LMFDB - Embedded newform 2312.2.a.s.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(1,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.4614129473$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.765367 q^{3} -1.84776 q^{5} +0.317025 q^{7} -2.41421 q^{9} +0.765367 q^{11} -1.41421 q^{15} +2.24264 q^{19} +0.242641 q^{21} +6.62567 q^{23} -1.58579 q^{25} -4.14386 q^{27} -5.54328 q^{29} +2.29610 q^{31} +0.585786 q^{33} -0.585786 q^{35} -0.765367 q^{37} +2.48181 q^{41} -10.2426 q^{43} +4.46088 q^{45} -12.8284 q^{47} -6.89949 q^{49} -3.07107 q^{53} -1.41421 q^{55} +1.71644 q^{57} +7.89949 q^{59} -9.87285 q^{61} -0.765367 q^{63} -9.65685 q^{67} +5.07107 q^{69} -9.68714 q^{71} -7.97069 q^{73} -1.21371 q^{75} +0.242641 q^{77} +14.2793 q^{79} +4.07107 q^{81} -10.2426 q^{83} -4.24264 q^{87} +1.65685 q^{89} +1.75736 q^{93} -4.14386 q^{95} -15.9958 q^{97} -1.84776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} - 8 q^{19} - 16 q^{21} - 12 q^{25} + 8 q^{33} - 8 q^{35} - 24 q^{43} - 40 q^{47} + 12 q^{49} + 16 q^{53} - 8 q^{59} - 16 q^{67} - 8 q^{69} - 16 q^{77} - 12 q^{81} - 24 q^{83} - 16 q^{89} + 24 q^{93}+O(q^{100}) $ 4 * q - 4 * q^9 - 8 * q^19 - 16 * q^21 - 12 * q^25 + 8 * q^33 - 8 * q^35 - 24 * q^43 - 40 * q^47 + 12 * q^49 + 16 * q^53 - 8 * q^59 - 16 * q^67 - 8 * q^69 - 16 * q^77 - 12 * q^81 - 24 * q^83 - 16 * q^89 + 24 * q^93

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
$3$	0.765367	0.441885	0.220942	−	0.975287i	$-0.429087\pi$
$3$	0.765367	0.441885	0.220942	+	0.975287i	$0.429087\pi$
$4$	0	0
$5$	−1.84776	−0.826343	−0.413171	−	0.910653i	$-0.635579\pi$
$5$	−1.84776	−0.826343	−0.413171	+	0.910653i	$0.635579\pi$
$6$	0	0
$7$	0.317025	0.119824	0.0599122	−	0.998204i	$-0.480918\pi$
$7$	0.317025	0.119824	0.0599122	+	0.998204i	$0.480918\pi$
$8$	0	0
$9$	−2.41421	−0.804738
$10$	0	0
$11$	0.765367	0.230767	0.115383	−	0.993321i	$-0.463190\pi$
$11$	0.765367	0.230767	0.115383	+	0.993321i	$0.463190\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.24264	0.514497	0.257249	−	0.966345i	$-0.417184\pi$
$19$	2.24264	0.514497	0.257249	+	0.966345i	$0.417184\pi$
$20$	0	0
$21$	0.242641	0.0529485
$22$	0	0
$23$	6.62567	1.38155	0.690774	−	0.723071i	$-0.257270\pi$
$23$	6.62567	1.38155	0.690774	+	0.723071i	$0.257270\pi$
$24$	0	0
$25$	−1.58579	−0.317157
$26$	0	0
$27$	−4.14386	−0.797486
$28$	0	0
$29$	−5.54328	−1.02936	−0.514680	−	0.857382i	$-0.672089\pi$
$29$	−5.54328	−1.02936	−0.514680	+	0.857382i	$0.672089\pi$
$30$	0	0
$31$	2.29610	0.412392	0.206196	−	0.978511i	$-0.433892\pi$
$31$	2.29610	0.412392	0.206196	+	0.978511i	$0.433892\pi$
$32$	0	0
$33$	0.585786	0.101972
$34$	0	0
$35$	−0.585786	−0.0990160
$36$	0	0
$37$	−0.765367	−0.125826	−0.0629128	−	0.998019i	$-0.520039\pi$
$37$	−0.765367	−0.125826	−0.0629128	+	0.998019i	$0.520039\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.48181	0.387594	0.193797	−	0.981042i	$-0.437920\pi$
$41$	2.48181	0.387594	0.193797	+	0.981042i	$0.437920\pi$
$42$	0	0
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	0	0
$45$	4.46088	0.664989
$46$	0	0
$47$	−12.8284	−1.87122	−0.935609	−	0.353037i	$-0.885149\pi$
$47$	−12.8284	−1.87122	−0.935609	+	0.353037i	$0.885149\pi$
$48$	0	0
$49$	−6.89949	−0.985642
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.07107	−0.421844	−0.210922	−	0.977503i	$-0.567647\pi$
$53$	−3.07107	−0.421844	−0.210922	+	0.977503i	$0.567647\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	1.71644	0.227348
$58$	0	0
$59$	7.89949	1.02843	0.514213	−	0.857662i	$-0.328084\pi$
$59$	7.89949	1.02843	0.514213	+	0.857662i	$0.328084\pi$
$60$	0	0
$61$	−9.87285	−1.26409	−0.632044	−	0.774932i	$-0.717784\pi$
$61$	−9.87285	−1.26409	−0.632044	+	0.774932i	$0.717784\pi$
$62$	0	0
$63$	−0.765367	−0.0964272
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	0	0
$69$	5.07107	0.610485
$70$	0	0
$71$	−9.68714	−1.14965	−0.574826	−	0.818276i	$-0.694930\pi$
$71$	−9.68714	−1.14965	−0.574826	+	0.818276i	$0.694930\pi$
$72$	0	0
$73$	−7.97069	−0.932899	−0.466450	−	0.884548i	$-0.654467\pi$
$73$	−7.97069	−0.932899	−0.466450	+	0.884548i	$0.654467\pi$
$74$	0	0
$75$	−1.21371	−0.140147
$76$	0	0
$77$	0.242641	0.0276515
$78$	0	0
$79$	14.2793	1.60655	0.803276	−	0.595608i	$-0.203089\pi$
$79$	14.2793	1.60655	0.803276	+	0.595608i	$0.203089\pi$
$80$	0	0
$81$	4.07107	0.452341
$82$	0	0
$83$	−10.2426	−1.12428	−0.562138	−	0.827043i	$-0.690021\pi$
$83$	−10.2426	−1.12428	−0.562138	+	0.827043i	$0.690021\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.24264	−0.454859
$88$	0	0
$89$	1.65685	0.175626	0.0878131	−	0.996137i	$-0.472012\pi$
$89$	1.65685	0.175626	0.0878131	+	0.996137i	$0.472012\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.75736	0.182230
$94$	0	0
$95$	−4.14386	−0.425151
$96$	0	0
$97$	−15.9958	−1.62413	−0.812063	−	0.583570i	$-0.801655\pi$
$97$	−15.9958	−1.62413	−0.812063	+	0.583570i	$0.801655\pi$
$98$	0	0
$99$	−1.84776	−0.185707
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	−0.448342	−0.0437537
$106$	0	0
$107$	14.4650	1.39839	0.699194	−	0.714932i	$-0.253542\pi$
$107$	14.4650	1.39839	0.699194	+	0.714932i	$0.253542\pi$
$108$	0	0
$109$	11.6662	1.11742	0.558710	−	0.829363i	$-0.311296\pi$
$109$	11.6662	1.11742	0.558710	+	0.829363i	$0.311296\pi$
$110$	0	0
$111$	−0.585786	−0.0556004
$112$	0	0
$113$	8.15640	0.767290	0.383645	−	0.923481i	$-0.374669\pi$
$113$	8.15640	0.767290	0.383645	+	0.923481i	$0.374669\pi$
$114$	0	0
$115$	−12.2426	−1.14163
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.4142	−0.946747
$122$	0	0
$123$	1.89949	0.171272
$124$	0	0
$125$	12.1689	1.08842
$126$	0	0
$127$	−1.75736	−0.155940	−0.0779702	−	0.996956i	$-0.524844\pi$
$127$	−1.75736	−0.155940	−0.0779702	+	0.996956i	$0.524844\pi$
$128$	0	0
$129$	−7.83938	−0.690219
$130$	0	0
$131$	−14.4650	−1.26382	−0.631909	−	0.775043i	$-0.717728\pi$
$131$	−14.4650	−1.26382	−0.631909	+	0.775043i	$0.717728\pi$
$132$	0	0
$133$	0.710974	0.0616493
$134$	0	0
$135$	7.65685	0.658997
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	−0.502734	−0.0426414	−0.0213207	−	0.999773i	$-0.506787\pi$
$139$	−0.502734	−0.0426414	−0.0213207	+	0.999773i	$0.506787\pi$
$140$	0	0
$141$	−9.81845	−0.826863
$142$	0	0
$143$	0	0
$144$	0	0
$145$	10.2426	0.850605
$146$	0	0
$147$	−5.28064	−0.435540
$148$	0	0
$149$	6.34315	0.519651	0.259825	−	0.965656i	$-0.416335\pi$
$149$	6.34315	0.519651	0.259825	+	0.965656i	$0.416335\pi$
$150$	0	0
$151$	−19.8995	−1.61940	−0.809699	−	0.586845i	$-0.800370\pi$
$151$	−19.8995	−1.61940	−0.809699	+	0.586845i	$0.800370\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.24264	−0.340777
$156$	0	0
$157$	15.3137	1.22217	0.611083	−	0.791566i	$-0.290734\pi$
$157$	15.3137	1.22217	0.611083	+	0.791566i	$0.290734\pi$
$158$	0	0
$159$	−2.35049	−0.186406
$160$	0	0
$161$	2.10051	0.165543
$162$	0	0
$163$	−4.64659	−0.363949	−0.181975	−	0.983303i	$-0.558249\pi$
$163$	−4.64659	−0.363949	−0.181975	+	0.983303i	$0.558249\pi$
$164$	0	0
$165$	−1.08239	−0.0842641
$166$	0	0
$167$	−0.765367	−0.0592259	−0.0296129	−	0.999561i	$-0.509427\pi$
$167$	−0.765367	−0.0592259	−0.0296129	+	0.999561i	$0.509427\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−5.41421	−0.414035
$172$	0	0
$173$	18.8715	1.43478	0.717388	−	0.696674i	$-0.245337\pi$
$173$	18.8715	1.43478	0.717388	+	0.696674i	$0.245337\pi$
$174$	0	0
$175$	−0.502734	−0.0380032
$176$	0	0
$177$	6.04601	0.454446
$178$	0	0
$179$	−18.2426	−1.36352	−0.681759	−	0.731576i	$-0.738785\pi$
$179$	−18.2426	−1.36352	−0.681759	+	0.731576i	$0.738785\pi$
$180$	0	0
$181$	8.15640	0.606261	0.303130	−	0.952949i	$-0.401968\pi$
$181$	8.15640	0.606261	0.303130	+	0.952949i	$0.401968\pi$
$182$	0	0
$183$	−7.55635	−0.558581
$184$	0	0
$185$	1.41421	0.103975
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.31371	−0.0955582
$190$	0	0
$191$	−8.82843	−0.638803	−0.319401	−	0.947620i	$-0.603482\pi$
$191$	−8.82843	−0.638803	−0.319401	+	0.947620i	$0.603482\pi$
$192$	0	0
$193$	−0.765367	−0.0550923	−0.0275462	−	0.999621i	$-0.508769\pi$
$193$	−0.765367	−0.0550923	−0.0275462	+	0.999621i	$0.508769\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.3003	0.876357	0.438179	−	0.898888i	$-0.355624\pi$
$197$	12.3003	0.876357	0.438179	+	0.898888i	$0.355624\pi$
$198$	0	0
$199$	22.4901	1.59428	0.797142	−	0.603792i	$-0.206344\pi$
$199$	22.4901	1.59428	0.797142	+	0.603792i	$0.206344\pi$
$200$	0	0
$201$	−7.39104	−0.521324
$202$	0	0
$203$	−1.75736	−0.123342
$204$	0	0
$205$	−4.58579	−0.320285
$206$	0	0
$207$	−15.9958	−1.11178
$208$	0	0
$209$	1.71644	0.118729
$210$	0	0
$211$	−26.1857	−1.80269	−0.901347	−	0.433097i	$-0.857421\pi$
$211$	−26.1857	−1.80269	−0.901347	+	0.433097i	$0.857421\pi$
$212$	0	0
$213$	−7.41421	−0.508014
$214$	0	0
$215$	18.9259	1.29074
$216$	0	0
$217$	0.727922	0.0494146
$218$	0	0
$219$	−6.10051	−0.412234
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.5563	1.17566	0.587830	−	0.808984i	$-0.299982\pi$
$223$	17.5563	1.17566	0.587830	+	0.808984i	$0.299982\pi$
$224$	0	0
$225$	3.82843	0.255228
$226$	0	0
$227$	3.56420	0.236564	0.118282	−	0.992980i	$-0.462261\pi$
$227$	3.56420	0.236564	0.118282	+	0.992980i	$0.462261\pi$
$228$	0	0
$229$	−9.89949	−0.654177	−0.327089	−	0.944994i	$-0.606068\pi$
$229$	−9.89949	−0.654177	−0.327089	+	0.944994i	$0.606068\pi$
$230$	0	0
$231$	0.185709	0.0122188
$232$	0	0
$233$	0.579658	0.0379746	0.0189873	−	0.999820i	$-0.493956\pi$
$233$	0.579658	0.0379746	0.0189873	+	0.999820i	$0.493956\pi$
$234$	0	0
$235$	23.7038	1.54627
$236$	0	0
$237$	10.9289	0.709910
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	0	0
$241$	14.0167	0.902895	0.451448	−	0.892298i	$-0.350908\pi$
$241$	14.0167	0.902895	0.451448	+	0.892298i	$0.350908\pi$
$242$	0	0
$243$	15.5474	0.997369
$244$	0	0
$245$	12.7486	0.814478
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−7.83938	−0.496800
$250$	0	0
$251$	−0.828427	−0.0522899	−0.0261449	−	0.999658i	$-0.508323\pi$
$251$	−0.828427	−0.0522899	−0.0261449	+	0.999658i	$0.508323\pi$
$252$	0	0
$253$	5.07107	0.318815
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.24264	0.514162	0.257081	−	0.966390i	$-0.417239\pi$
$257$	8.24264	0.514162	0.257081	+	0.966390i	$0.417239\pi$
$258$	0	0
$259$	−0.242641	−0.0150770
$260$	0	0
$261$	13.3827	0.828366
$262$	0	0
$263$	3.89949	0.240453	0.120227	−	0.992746i	$-0.461638\pi$
$263$	3.89949	0.240453	0.120227	+	0.992746i	$0.461638\pi$
$264$	0	0
$265$	5.67459	0.348588
$266$	0	0
$267$	1.26810	0.0776065
$268$	0	0
$269$	−14.0167	−0.854614	−0.427307	−	0.904107i	$-0.640538\pi$
$269$	−14.0167	−0.854614	−0.427307	+	0.904107i	$0.640538\pi$
$270$	0	0
$271$	11.3137	0.687259	0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137	0.687259	0.343629	+	0.939105i	$0.388344\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.21371	−0.0731894
$276$	0	0
$277$	−9.23880	−0.555105	−0.277553	−	0.960710i	$-0.589523\pi$
$277$	−9.23880	−0.555105	−0.277553	+	0.960710i	$0.589523\pi$
$278$	0	0
$279$	−5.54328	−0.331867
$280$	0	0
$281$	3.27208	0.195196	0.0975979	−	0.995226i	$-0.468884\pi$
$281$	3.27208	0.195196	0.0975979	+	0.995226i	$0.468884\pi$
$282$	0	0
$283$	−26.2626	−1.56115	−0.780574	−	0.625063i	$-0.785073\pi$
$283$	−26.2626	−1.56115	−0.780574	+	0.625063i	$0.785073\pi$
$284$	0	0
$285$	−3.17157	−0.187868
$286$	0	0
$287$	0.786797	0.0464431
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−12.2426	−0.717676
$292$	0	0
$293$	−7.31371	−0.427271	−0.213636	−	0.976913i	$-0.568531\pi$
$293$	−7.31371	−0.427271	−0.213636	+	0.976913i	$0.568531\pi$
$294$	0	0
$295$	−14.5964	−0.849833
$296$	0	0
$297$	−3.17157	−0.184033
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.24718	−0.187164
$302$	0	0
$303$	−7.65367	−0.439692
$304$	0	0
$305$	18.2426	1.04457
$306$	0	0
$307$	−26.6274	−1.51971	−0.759853	−	0.650094i	$-0.774729\pi$
$307$	−26.6274	−1.51971	−0.759853	+	0.650094i	$0.774729\pi$
$308$	0	0
$309$	−1.79337	−0.102021
$310$	0	0
$311$	4.01254	0.227530	0.113765	−	0.993508i	$-0.463709\pi$
$311$	4.01254	0.227530	0.113765	+	0.993508i	$0.463709\pi$
$312$	0	0
$313$	4.27518	0.241647	0.120824	−	0.992674i	$-0.461446\pi$
$313$	4.27518	0.241647	0.120824	+	0.992674i	$0.461446\pi$
$314$	0	0
$315$	1.41421	0.0796819
$316$	0	0
$317$	−15.8101	−0.887982	−0.443991	−	0.896031i	$-0.646438\pi$
$317$	−15.8101	−0.887982	−0.443991	+	0.896031i	$0.646438\pi$
$318$	0	0
$319$	−4.24264	−0.237542
$320$	0	0
$321$	11.0711	0.617927
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.92893	0.493771
$328$	0	0
$329$	−4.06694	−0.224217
$330$	0	0
$331$	−13.7574	−0.756173	−0.378086	−	0.925770i	$-0.623418\pi$
$331$	−13.7574	−0.756173	−0.378086	+	0.925770i	$0.623418\pi$
$332$	0	0
$333$	1.84776	0.101257
$334$	0	0
$335$	17.8435	0.974897
$336$	0	0
$337$	15.5474	0.846923	0.423461	−	0.905914i	$-0.360815\pi$
$337$	15.5474	0.846923	0.423461	+	0.905914i	$0.360815\pi$
$338$	0	0
$339$	6.24264	0.339054
$340$	0	0
$341$	1.75736	0.0951663
$342$	0	0
$343$	−4.40649	−0.237928
$344$	0	0
$345$	−9.37011	−0.504470
$346$	0	0
$347$	−5.80591	−0.311678	−0.155839	−	0.987783i	$-0.549808\pi$
$347$	−5.80591	−0.311678	−0.155839	+	0.987783i	$0.549808\pi$
$348$	0	0
$349$	21.8995	1.17225	0.586127	−	0.810220i	$-0.300652\pi$
$349$	21.8995	1.17225	0.586127	+	0.810220i	$0.300652\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.9706	1.32905	0.664524	−	0.747266i	$-0.268634\pi$
$353$	24.9706	1.32905	0.664524	+	0.747266i	$0.268634\pi$
$354$	0	0
$355$	17.8995	0.950007
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.2426	1.17392	0.586961	−	0.809615i	$-0.300324\pi$
$359$	22.2426	1.17392	0.586961	+	0.809615i	$0.300324\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	0	0
$363$	−7.97069	−0.418353
$364$	0	0
$365$	14.7279	0.770895
$366$	0	0
$367$	−2.11039	−0.110162	−0.0550808	−	0.998482i	$-0.517542\pi$
$367$	−2.11039	−0.110162	−0.0550808	+	0.998482i	$0.517542\pi$
$368$	0	0
$369$	−5.99162	−0.311911
$370$	0	0
$371$	−0.973606	−0.0505471
$372$	0	0
$373$	−35.9411	−1.86096	−0.930480	−	0.366342i	$-0.880610\pi$
$373$	−35.9411	−1.86096	−0.930480	+	0.366342i	$0.880610\pi$
$374$	0	0
$375$	9.31371	0.480958
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−15.0991	−0.775589	−0.387794	−	0.921746i	$-0.626763\pi$
$379$	−15.0991	−0.775589	−0.387794	+	0.921746i	$0.626763\pi$
$380$	0	0
$381$	−1.34502	−0.0689077
$382$	0	0
$383$	10.7279	0.548171	0.274086	−	0.961705i	$-0.411625\pi$
$383$	10.7279	0.548171	0.274086	+	0.961705i	$0.411625\pi$
$384$	0	0
$385$	−0.448342	−0.0228496
$386$	0	0
$387$	24.7279	1.25699
$388$	0	0
$389$	28.7279	1.45656	0.728282	−	0.685278i	$-0.240319\pi$
$389$	28.7279	1.45656	0.728282	+	0.685278i	$0.240319\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−11.0711	−0.558461
$394$	0	0
$395$	−26.3848	−1.32756
$396$	0	0
$397$	−29.0614	−1.45855	−0.729275	−	0.684221i	$-0.760143\pi$
$397$	−29.0614	−1.45855	−0.729275	+	0.684221i	$0.760143\pi$
$398$	0	0
$399$	0.544156	0.0272419
$400$	0	0
$401$	27.0823	1.35243	0.676214	−	0.736706i	$-0.263620\pi$
$401$	27.0823	1.35243	0.676214	+	0.736706i	$0.263620\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−7.52235	−0.373789
$406$	0	0
$407$	−0.585786	−0.0290364
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	10.7151	0.528539
$412$	0	0
$413$	2.50434	0.123231
$414$	0	0
$415$	18.9259	0.929037
$416$	0	0
$417$	−0.384776	−0.0188426
$418$	0	0
$419$	−6.62567	−0.323685	−0.161843	−	0.986817i	$-0.551744\pi$
$419$	−6.62567	−0.323685	−0.161843	+	0.986817i	$0.551744\pi$
$420$	0	0
$421$	11.3137	0.551396	0.275698	−	0.961244i	$-0.411091\pi$
$421$	11.3137	0.551396	0.275698	+	0.961244i	$0.411091\pi$
$422$	0	0
$423$	30.9706	1.50584
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.12994	−0.151469
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.29610	0.110599	0.0552996	−	0.998470i	$-0.482389\pi$
$431$	2.29610	0.110599	0.0552996	+	0.998470i	$0.482389\pi$
$432$	0	0
$433$	5.41421	0.260190	0.130095	−	0.991502i	$-0.458472\pi$
$433$	5.41421	0.260190	0.130095	+	0.991502i	$0.458472\pi$
$434$	0	0
$435$	7.83938	0.375869
$436$	0	0
$437$	14.8590	0.710802
$438$	0	0
$439$	−9.42450	−0.449807	−0.224904	−	0.974381i	$-0.572207\pi$
$439$	−9.42450	−0.449807	−0.224904	+	0.974381i	$0.572207\pi$
$440$	0	0
$441$	16.6569	0.793184
$442$	0	0
$443$	−40.2843	−1.91396	−0.956982	−	0.290148i	$-0.906295\pi$
$443$	−40.2843	−1.91396	−0.956982	+	0.290148i	$0.906295\pi$
$444$	0	0
$445$	−3.06147	−0.145127
$446$	0	0
$447$	4.85483	0.229626
$448$	0	0
$449$	−17.8979	−0.844656	−0.422328	−	0.906443i	$-0.638787\pi$
$449$	−17.8979	−0.844656	−0.422328	+	0.906443i	$0.638787\pi$
$450$	0	0
$451$	1.89949	0.0894437
$452$	0	0
$453$	−15.2304	−0.715587
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.92893	−0.230566	−0.115283	−	0.993333i	$-0.536777\pi$
$457$	−4.92893	−0.230566	−0.115283	+	0.993333i	$0.536777\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.55635	0.165636	0.0828178	−	0.996565i	$-0.473608\pi$
$461$	3.55635	0.165636	0.0828178	+	0.996565i	$0.473608\pi$
$462$	0	0
$463$	24.1421	1.12198	0.560990	−	0.827823i	$-0.310421\pi$
$463$	24.1421	1.12198	0.560990	+	0.827823i	$0.310421\pi$
$464$	0	0
$465$	−3.24718	−0.150584
$466$	0	0
$467$	13.5563	0.627313	0.313657	−	0.949537i	$-0.398446\pi$
$467$	13.5563	0.627313	0.313657	+	0.949537i	$0.398446\pi$
$468$	0	0
$469$	−3.06147	−0.141365
$470$	0	0
$471$	11.7206	0.540057
$472$	0	0
$473$	−7.83938	−0.360455
$474$	0	0
$475$	−3.55635	−0.163176
$476$	0	0
$477$	7.41421	0.339474
$478$	0	0
$479$	−37.9832	−1.73550	−0.867748	−	0.497005i	$-0.834433\pi$
$479$	−37.9832	−1.73550	−0.867748	+	0.497005i	$0.834433\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.60766	0.0731509
$484$	0	0
$485$	29.5563	1.34208
$486$	0	0
$487$	−17.6034	−0.797688	−0.398844	−	0.917019i	$-0.630588\pi$
$487$	−17.6034	−0.797688	−0.398844	+	0.917019i	$0.630588\pi$
$488$	0	0
$489$	−3.55635	−0.160824
$490$	0	0
$491$	20.3848	0.919952	0.459976	−	0.887931i	$-0.347858\pi$
$491$	20.3848	0.919952	0.459976	+	0.887931i	$0.347858\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	3.41421	0.153457
$496$	0	0
$497$	−3.07107	−0.137756
$498$	0	0
$499$	−16.8155	−0.752767	−0.376383	−	0.926464i	$-0.622832\pi$
$499$	−16.8155	−0.752767	−0.376383	+	0.926464i	$0.622832\pi$
$500$	0	0
$501$	−0.585786	−0.0261710
$502$	0	0
$503$	−0.317025	−0.0141355	−0.00706773	−	0.999975i	$-0.502250\pi$
$503$	−0.317025	−0.0141355	−0.00706773	+	0.999975i	$0.502250\pi$
$504$	0	0
$505$	18.4776	0.822242
$506$	0	0
$507$	−9.94977	−0.441885
$508$	0	0
$509$	38.2843	1.69692	0.848460	−	0.529259i	$-0.177530\pi$
$509$	38.2843	1.69692	0.848460	+	0.529259i	$0.177530\pi$
$510$	0	0
$511$	−2.52691	−0.111784
$512$	0	0
$513$	−9.29319	−0.410304
$514$	0	0
$515$	4.32957	0.190784
$516$	0	0
$517$	−9.81845	−0.431815
$518$	0	0
$519$	14.4437	0.634006
$520$	0	0
$521$	−39.2513	−1.71963	−0.859815	−	0.510606i	$-0.829421\pi$
$521$	−39.2513	−1.71963	−0.859815	+	0.510606i	$0.829421\pi$
$522$	0	0
$523$	−4.14214	−0.181123	−0.0905615	−	0.995891i	$-0.528866\pi$
$523$	−4.14214	−0.181123	−0.0905615	+	0.995891i	$0.528866\pi$
$524$	0	0
$525$	−0.384776	−0.0167930
$526$	0	0
$527$	0	0
$528$	0	0
$529$	20.8995	0.908674
$530$	0	0
$531$	−19.0711	−0.827614
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−26.7279	−1.15555
$536$	0	0
$537$	−13.9623	−0.602518
$538$	0	0
$539$	−5.28064	−0.227453
$540$	0	0
$541$	25.9999	1.11782	0.558912	−	0.829227i	$-0.311219\pi$
$541$	25.9999	1.11782	0.558912	+	0.829227i	$0.311219\pi$
$542$	0	0
$543$	6.24264	0.267897
$544$	0	0
$545$	−21.5563	−0.923373
$546$	0	0
$547$	45.1116	1.92883	0.964416	−	0.264389i	$-0.0851704\pi$
$547$	45.1116	1.92883	0.964416	+	0.264389i	$0.0851704\pi$
$548$	0	0
$549$	23.8352	1.01726
$550$	0	0
$551$	−12.4316	−0.529603
$552$	0	0
$553$	4.52691	0.192504
$554$	0	0
$555$	1.08239	0.0459450
$556$	0	0
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.0416	−1.09752	−0.548762	−	0.835979i	$-0.684901\pi$
$563$	−26.0416	−1.09752	−0.548762	+	0.835979i	$0.684901\pi$
$564$	0	0
$565$	−15.0711	−0.634045
$566$	0	0
$567$	1.29063	0.0542014
$568$	0	0
$569$	−30.1838	−1.26537	−0.632685	−	0.774410i	$-0.718047\pi$
$569$	−30.1838	−1.26537	−0.632685	+	0.774410i	$0.718047\pi$
$570$	0	0
$571$	32.3086	1.35207	0.676036	−	0.736868i	$-0.263696\pi$
$571$	32.3086	1.35207	0.676036	+	0.736868i	$0.263696\pi$
$572$	0	0
$573$	−6.75699	−0.282277
$574$	0	0
$575$	−10.5069	−0.438168
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	−0.585786	−0.0243445
$580$	0	0
$581$	−3.24718	−0.134716
$582$	0	0
$583$	−2.35049	−0.0973475
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.87006	−0.366106	−0.183053	−	0.983103i	$-0.558598\pi$
$587$	−8.87006	−0.366106	−0.183053	+	0.983103i	$0.558598\pi$
$588$	0	0
$589$	5.14933	0.212174
$590$	0	0
$591$	9.41421	0.387249
$592$	0	0
$593$	21.6985	0.891050	0.445525	−	0.895270i	$-0.353017\pi$
$593$	21.6985	0.891050	0.445525	+	0.895270i	$0.353017\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.2132	0.704490
$598$	0	0
$599$	44.8284	1.83164	0.915820	−	0.401589i	$-0.131542\pi$
$599$	44.8284	1.83164	0.915820	+	0.401589i	$0.131542\pi$
$600$	0	0
$601$	−30.3295	−1.23717	−0.618583	−	0.785719i	$-0.712293\pi$
$601$	−30.3295	−1.23717	−0.618583	+	0.785719i	$0.712293\pi$
$602$	0	0
$603$	23.3137	0.949408
$604$	0	0
$605$	19.2430	0.782337
$606$	0	0
$607$	39.0656	1.58562	0.792811	−	0.609467i	$-0.208617\pi$
$607$	39.0656	1.58562	0.792811	+	0.609467i	$0.208617\pi$
$608$	0	0
$609$	−1.34502	−0.0545031
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−29.3137	−1.18397	−0.591985	−	0.805949i	$-0.701655\pi$
$613$	−29.3137	−1.18397	−0.591985	+	0.805949i	$0.701655\pi$
$614$	0	0
$615$	−3.50981	−0.141529
$616$	0	0
$617$	24.6549	0.992570	0.496285	−	0.868160i	$-0.334697\pi$
$617$	24.6549	0.992570	0.496285	+	0.868160i	$0.334697\pi$
$618$	0	0
$619$	2.11039	0.0848238	0.0424119	−	0.999100i	$-0.486496\pi$
$619$	2.11039	0.0848238	0.0424119	+	0.999100i	$0.486496\pi$
$620$	0	0
$621$	−27.4558	−1.10177
$622$	0	0
$623$	0.525265	0.0210443
$624$	0	0
$625$	−14.5563	−0.582254
$626$	0	0
$627$	1.31371	0.0524645
$628$	0	0
$629$	0	0
$630$	0	0
$631$	24.5858	0.978745	0.489372	−	0.872075i	$-0.337226\pi$
$631$	24.5858	0.978745	0.489372	+	0.872075i	$0.337226\pi$
$632$	0	0
$633$	−20.0416	−0.796583
$634$	0	0
$635$	3.24718	0.128860
$636$	0	0
$637$	0	0
$638$	0	0
$639$	23.3868	0.925168
$640$	0	0
$641$	11.7750	0.465084	0.232542	−	0.972586i	$-0.425296\pi$
$641$	11.7750	0.465084	0.232542	+	0.972586i	$0.425296\pi$
$642$	0	0
$643$	39.2513	1.54792	0.773960	−	0.633235i	$-0.218273\pi$
$643$	39.2513	1.54792	0.773960	+	0.633235i	$0.218273\pi$
$644$	0	0
$645$	14.4853	0.570357
$646$	0	0
$647$	45.2548	1.77915	0.889576	−	0.456788i	$-0.151000\pi$
$647$	45.2548	1.77915	0.889576	+	0.456788i	$0.151000\pi$
$648$	0	0
$649$	6.04601	0.237327
$650$	0	0
$651$	0.557127	0.0218355
$652$	0	0
$653$	−25.2890	−0.989634	−0.494817	−	0.868997i	$-0.664765\pi$
$653$	−25.2890	−0.989634	−0.494817	+	0.868997i	$0.664765\pi$
$654$	0	0
$655$	26.7279	1.04435
$656$	0	0
$657$	19.2430	0.750739
$658$	0	0
$659$	22.7696	0.886976	0.443488	−	0.896280i	$-0.353741\pi$
$659$	22.7696	0.886976	0.443488	+	0.896280i	$0.353741\pi$
$660$	0	0
$661$	−26.5858	−1.03407	−0.517034	−	0.855965i	$-0.672964\pi$
$661$	−26.5858	−1.03407	−0.517034	+	0.855965i	$0.672964\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.31371	−0.0509434
$666$	0	0
$667$	−36.7279	−1.42211
$668$	0	0
$669$	13.4370	0.519506
$670$	0	0
$671$	−7.55635	−0.291710
$672$	0	0
$673$	15.3617	0.592151	0.296076	−	0.955165i	$-0.404322\pi$
$673$	15.3617	0.592151	0.296076	+	0.955165i	$0.404322\pi$
$674$	0	0
$675$	6.57128	0.252929
$676$	0	0
$677$	−10.2124	−0.392494	−0.196247	−	0.980554i	$-0.562876\pi$
$677$	−10.2124	−0.392494	−0.196247	+	0.980554i	$0.562876\pi$
$678$	0	0
$679$	−5.07107	−0.194610
$680$	0	0
$681$	2.72792	0.104534
$682$	0	0
$683$	33.9162	1.29777	0.648885	−	0.760887i	$-0.275236\pi$
$683$	33.9162	1.29777	0.648885	+	0.760887i	$0.275236\pi$
$684$	0	0
$685$	−25.8686	−0.988389
$686$	0	0
$687$	−7.57675	−0.289071
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−32.3855	−1.23200	−0.616001	−	0.787745i	$-0.711248\pi$
$691$	−32.3855	−1.23200	−0.616001	+	0.787745i	$0.711248\pi$
$692$	0	0
$693$	−0.585786	−0.0222522
$694$	0	0
$695$	0.928932	0.0352364
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0.443651	0.0167804
$700$	0	0
$701$	28.6863	1.08347	0.541733	−	0.840551i	$-0.317768\pi$
$701$	28.6863	1.08347	0.541733	+	0.840551i	$0.317768\pi$
$702$	0	0
$703$	−1.71644	−0.0647369
$704$	0	0
$705$	18.1421	0.683272
$706$	0	0
$707$	−3.17025	−0.119230
$708$	0	0
$709$	37.0096	1.38992	0.694962	−	0.719047i	$-0.255421\pi$
$709$	37.0096	1.38992	0.694962	+	0.719047i	$0.255421\pi$
$710$	0	0
$711$	−34.4734	−1.29285
$712$	0	0
$713$	15.2132	0.569739
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.65914	0.323381
$718$	0	0
$719$	−19.4287	−0.724567	−0.362284	−	0.932068i	$-0.618003\pi$
$719$	−19.4287	−0.724567	−0.362284	+	0.932068i	$0.618003\pi$
$720$	0	0
$721$	−0.742837	−0.0276647
$722$	0	0
$723$	10.7279	0.398976
$724$	0	0
$725$	8.79045	0.326469
$726$	0	0
$727$	17.7990	0.660128	0.330064	−	0.943959i	$-0.392930\pi$
$727$	17.7990	0.660128	0.330064	+	0.943959i	$0.392930\pi$
$728$	0	0
$729$	−0.313708	−0.0116188
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−24.7279	−0.913347	−0.456673	−	0.889634i	$-0.650959\pi$
$733$	−24.7279	−0.913347	−0.456673	+	0.889634i	$0.650959\pi$
$734$	0	0
$735$	9.75736	0.359906
$736$	0	0
$737$	−7.39104	−0.272252
$738$	0	0
$739$	−9.07107	−0.333685	−0.166842	−	0.985984i	$-0.553357\pi$
$739$	−9.07107	−0.333685	−0.166842	+	0.985984i	$0.553357\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.3086	−1.18529	−0.592644	−	0.805465i	$-0.701916\pi$
$743$	−32.3086	−1.18529	−0.592644	+	0.805465i	$0.701916\pi$
$744$	0	0
$745$	−11.7206	−0.429410
$746$	0	0
$747$	24.7279	0.904747
$748$	0	0
$749$	4.58579	0.167561
$750$	0	0
$751$	24.2066	0.883311	0.441655	−	0.897185i	$-0.354391\pi$
$751$	24.2066	0.883311	0.441655	+	0.897185i	$0.354391\pi$
$752$	0	0
$753$	−0.634051	−0.0231061
$754$	0	0
$755$	36.7695	1.33818
$756$	0	0
$757$	−14.1005	−0.512492	−0.256246	−	0.966612i	$-0.582486\pi$
$757$	−14.1005	−0.512492	−0.256246	+	0.966612i	$0.582486\pi$
$758$	0	0
$759$	3.88123	0.140880
$760$	0	0
$761$	3.02944	0.109817	0.0549085	−	0.998491i	$-0.482513\pi$
$761$	3.02944	0.109817	0.0549085	+	0.998491i	$0.482513\pi$
$762$	0	0
$763$	3.69848	0.133894
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	43.5980	1.57218	0.786092	−	0.618110i	$-0.212101\pi$
$769$	43.5980	1.57218	0.786092	+	0.618110i	$0.212101\pi$
$770$	0	0
$771$	6.30864	0.227200
$772$	0	0
$773$	−32.9289	−1.18437	−0.592186	−	0.805802i	$-0.701735\pi$
$773$	−32.9289	−1.18437	−0.592186	+	0.805802i	$0.701735\pi$
$774$	0	0
$775$	−3.64113	−0.130793
$776$	0	0
$777$	−0.185709	−0.00666228
$778$	0	0
$779$	5.56581	0.199416
$780$	0	0
$781$	−7.41421	−0.265301
$782$	0	0
$783$	22.9706	0.820901
$784$	0	0
$785$	−28.2960	−1.00993
$786$	0	0
$787$	−11.2948	−0.402616	−0.201308	−	0.979528i	$-0.564519\pi$
$787$	−11.2948	−0.402616	−0.201308	+	0.979528i	$0.564519\pi$
$788$	0	0
$789$	2.98454	0.106253
$790$	0	0
$791$	2.58579	0.0919400
$792$	0	0
$793$	0	0
$794$	0	0
$795$	4.34315	0.154036
$796$	0	0
$797$	13.4142	0.475156	0.237578	−	0.971368i	$-0.423646\pi$
$797$	13.4142	0.475156	0.237578	+	0.971368i	$0.423646\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	−6.10051	−0.215282
$804$	0	0
$805$	−3.88123	−0.136795
$806$	0	0
$807$	−10.7279	−0.377641
$808$	0	0
$809$	−15.9958	−0.562382	−0.281191	−	0.959652i	$-0.590729\pi$
$809$	−15.9958	−0.562382	−0.281191	+	0.959652i	$0.590729\pi$
$810$	0	0
$811$	−42.7611	−1.50154	−0.750772	−	0.660561i	$-0.770319\pi$
$811$	−42.7611	−1.50154	−0.750772	+	0.660561i	$0.770319\pi$
$812$	0	0
$813$	8.65914	0.303689
$814$	0	0
$815$	8.58579	0.300747
$816$	0	0
$817$	−22.9706	−0.803638
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.0961	1.67857	0.839283	−	0.543694i	$-0.182975\pi$
$821$	48.0961	1.67857	0.839283	+	0.543694i	$0.182975\pi$
$822$	0	0
$823$	9.50143	0.331199	0.165599	−	0.986193i	$-0.447044\pi$
$823$	9.50143	0.331199	0.165599	+	0.986193i	$0.447044\pi$
$824$	0	0
$825$	−0.928932	−0.0323413
$826$	0	0
$827$	8.15640	0.283626	0.141813	−	0.989893i	$-0.454707\pi$
$827$	8.15640	0.283626	0.141813	+	0.989893i	$0.454707\pi$
$828$	0	0
$829$	−53.9411	−1.87345	−0.936726	−	0.350062i	$-0.886160\pi$
$829$	−53.9411	−1.87345	−0.936726	+	0.350062i	$0.886160\pi$
$830$	0	0
$831$	−7.07107	−0.245293
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.41421	0.0489409
$836$	0	0
$837$	−9.51472	−0.328877
$838$	0	0
$839$	34.9217	1.20563	0.602816	−	0.797880i	$-0.294045\pi$
$839$	34.9217	1.20563	0.602816	+	0.797880i	$0.294045\pi$
$840$	0	0
$841$	1.72792	0.0595835
$842$	0	0
$843$	2.50434	0.0862541
$844$	0	0
$845$	24.0209	0.826343
$846$	0	0
$847$	−3.30157	−0.113443
$848$	0	0
$849$	−20.1005	−0.689848
$850$	0	0
$851$	−5.07107	−0.173834
$852$	0	0
$853$	15.8101	0.541327	0.270663	−	0.962674i	$-0.412757\pi$
$853$	15.8101	0.541327	0.270663	+	0.962674i	$0.412757\pi$
$854$	0	0
$855$	10.0042	0.342135
$856$	0	0
$857$	−7.97069	−0.272274	−0.136137	−	0.990690i	$-0.543469\pi$
$857$	−7.97069	−0.272274	−0.136137	+	0.990690i	$0.543469\pi$
$858$	0	0
$859$	42.0416	1.43444	0.717221	−	0.696846i	$-0.245414\pi$
$859$	42.0416	1.43444	0.717221	+	0.696846i	$0.245414\pi$
$860$	0	0
$861$	0.602188	0.0205225
$862$	0	0
$863$	15.8579	0.539808	0.269904	−	0.962887i	$-0.413008\pi$
$863$	15.8579	0.539808	0.269904	+	0.962887i	$0.413008\pi$
$864$	0	0
$865$	−34.8701	−1.18562
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10.9289	0.370739
$870$	0	0
$871$	0	0
$872$	0	0
$873$	38.6172	1.30700
$874$	0	0
$875$	3.85786	0.130420
$876$	0	0
$877$	−7.33664	−0.247741	−0.123870	−	0.992298i	$-0.539531\pi$
$877$	−7.33664	−0.247741	−0.123870	+	0.992298i	$0.539531\pi$
$878$	0	0
$879$	−5.59767	−0.188805
$880$	0	0
$881$	24.2066	0.815540	0.407770	−	0.913085i	$-0.366306\pi$
$881$	24.2066	0.815540	0.407770	+	0.913085i	$0.366306\pi$
$882$	0	0
$883$	−41.2548	−1.38834	−0.694168	−	0.719813i	$-0.744227\pi$
$883$	−41.2548	−1.38834	−0.694168	+	0.719813i	$0.744227\pi$
$884$	0	0
$885$	−11.1716	−0.375528
$886$	0	0
$887$	−15.2848	−0.513214	−0.256607	−	0.966516i	$-0.582605\pi$
$887$	−15.2848	−0.513214	−0.256607	+	0.966516i	$0.582605\pi$
$888$	0	0
$889$	−0.557127	−0.0186855
$890$	0	0
$891$	3.11586	0.104385
$892$	0	0
$893$	−28.7696	−0.962736
$894$	0	0
$895$	33.7080	1.12673
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.7279	−0.424500
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−2.48528	−0.0827050
$904$	0	0
$905$	−15.0711	−0.500979
$906$	0	0
$907$	−23.4637	−0.779101	−0.389550	−	0.921005i	$-0.627370\pi$
$907$	−23.4637	−0.779101	−0.389550	+	0.921005i	$0.627370\pi$
$908$	0	0
$909$	24.1421	0.800744
$910$	0	0
$911$	−29.1383	−0.965396	−0.482698	−	0.875787i	$-0.660343\pi$
$911$	−29.1383	−0.965396	−0.482698	+	0.875787i	$0.660343\pi$
$912$	0	0
$913$	−7.83938	−0.259446
$914$	0	0
$915$	13.9623	0.461580
$916$	0	0
$917$	−4.58579	−0.151436
$918$	0	0
$919$	44.2843	1.46080	0.730402	−	0.683018i	$-0.239333\pi$
$919$	44.2843	1.46080	0.730402	+	0.683018i	$0.239333\pi$
$920$	0	0
$921$	−20.3797	−0.671535
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.21371	0.0399065
$926$	0	0
$927$	5.65685	0.185795
$928$	0	0
$929$	27.7933	0.911869	0.455934	−	0.890013i	$-0.349305\pi$
$929$	27.7933	0.911869	0.455934	+	0.890013i	$0.349305\pi$
$930$	0	0
$931$	−15.4731	−0.507110
$932$	0	0
$933$	3.07107	0.100542
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.443651	−0.0144934	−0.00724672	−	0.999974i	$-0.502307\pi$
$937$	−0.443651	−0.0144934	−0.00724672	+	0.999974i	$0.502307\pi$
$938$	0	0
$939$	3.27208	0.106780
$940$	0	0
$941$	−38.9886	−1.27099	−0.635497	−	0.772103i	$-0.719205\pi$
$941$	−38.9886	−1.27099	−0.635497	+	0.772103i	$0.719205\pi$
$942$	0	0
$943$	16.4437	0.535479
$944$	0	0
$945$	2.42742	0.0789639
$946$	0	0
$947$	−51.4202	−1.67093	−0.835466	−	0.549541i	$-0.814802\pi$
$947$	−51.4202	−1.67093	−0.835466	+	0.549541i	$0.814802\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−12.1005	−0.392386
$952$	0	0
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	16.3128	0.527870
$956$	0	0
$957$	−3.24718	−0.104966
$958$	0	0
$959$	4.43835	0.143322
$960$	0	0
$961$	−25.7279	−0.829933
$962$	0	0
$963$	−34.9217	−1.12534
$964$	0	0
$965$	1.41421	0.0455251
$966$	0	0
$967$	−34.5269	−1.11031	−0.555155	−	0.831747i	$-0.687341\pi$
$967$	−34.5269	−1.11031	−0.555155	+	0.831747i	$0.687341\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−54.7279	−1.75630	−0.878151	−	0.478383i	$-0.841223\pi$
$971$	−54.7279	−1.75630	−0.878151	+	0.478383i	$0.841223\pi$
$972$	0	0
$973$	−0.159380	−0.00510947
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.3848	1.10007	0.550033	−	0.835143i	$-0.314615\pi$
$977$	34.3848	1.10007	0.550033	+	0.835143i	$0.314615\pi$
$978$	0	0
$979$	1.26810	0.0405287
$980$	0	0
$981$	−28.1647	−0.899231
$982$	0	0
$983$	−38.9886	−1.24354	−0.621772	−	0.783198i	$-0.713587\pi$
$983$	−38.9886	−1.24354	−0.621772	+	0.783198i	$0.713587\pi$
$984$	0	0
$985$	−22.7279	−0.724172
$986$	0	0
$987$	−3.11270	−0.0990783
$988$	0	0
$989$	−67.8644	−2.15796
$990$	0	0
$991$	−50.7862	−1.61328	−0.806638	−	0.591046i	$-0.798715\pi$
$991$	−50.7862	−1.61328	−0.806638	+	0.591046i	$0.798715\pi$
$992$	0	0
$993$	−10.5294	−0.334141
$994$	0	0
$995$	−41.5563	−1.31742
$996$	0	0
$997$	8.68167	0.274951	0.137476	−	0.990505i	$-0.456101\pi$
$997$	8.68167	0.274951	0.137476	+	0.990505i	$0.456101\pi$
$998$	0	0
$999$	3.17157	0.100344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.s.1.3		4
4.3	odd	2		4624.2.a.bm.1.2		4
17.4	even	4		2312.2.b.j.577.2		4
17.5	odd	16		136.2.n.a.25.1	✓	4
17.7	odd	16		136.2.n.a.49.1	yes	4
17.13	even	4		2312.2.b.j.577.3		4
17.16	even	2	inner	2312.2.a.s.1.2		4
51.5	even	16		1224.2.bq.a.433.1		4
51.41	even	16		1224.2.bq.a.865.1		4
68.7	even	16		272.2.v.e.49.1		4
68.39	even	16		272.2.v.e.161.1		4
68.67	odd	2		4624.2.a.bm.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.a.25.1	✓	4	17.5	odd	16
136.2.n.a.49.1	yes	4	17.7	odd	16
272.2.v.e.49.1		4	68.7	even	16
272.2.v.e.161.1		4	68.39	even	16
1224.2.bq.a.433.1		4	51.5	even	16
1224.2.bq.a.865.1		4	51.41	even	16
2312.2.a.s.1.2		4	17.16	even	2	inner
2312.2.a.s.1.3		4	1.1	even	1	trivial
2312.2.b.j.577.2		4	17.4	even	4
2312.2.b.j.577.3		4	17.13	even	4
4624.2.a.bm.1.2		4	4.3	odd	2
4624.2.a.bm.1.3		4	68.67	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+0.765367 q^{3} -1.84776 q^{5} +0.317025 q^{7} -2.41421 q^{9} +0.765367 q^{11} -1.41421 q^{15} +2.24264 q^{19} +0.242641 q^{21} +6.62567 q^{23} -1.58579 q^{25} -4.14386 q^{27} -5.54328 q^{29} +2.29610 q^{31} +0.585786 q^{33} -0.585786 q^{35} -0.765367 q^{37} +2.48181 q^{41} -10.2426 q^{43} +4.46088 q^{45} -12.8284 q^{47} -6.89949 q^{49} -3.07107 q^{53} -1.41421 q^{55} +1.71644 q^{57} +7.89949 q^{59} -9.87285 q^{61} -0.765367 q^{63} -9.65685 q^{67} +5.07107 q^{69} -9.68714 q^{71} -7.97069 q^{73} -1.21371 q^{75} +0.242641 q^{77} +14.2793 q^{79} +4.07107 q^{81} -10.2426 q^{83} -4.24264 q^{87} +1.65685 q^{89} +1.75736 q^{93} -4.14386 q^{95} -15.9958 q^{97} -1.84776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} - 8 q^{19} - 16 q^{21} - 12 q^{25} + 8 q^{33} - 8 q^{35} - 24 q^{43} - 40 q^{47} + 12 q^{49} + 16 q^{53} - 8 q^{59} - 16 q^{67} - 8 q^{69} - 16 q^{77} - 12 q^{81} - 24 q^{83} - 16 q^{89} + 24 q^{93}+O(q^{100}) \) 4 * q - 4 * q^9 - 8 * q^19 - 16 * q^21 - 12 * q^25 + 8 * q^33 - 8 * q^35 - 24 * q^43 - 40 * q^47 + 12 * q^49 + 16 * q^53 - 8 * q^59 - 16 * q^67 - 8 * q^69 - 16 * q^77 - 12 * q^81 - 24 * q^83 - 16 * q^89 + 24 * q^93

Introduction

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