0.906897i	$0.361555\pi$
$828$	0	0
$829$	25.9285	0.900535	0.450267	−	0.892894i	$-0.351329\pi$
$829$	25.9285	0.900535	0.450267	+	0.892894i	$0.351329\pi$
$830$	0	0
$831$	10.1216	0.351114
$832$	0	0
$833$	−8.47821	−0.293753
$834$	0	0
$835$	22.0493	0.763047
$836$	0	0
$837$	54.0157	1.86705
$838$	0	0
$839$	−3.93634	−0.135897	−0.0679487	−	0.997689i	$-0.521645\pi$
$839$	−3.93634	−0.135897	−0.0679487	+	0.997689i	$0.521645\pi$
$840$	0	0
$841$	29.3723	1.01284
$842$	0	0
$843$	48.0603	1.65529
$844$	0	0
$845$	−54.2699	−1.86694
$846$	0	0
$847$	7.34777	0.252472
$848$	0	0
$849$	3.21722	0.110415
$850$	0	0
$851$	−4.60346	−0.157805
$852$	0	0
$853$	−18.6413	−0.638265	−0.319132	−	0.947710i	$-0.603391\pi$
$853$	−18.6413	−0.638265	−0.319132	+	0.947710i	$0.603391\pi$
$854$	0	0
$855$	49.4209	1.69016
$856$	0	0
$857$	−24.9463	−0.852149	−0.426075	−	0.904688i	$-0.640104\pi$
$857$	−24.9463	−0.852149	−0.426075	+	0.904688i	$0.640104\pi$
$858$	0	0
$859$	−34.9393	−1.19211	−0.596056	−	0.802943i	$-0.703266\pi$
$859$	−34.9393	−1.19211	−0.596056	+	0.802943i	$0.703266\pi$
$860$	0	0
$861$	21.5381	0.734018
$862$	0	0
$863$	−0.295095	−0.0100452	−0.00502258	−	0.999987i	$-0.501599\pi$
$863$	−0.295095	−0.0100452	−0.00502258	+	0.999987i	$0.501599\pi$
$864$	0	0
$865$	−22.7405	−0.773201
$866$	0	0
$867$	−41.3846	−1.40549
$868$	0	0
$869$	−30.1186	−1.02170
$870$	0	0
$871$	19.7157	0.668041
$872$	0	0
$873$	34.9184	1.18181
$874$	0	0
$875$	−15.2585	−0.515833
$876$	0	0
$877$	8.32050	0.280963	0.140482	−	0.990083i	$-0.455135\pi$
$877$	8.32050	0.280963	0.140482	+	0.990083i	$0.455135\pi$
$878$	0	0
$879$	7.47581	0.252153
$880$	0	0
$881$	17.7934	0.599475	0.299737	−	0.954022i	$-0.403101\pi$
$881$	17.7934	0.599475	0.299737	+	0.954022i	$0.403101\pi$
$882$	0	0
$883$	−0.363619	−0.0122368	−0.00611838	−	0.999981i	$-0.501948\pi$
$883$	−0.363619	−0.0122368	−0.00611838	+	0.999981i	$0.501948\pi$
$884$	0	0
$885$	−10.5579	−0.354899
$886$	0	0
$887$	−10.1284	−0.340077	−0.170038	−	0.985437i	$-0.554389\pi$
$887$	−10.1284	−0.340077	−0.170038	+	0.985437i	$0.554389\pi$
$888$	0	0
$889$	17.1792	0.576171
$890$	0	0
$891$	5.39043	0.180586
$892$	0	0
$893$	58.6644	1.96313
$894$	0	0
$895$	1.37051	0.0458111
$896$	0	0
$897$	17.8832	0.597103
$898$	0	0
$899$	−65.9853	−2.20073
$900$	0	0
$901$	3.38801	0.112871
$902$	0	0
$903$	−29.1973	−0.971626
$904$	0	0
$905$	−12.7637	−0.424279
$906$	0	0
$907$	23.7535	0.788721	0.394360	−	0.918956i	$-0.370966\pi$
$907$	23.7535	0.788721	0.394360	+	0.918956i	$0.370966\pi$
$908$	0	0
$909$	−49.6919	−1.64818
$910$	0	0
$911$	−49.5052	−1.64018	−0.820090	−	0.572235i	$-0.806077\pi$
$911$	−49.5052	−1.64018	−0.820090	+	0.572235i	$0.806077\pi$
$912$	0	0
$913$	−2.25829	−0.0747386
$914$	0	0
$915$	2.13170	0.0704720
$916$	0	0
$917$	−13.4269	−0.443396
$918$	0	0
$919$	11.8657	0.391414	0.195707	−	0.980662i	$-0.437300\pi$
$919$	11.8657	0.391414	0.195707	+	0.980662i	$0.437300\pi$
$920$	0	0
$921$	30.9275	1.01910
$922$	0	0
$923$	113.723	3.74324
$924$	0	0
$925$	−13.1760	−0.433225
$926$	0	0
$927$	7.48902	0.245972
$928$	0	0
$929$	−36.0228	−1.18187	−0.590934	−	0.806720i	$-0.701241\pi$
$929$	−36.0228	−1.18187	−0.590934	+	0.806720i	$0.701241\pi$
$930$	0	0
$931$	32.6471	1.06997
$932$	0	0
$933$	−42.3709	−1.38716
$934$	0	0
$935$	−5.71008	−0.186740
$936$	0	0
$937$	35.7179	1.16685	0.583427	−	0.812165i	$-0.301711\pi$
$937$	35.7179	1.16685	0.583427	+	0.812165i	$0.301711\pi$
$938$	0	0
$939$	−52.2914	−1.70647
$940$	0	0
$941$	26.5108	0.864228	0.432114	−	0.901819i	$-0.357768\pi$
$941$	26.5108	0.864228	0.432114	+	0.901819i	$0.357768\pi$
$942$	0	0
$943$	5.25117	0.171002
$944$	0	0
$945$	12.5825	0.409309
$946$	0	0
$947$	2.41234	0.0783906	0.0391953	−	0.999232i	$-0.487521\pi$
$947$	2.41234	0.0783906	0.0391953	+	0.999232i	$0.487521\pi$
$948$	0	0
$949$	39.5007	1.28225
$950$	0	0
$951$	1.18717	0.0384965
$952$	0	0
$953$	23.6224	0.765205	0.382602	−	0.923913i	$-0.375028\pi$
$953$	23.6224	0.765205	0.382602	+	0.923913i	$0.375028\pi$
$954$	0	0
$955$	33.2885	1.07719
$956$	0	0
$957$	−50.4358	−1.63036
$958$	0	0
$959$	−19.3567	−0.625061
$960$	0	0
$961$	43.5912	1.40617
$962$	0	0
$963$	−66.1813	−2.13266
$964$	0	0
$965$	15.9781	0.514354
$966$	0	0
$967$	37.7599	1.21428	0.607138	−	0.794597i	$-0.292318\pi$
$967$	37.7599	1.21428	0.607138	+	0.794597i	$0.292318\pi$
$968$	0	0
$969$	−27.9341	−0.897374
$970$	0	0
$971$	9.10742	0.292271	0.146135	−	0.989265i	$-0.453316\pi$
$971$	9.10742	0.292271	0.146135	+	0.989265i	$0.453316\pi$
$972$	0	0
$973$	−15.5834	−0.499580
$974$	0	0
$975$	51.1854	1.63924
$976$	0	0
$977$	3.87547	0.123987	0.0619936	−	0.998077i	$-0.480254\pi$
$977$	3.87547	0.123987	0.0619936	+	0.998077i	$0.480254\pi$
$978$	0	0
$979$	4.00276	0.127929
$980$	0	0
$981$	19.3507	0.617819
$982$	0	0
$983$	17.2435	0.549982	0.274991	−	0.961447i	$-0.411325\pi$
$983$	17.2435	0.549982	0.274991	+	0.961447i	$0.411325\pi$
$984$	0	0
$985$	41.4050	1.31927
$986$	0	0
$987$	35.4339	1.12787
$988$	0	0
$989$	−7.11853	−0.226356
$990$	0	0
$991$	−39.5794	−1.25728	−0.628641	−	0.777696i	$-0.716388\pi$
$991$	−39.5794	−1.25728	−0.628641	+	0.777696i	$0.716388\pi$
$992$	0	0
$993$	40.9443	1.29933
$994$	0	0
$995$	33.7248	1.06915
$996$	0	0
$997$	−4.84695	−0.153505	−0.0767523	−	0.997050i	$-0.524455\pi$
$997$	−4.84695	−0.153505	−0.0767523	+	0.997050i	$0.524455\pi$
$998$	0	0
$999$	31.8859	1.00882

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.k.1.14		15
4.3	odd	2		2152.2.a.c.1.2	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2152.2.a.c.1.2	✓	15	4.3	odd	2
4304.2.a.k.1.14		15	1.1	even	1	trivial

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

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4304.2.a.k.1.14

Level	$4304$
Weight	$2$
Character	4304.1
Self dual	yes
Analytic conductor	$34.368$
Analytic rank	$1$
Dimension	$15$
CM	no
Inner twists	$1$