LMFDB - Embedded newform 136.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(1,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("136.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.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:	$8.02425976078$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	136.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46410 q^{3} +0.928203 q^{5} -7.60770 q^{7} -24.8564 q^{9} +O(q^{10})$	$q-1.46410 q^{3} +0.928203 q^{5} -7.60770 q^{7} -24.8564 q^{9} +35.0333 q^{11} -83.5692 q^{13} -1.35898 q^{15} +17.0000 q^{17} -115.636 q^{19} +11.1384 q^{21} -64.6025 q^{23} -124.138 q^{25} +75.9230 q^{27} -288.067 q^{29} +186.172 q^{31} -51.2923 q^{33} -7.06149 q^{35} +241.769 q^{37} +122.354 q^{39} -43.8564 q^{41} +257.646 q^{43} -23.0718 q^{45} -40.1539 q^{47} -285.123 q^{49} -24.8897 q^{51} +214.995 q^{53} +32.5180 q^{55} +169.303 q^{57} +585.913 q^{59} +331.759 q^{61} +189.100 q^{63} -77.5692 q^{65} +819.426 q^{67} +94.5847 q^{69} -961.079 q^{71} -342.420 q^{73} +181.751 q^{75} -266.523 q^{77} -513.290 q^{79} +559.964 q^{81} +270.887 q^{83} +15.7795 q^{85} +421.759 q^{87} +728.082 q^{89} +635.769 q^{91} -272.574 q^{93} -107.334 q^{95} +211.990 q^{97} -870.802 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 12 * q^5 - 36 * q^7 - 22 * q^9	$ 2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9} - 20 q^{11} - 84 q^{13} - 72 q^{15} + 34 q^{17} + 32 q^{19} - 144 q^{21} + 44 q^{23} - 82 q^{25} - 56 q^{27} - 396 q^{29} + 116 q^{31} - 352 q^{33} + 360 q^{35} - 140 q^{37} + 120 q^{39} - 60 q^{41} + 640 q^{43} - 60 q^{45} - 496 q^{47} + 178 q^{49} + 68 q^{51} + 236 q^{53} + 744 q^{55} + 976 q^{57} + 576 q^{59} - 348 q^{61} + 108 q^{63} - 72 q^{65} + 1528 q^{67} + 688 q^{69} - 876 q^{71} - 380 q^{73} + 412 q^{75} + 1296 q^{77} + 172 q^{79} - 238 q^{81} - 1024 q^{83} - 204 q^{85} - 168 q^{87} - 844 q^{89} + 648 q^{91} - 656 q^{93} - 2016 q^{95} + 36 q^{97} - 1028 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 12 * q^5 - 36 * q^7 - 22 * q^9 - 20 * q^11 - 84 * q^13 - 72 * q^15 + 34 * q^17 + 32 * q^19 - 144 * q^21 + 44 * q^23 - 82 * q^25 - 56 * q^27 - 396 * q^29 + 116 * q^31 - 352 * q^33 + 360 * q^35 - 140 * q^37 + 120 * q^39 - 60 * q^41 + 640 * q^43 - 60 * q^45 - 496 * q^47 + 178 * q^49 + 68 * q^51 + 236 * q^53 + 744 * q^55 + 976 * q^57 + 576 * q^59 - 348 * q^61 + 108 * q^63 - 72 * q^65 + 1528 * q^67 + 688 * q^69 - 876 * q^71 - 380 * q^73 + 412 * q^75 + 1296 * q^77 + 172 * q^79 - 238 * q^81 - 1024 * q^83 - 204 * q^85 - 168 * q^87 - 844 * q^89 + 648 * q^91 - 656 * q^93 - 2016 * q^95 + 36 * q^97 - 1028 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.46410	−0.281766	−0.140883	−	0.990026i	$-0.544994\pi$
$3$	−1.46410	−0.281766	−0.140883	+	0.990026i	$0.544994\pi$
$4$	0	0
$5$	0.928203	0.0830210	0.0415105	−	0.999138i	$-0.486783\pi$
$5$	0.928203	0.0830210	0.0415105	+	0.999138i	$0.486783\pi$
$6$	0	0
$7$	−7.60770	−0.410777	−0.205388	−	0.978681i	$-0.565846\pi$
$7$	−7.60770	−0.410777	−0.205388	+	0.978681i	$0.565846\pi$
$8$	0	0
$9$	−24.8564	−0.920608
$10$	0	0
$11$	35.0333	0.960268	0.480134	−	0.877195i	$-0.340588\pi$
$11$	35.0333	0.960268	0.480134	+	0.877195i	$0.340588\pi$
$12$	0	0
$13$	−83.5692	−1.78292	−0.891459	−	0.453102i	$-0.850317\pi$
$13$	−83.5692	−1.78292	−0.891459	+	0.453102i	$0.850317\pi$
$14$	0	0
$15$	−1.35898	−0.0233925
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−115.636	−1.39625	−0.698123	−	0.715978i	$-0.745981\pi$
$19$	−115.636	−1.39625	−0.698123	+	0.715978i	$0.745981\pi$
$20$	0	0
$21$	11.1384	0.115743
$22$	0	0
$23$	−64.6025	−0.585677	−0.292838	−	0.956162i	$-0.594600\pi$
$23$	−64.6025	−0.585677	−0.292838	+	0.956162i	$0.594600\pi$
$24$	0	0
$25$	−124.138	−0.993108
$26$	0	0
$27$	75.9230	0.541163
$28$	0	0
$29$	−288.067	−1.84457	−0.922287	−	0.386506i	$-0.873682\pi$
$29$	−288.067	−1.84457	−0.922287	+	0.386506i	$0.873682\pi$
$30$	0	0
$31$	186.172	1.07863	0.539313	−	0.842105i	$-0.318684\pi$
$31$	186.172	1.07863	0.539313	+	0.842105i	$0.318684\pi$
$32$	0	0
$33$	−51.2923	−0.270571
$34$	0	0
$35$	−7.06149	−0.0341031
$36$	0	0
$37$	241.769	1.07423	0.537116	−	0.843508i	$-0.319514\pi$
$37$	241.769	1.07423	0.537116	+	0.843508i	$0.319514\pi$
$38$	0	0
$39$	122.354	0.502366
$40$	0	0
$41$	−43.8564	−0.167054	−0.0835271	−	0.996506i	$-0.526619\pi$
$41$	−43.8564	−0.167054	−0.0835271	+	0.996506i	$0.526619\pi$
$42$	0	0
$43$	257.646	0.913737	0.456868	−	0.889534i	$-0.348971\pi$
$43$	257.646	0.913737	0.456868	+	0.889534i	$0.348971\pi$
$44$	0	0
$45$	−23.0718	−0.0764298
$46$	0	0
$47$	−40.1539	−0.124618	−0.0623090	−	0.998057i	$-0.519846\pi$
$47$	−40.1539	−0.124618	−0.0623090	+	0.998057i	$0.519846\pi$
$48$	0	0
$49$	−285.123	−0.831262
$50$	0	0
$51$	−24.8897	−0.0683384
$52$	0	0
$53$	214.995	0.557204	0.278602	−	0.960407i	$-0.410129\pi$
$53$	214.995	0.557204	0.278602	+	0.960407i	$0.410129\pi$
$54$	0	0
$55$	32.5180	0.0797224
$56$	0	0
$57$	169.303	0.393416
$58$	0	0
$59$	585.913	1.29287	0.646435	−	0.762969i	$-0.276259\pi$
$59$	585.913	1.29287	0.646435	+	0.762969i	$0.276259\pi$
$60$	0	0
$61$	331.759	0.696350	0.348175	−	0.937430i	$-0.386801\pi$
$61$	331.759	0.696350	0.348175	+	0.937430i	$0.386801\pi$
$62$	0	0
$63$	189.100	0.378164
$64$	0	0
$65$	−77.5692	−0.148020
$66$	0	0
$67$	819.426	1.49416	0.747080	−	0.664734i	$-0.231455\pi$
$67$	819.426	1.49416	0.747080	+	0.664734i	$0.231455\pi$
$68$	0	0
$69$	94.5847	0.165024
$70$	0	0
$71$	−961.079	−1.60647	−0.803233	−	0.595665i	$-0.796889\pi$
$71$	−961.079	−1.60647	−0.803233	+	0.595665i	$0.796889\pi$
$72$	0	0
$73$	−342.420	−0.549004	−0.274502	−	0.961587i	$-0.588513\pi$
$73$	−342.420	−0.549004	−0.274502	+	0.961587i	$0.588513\pi$
$74$	0	0
$75$	181.751	0.279824
$76$	0	0
$77$	−266.523	−0.394456
$78$	0	0
$79$	−513.290	−0.731007	−0.365504	−	0.930810i	$-0.619103\pi$
$79$	−513.290	−0.731007	−0.365504	+	0.930810i	$0.619103\pi$
$80$	0	0
$81$	559.964	0.768126
$82$	0	0
$83$	270.887	0.358237	0.179119	−	0.983827i	$-0.442675\pi$
$83$	270.887	0.358237	0.179119	+	0.983827i	$0.442675\pi$
$84$	0	0
$85$	15.7795	0.0201356
$86$	0	0
$87$	421.759	0.519739
$88$	0	0
$89$	728.082	0.867152	0.433576	−	0.901117i	$-0.357252\pi$
$89$	728.082	0.867152	0.433576	+	0.901117i	$0.357252\pi$
$90$	0	0
$91$	635.769	0.732381
$92$	0	0
$93$	−272.574	−0.303921
$94$	0	0
$95$	−107.334	−0.115918
$96$	0	0
$97$	211.990	0.221900	0.110950	−	0.993826i	$-0.464611\pi$
$97$	211.990	0.221900	0.110950	+	0.993826i	$0.464611\pi$
$98$	0	0
$99$	−870.802	−0.884030
$100$	0	0
$101$	−865.538	−0.852716	−0.426358	−	0.904555i	$-0.640204\pi$
$101$	−865.538	−0.852716	−0.426358	+	0.904555i	$0.640204\pi$
$102$	0	0
$103$	−772.246	−0.738754	−0.369377	−	0.929280i	$-0.620429\pi$
$103$	−772.246	−0.738754	−0.369377	+	0.929280i	$0.620429\pi$
$104$	0	0
$105$	10.3387	0.00960912
$106$	0	0
$107$	−2200.23	−1.98789	−0.993946	−	0.109866i	$-0.964958\pi$
$107$	−2200.23	−1.98789	−0.993946	+	0.109866i	$0.964958\pi$
$108$	0	0
$109$	−39.9127	−0.0350729	−0.0175365	−	0.999846i	$-0.505582\pi$
$109$	−39.9127	−0.0350729	−0.0175365	+	0.999846i	$0.505582\pi$
$110$	0	0
$111$	−353.975	−0.302683
$112$	0	0
$113$	−823.723	−0.685746	−0.342873	−	0.939382i	$-0.611400\pi$
$113$	−823.723	−0.685746	−0.342873	+	0.939382i	$0.611400\pi$
$114$	0	0
$115$	−59.9643	−0.0486235
$116$	0	0
$117$	2077.23	1.64137
$118$	0	0
$119$	−129.331	−0.0996280
$120$	0	0
$121$	−103.666	−0.0778861
$122$	0	0
$123$	64.2102	0.0470703
$124$	0	0
$125$	−231.251	−0.165470
$126$	0	0
$127$	−2112.78	−1.47622	−0.738108	−	0.674683i	$-0.764280\pi$
$127$	−2112.78	−1.47622	−0.738108	+	0.674683i	$0.764280\pi$
$128$	0	0
$129$	−377.220	−0.257460
$130$	0	0
$131$	−921.310	−0.614468	−0.307234	−	0.951634i	$-0.599403\pi$
$131$	−921.310	−0.614468	−0.307234	+	0.951634i	$0.599403\pi$
$132$	0	0
$133$	879.722	0.573546
$134$	0	0
$135$	70.4720	0.0449279
$136$	0	0
$137$	688.656	0.429459	0.214729	−	0.976674i	$-0.431113\pi$
$137$	688.656	0.429459	0.214729	+	0.976674i	$0.431113\pi$
$138$	0	0
$139$	2132.04	1.30099	0.650495	−	0.759511i	$-0.274562\pi$
$139$	2132.04	1.30099	0.650495	+	0.759511i	$0.274562\pi$
$140$	0	0
$141$	58.7894	0.0351132
$142$	0	0
$143$	−2927.71	−1.71208
$144$	0	0
$145$	−267.384	−0.153138
$146$	0	0
$147$	417.449	0.234222
$148$	0	0
$149$	−2505.65	−1.37766	−0.688829	−	0.724924i	$-0.741875\pi$
$149$	−2505.65	−1.37766	−0.688829	+	0.724924i	$0.741875\pi$
$150$	0	0
$151$	−1272.06	−0.685553	−0.342776	−	0.939417i	$-0.611367\pi$
$151$	−1272.06	−0.685553	−0.342776	+	0.939417i	$0.611367\pi$
$152$	0	0
$153$	−422.559	−0.223280
$154$	0	0
$155$	172.805	0.0895487
$156$	0	0
$157$	178.328	0.0906507	0.0453253	−	0.998972i	$-0.485568\pi$
$157$	178.328	0.0906507	0.0453253	+	0.998972i	$0.485568\pi$
$158$	0	0
$159$	−314.774	−0.157001
$160$	0	0
$161$	491.476	0.240582
$162$	0	0
$163$	3511.89	1.68756	0.843782	−	0.536687i	$-0.180324\pi$
$163$	3511.89	1.68756	0.843782	+	0.536687i	$0.180324\pi$
$164$	0	0
$165$	−47.6097	−0.0224631
$166$	0	0
$167$	−432.854	−0.200570	−0.100285	−	0.994959i	$-0.531976\pi$
$167$	−432.854	−0.200570	−0.100285	+	0.994959i	$0.531976\pi$
$168$	0	0
$169$	4786.81	2.17880
$170$	0	0
$171$	2874.29	1.28540
$172$	0	0
$173$	2338.15	1.02755	0.513775	−	0.857925i	$-0.328247\pi$
$173$	2338.15	1.02755	0.513775	+	0.857925i	$0.328247\pi$
$174$	0	0
$175$	944.407	0.407946
$176$	0	0
$177$	−857.836	−0.364287
$178$	0	0
$179$	−1011.75	−0.422467	−0.211234	−	0.977436i	$-0.567748\pi$
$179$	−1011.75	−0.422467	−0.211234	+	0.977436i	$0.567748\pi$
$180$	0	0
$181$	−1983.64	−0.814599	−0.407300	−	0.913295i	$-0.633529\pi$
$181$	−1983.64	−0.814599	−0.407300	+	0.913295i	$0.633529\pi$
$182$	0	0
$183$	−485.729	−0.196208
$184$	0	0
$185$	224.411	0.0891839
$186$	0	0
$187$	595.566	0.232899
$188$	0	0
$189$	−577.599	−0.222297
$190$	0	0
$191$	2156.05	0.816787	0.408394	−	0.912806i	$-0.366089\pi$
$191$	2156.05	0.816787	0.408394	+	0.912806i	$0.366089\pi$
$192$	0	0
$193$	1957.28	0.729991	0.364995	−	0.931009i	$-0.381071\pi$
$193$	1957.28	0.729991	0.364995	+	0.931009i	$0.381071\pi$
$194$	0	0
$195$	113.569	0.0417070
$196$	0	0
$197$	1822.13	0.658991	0.329496	−	0.944157i	$-0.393121\pi$
$197$	1822.13	0.658991	0.329496	+	0.944157i	$0.393121\pi$
$198$	0	0
$199$	−3990.55	−1.42152	−0.710759	−	0.703435i	$-0.751649\pi$
$199$	−3990.55	−1.42152	−0.710759	+	0.703435i	$0.751649\pi$
$200$	0	0
$201$	−1199.72	−0.421004
$202$	0	0
$203$	2191.52	0.757708
$204$	0	0
$205$	−40.7077	−0.0138690
$206$	0	0
$207$	1605.79	0.539178
$208$	0	0
$209$	−4051.11	−1.34077
$210$	0	0
$211$	5971.53	1.94833	0.974163	−	0.225846i	$-0.0725148\pi$
$211$	5971.53	1.94833	0.974163	+	0.225846i	$0.0725148\pi$
$212$	0	0
$213$	1407.12	0.452648
$214$	0	0
$215$	239.148	0.0758593
$216$	0	0
$217$	−1416.34	−0.443075
$218$	0	0
$219$	501.338	0.154691
$220$	0	0
$221$	−1420.68	−0.432421
$222$	0	0
$223$	1854.26	0.556819	0.278409	−	0.960463i	$-0.410193\pi$
$223$	1854.26	0.556819	0.278409	+	0.960463i	$0.410193\pi$
$224$	0	0
$225$	3085.64	0.914262
$226$	0	0
$227$	1958.41	0.572617	0.286309	−	0.958137i	$-0.407572\pi$
$227$	1958.41	0.572617	0.286309	+	0.958137i	$0.407572\pi$
$228$	0	0
$229$	−3406.66	−0.983049	−0.491524	−	0.870864i	$-0.663560\pi$
$229$	−3406.66	−0.983049	−0.491524	+	0.870864i	$0.663560\pi$
$230$	0	0
$231$	390.217	0.111144
$232$	0	0
$233$	−1119.92	−0.314885	−0.157443	−	0.987528i	$-0.550325\pi$
$233$	−1119.92	−0.314885	−0.157443	+	0.987528i	$0.550325\pi$
$234$	0	0
$235$	−37.2710	−0.0103459
$236$	0	0
$237$	751.508	0.205973
$238$	0	0
$239$	−4839.65	−1.30984	−0.654918	−	0.755700i	$-0.727297\pi$
$239$	−4839.65	−1.30984	−0.654918	+	0.755700i	$0.727297\pi$
$240$	0	0
$241$	−6067.19	−1.62167	−0.810834	−	0.585276i	$-0.800986\pi$
$241$	−6067.19	−1.62167	−0.810834	+	0.585276i	$0.800986\pi$
$242$	0	0
$243$	−2869.77	−0.757595
$244$	0	0
$245$	−264.652	−0.0690122
$246$	0	0
$247$	9663.60	2.48939
$248$	0	0
$249$	−396.606	−0.100939
$250$	0	0
$251$	−2963.93	−0.745345	−0.372672	−	0.927963i	$-0.621558\pi$
$251$	−2963.93	−0.745345	−0.372672	+	0.927963i	$0.621558\pi$
$252$	0	0
$253$	−2263.24	−0.562406
$254$	0	0
$255$	−23.1027	−0.00567352
$256$	0	0
$257$	3756.86	0.911854	0.455927	−	0.890017i	$-0.349308\pi$
$257$	3756.86	0.911854	0.455927	+	0.890017i	$0.349308\pi$
$258$	0	0
$259$	−1839.31	−0.441270
$260$	0	0
$261$	7160.30	1.69813
$262$	0	0
$263$	−4883.27	−1.14492	−0.572462	−	0.819931i	$-0.694012\pi$
$263$	−4883.27	−1.14492	−0.572462	+	0.819931i	$0.694012\pi$
$264$	0	0
$265$	199.559	0.0462596
$266$	0	0
$267$	−1065.99	−0.244334
$268$	0	0
$269$	−4279.27	−0.969931	−0.484966	−	0.874533i	$-0.661168\pi$
$269$	−4279.27	−0.969931	−0.484966	+	0.874533i	$0.661168\pi$
$270$	0	0
$271$	6381.54	1.43045	0.715223	−	0.698896i	$-0.246325\pi$
$271$	6381.54	1.43045	0.715223	+	0.698896i	$0.246325\pi$
$272$	0	0
$273$	−930.831	−0.206361
$274$	0	0
$275$	−4348.98	−0.953649
$276$	0	0
$277$	−7522.44	−1.63169	−0.815847	−	0.578268i	$-0.803729\pi$
$277$	−7522.44	−1.63169	−0.815847	+	0.578268i	$0.803729\pi$
$278$	0	0
$279$	−4627.56	−0.992992
$280$	0	0
$281$	−6559.87	−1.39263	−0.696315	−	0.717737i	$-0.745178\pi$
$281$	−6559.87	−1.39263	−0.696315	+	0.717737i	$0.745178\pi$
$282$	0	0
$283$	−1759.12	−0.369501	−0.184751	−	0.982785i	$-0.559148\pi$
$283$	−1759.12	−0.369501	−0.184751	+	0.982785i	$0.559148\pi$
$284$	0	0
$285$	157.147	0.0326618
$286$	0	0
$287$	333.646	0.0686220
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−310.374	−0.0625240
$292$	0	0
$293$	3079.42	0.613998	0.306999	−	0.951710i	$-0.400675\pi$
$293$	3079.42	0.613998	0.306999	+	0.951710i	$0.400675\pi$
$294$	0	0
$295$	543.846	0.107335
$296$	0	0
$297$	2659.84	0.519661
$298$	0	0
$299$	5398.78	1.04421
$300$	0	0
$301$	−1960.09	−0.375342
$302$	0	0
$303$	1267.24	0.240267
$304$	0	0
$305$	307.940	0.0578117
$306$	0	0
$307$	7231.51	1.34438	0.672189	−	0.740380i	$-0.265354\pi$
$307$	7231.51	1.34438	0.672189	+	0.740380i	$0.265354\pi$
$308$	0	0
$309$	1130.65	0.208156
$310$	0	0
$311$	−8545.03	−1.55802	−0.779010	−	0.627012i	$-0.784278\pi$
$311$	−8545.03	−1.55802	−0.779010	+	0.627012i	$0.784278\pi$
$312$	0	0
$313$	−5808.14	−1.04887	−0.524434	−	0.851451i	$-0.675723\pi$
$313$	−5808.14	−1.04887	−0.524434	+	0.851451i	$0.675723\pi$
$314$	0	0
$315$	175.523	0.0313956
$316$	0	0
$317$	−3478.13	−0.616249	−0.308125	−	0.951346i	$-0.599701\pi$
$317$	−3478.13	−0.616249	−0.308125	+	0.951346i	$0.599701\pi$
$318$	0	0
$319$	−10091.9	−1.77128
$320$	0	0
$321$	3221.36	0.560122
$322$	0	0
$323$	−1965.81	−0.338640
$324$	0	0
$325$	10374.2	1.77063
$326$	0	0
$327$	58.4363	0.00988237
$328$	0	0
$329$	305.479	0.0511902
$330$	0	0
$331$	−2729.62	−0.453273	−0.226636	−	0.973979i	$-0.572773\pi$
$331$	−2729.62	−0.453273	−0.226636	+	0.973979i	$0.572773\pi$
$332$	0	0
$333$	−6009.51	−0.988947
$334$	0	0
$335$	760.594	0.124047
$336$	0	0
$337$	−3408.52	−0.550962	−0.275481	−	0.961307i	$-0.588837\pi$
$337$	−3408.52	−0.550962	−0.275481	+	0.961307i	$0.588837\pi$
$338$	0	0
$339$	1206.01	0.193220
$340$	0	0
$341$	6522.22	1.03577
$342$	0	0
$343$	4778.57	0.752240
$344$	0	0
$345$	87.7938	0.0137005
$346$	0	0
$347$	1393.38	0.215564	0.107782	−	0.994175i	$-0.465625\pi$
$347$	1393.38	0.215564	0.107782	+	0.994175i	$0.465625\pi$
$348$	0	0
$349$	−390.298	−0.0598630	−0.0299315	−	0.999552i	$-0.509529\pi$
$349$	−390.298	−0.0598630	−0.0299315	+	0.999552i	$0.509529\pi$
$350$	0	0
$351$	−6344.83	−0.964849
$352$	0	0
$353$	−6504.99	−0.980810	−0.490405	−	0.871495i	$-0.663151\pi$
$353$	−6504.99	−0.980810	−0.490405	+	0.871495i	$0.663151\pi$
$354$	0	0
$355$	−892.077	−0.133371
$356$	0	0
$357$	189.353	0.0280718
$358$	0	0
$359$	2604.52	0.382900	0.191450	−	0.981502i	$-0.438681\pi$
$359$	2604.52	0.382900	0.191450	+	0.981502i	$0.438681\pi$
$360$	0	0
$361$	6512.65	0.949505
$362$	0	0
$363$	151.778	0.0219457
$364$	0	0
$365$	−317.836	−0.0455789
$366$	0	0
$367$	10882.8	1.54790	0.773948	−	0.633249i	$-0.218279\pi$
$367$	10882.8	1.54790	0.773948	+	0.633249i	$0.218279\pi$
$368$	0	0
$369$	1090.11	0.153791
$370$	0	0
$371$	−1635.62	−0.228887
$372$	0	0
$373$	−2983.40	−0.414141	−0.207071	−	0.978326i	$-0.566393\pi$
$373$	−2983.40	−0.414141	−0.207071	+	0.978326i	$0.566393\pi$
$374$	0	0
$375$	338.575	0.0466238
$376$	0	0
$377$	24073.5	3.28872
$378$	0	0
$379$	−3227.09	−0.437373	−0.218686	−	0.975795i	$-0.570177\pi$
$379$	−3227.09	−0.437373	−0.218686	+	0.975795i	$0.570177\pi$
$380$	0	0
$381$	3093.33	0.415948
$382$	0	0
$383$	6857.86	0.914936	0.457468	−	0.889226i	$-0.348756\pi$
$383$	6857.86	0.914936	0.457468	+	0.889226i	$0.348756\pi$
$384$	0	0
$385$	−247.387	−0.0327481
$386$	0	0
$387$	−6404.16	−0.841193
$388$	0	0
$389$	10890.1	1.41940	0.709702	−	0.704502i	$-0.248830\pi$
$389$	10890.1	1.41940	0.709702	+	0.704502i	$0.248830\pi$
$390$	0	0
$391$	−1098.24	−0.142047
$392$	0	0
$393$	1348.89	0.173136
$394$	0	0
$395$	−476.437	−0.0606890
$396$	0	0
$397$	2759.86	0.348901	0.174450	−	0.984666i	$-0.444185\pi$
$397$	2759.86	0.348901	0.174450	+	0.984666i	$0.444185\pi$
$398$	0	0
$399$	−1288.00	−0.161606
$400$	0	0
$401$	1961.90	0.244321	0.122160	−	0.992510i	$-0.461018\pi$
$401$	1961.90	0.244321	0.122160	+	0.992510i	$0.461018\pi$
$402$	0	0
$403$	−15558.2	−1.92310
$404$	0	0
$405$	519.760	0.0637706
$406$	0	0
$407$	8469.98	1.03155
$408$	0	0
$409$	8258.94	0.998480	0.499240	−	0.866464i	$-0.333613\pi$
$409$	8258.94	0.998480	0.499240	+	0.866464i	$0.333613\pi$
$410$	0	0
$411$	−1008.26	−0.121007
$412$	0	0
$413$	−4457.45	−0.531081
$414$	0	0
$415$	251.438	0.0297412
$416$	0	0
$417$	−3121.53	−0.366575
$418$	0	0
$419$	6590.86	0.768459	0.384230	−	0.923238i	$-0.374467\pi$
$419$	6590.86	0.768459	0.384230	+	0.923238i	$0.374467\pi$
$420$	0	0
$421$	−2451.24	−0.283768	−0.141884	−	0.989883i	$-0.545316\pi$
$421$	−2451.24	−0.283768	−0.141884	+	0.989883i	$0.545316\pi$
$422$	0	0
$423$	998.082	0.114724
$424$	0	0
$425$	−2110.35	−0.240864
$426$	0	0
$427$	−2523.92	−0.286045
$428$	0	0
$429$	4286.46	0.482406
$430$	0	0
$431$	12065.2	1.34840	0.674199	−	0.738550i	$-0.264489\pi$
$431$	12065.2	1.34840	0.674199	+	0.738550i	$0.264489\pi$
$432$	0	0
$433$	−7543.04	−0.837171	−0.418586	−	0.908177i	$-0.637474\pi$
$433$	−7543.04	−0.837171	−0.418586	+	0.908177i	$0.637474\pi$
$434$	0	0
$435$	391.478	0.0431493
$436$	0	0
$437$	7470.37	0.817749
$438$	0	0
$439$	−4914.93	−0.534343	−0.267172	−	0.963649i	$-0.586089\pi$
$439$	−4914.93	−0.534343	−0.267172	+	0.963649i	$0.586089\pi$
$440$	0	0
$441$	7087.13	0.765266
$442$	0	0
$443$	−14440.1	−1.54869	−0.774346	−	0.632762i	$-0.781921\pi$
$443$	−14440.1	−1.54869	−0.774346	+	0.632762i	$0.781921\pi$
$444$	0	0
$445$	675.808	0.0719918
$446$	0	0
$447$	3668.53	0.388178
$448$	0	0
$449$	−9707.39	−1.02031	−0.510156	−	0.860082i	$-0.670412\pi$
$449$	−9707.39	−1.02031	−0.510156	+	0.860082i	$0.670412\pi$
$450$	0	0
$451$	−1536.44	−0.160417
$452$	0	0
$453$	1862.42	0.193166
$454$	0	0
$455$	590.123	0.0608031
$456$	0	0
$457$	4495.06	0.460109	0.230055	−	0.973178i	$-0.426110\pi$
$457$	4495.06	0.460109	0.230055	+	0.973178i	$0.426110\pi$
$458$	0	0
$459$	1290.69	0.131251
$460$	0	0
$461$	−1092.31	−0.110355	−0.0551777	−	0.998477i	$-0.517573\pi$
$461$	−1092.31	−0.110355	−0.0551777	+	0.998477i	$0.517573\pi$
$462$	0	0
$463$	8212.83	0.824368	0.412184	−	0.911101i	$-0.364766\pi$
$463$	8212.83	0.824368	0.412184	+	0.911101i	$0.364766\pi$
$464$	0	0
$465$	−253.004	−0.0252318
$466$	0	0
$467$	−8826.37	−0.874595	−0.437297	−	0.899317i	$-0.644064\pi$
$467$	−8826.37	−0.874595	−0.437297	+	0.899317i	$0.644064\pi$
$468$	0	0
$469$	−6233.94	−0.613767
$470$	0	0
$471$	−261.091	−0.0255423
$472$	0	0
$473$	9026.20	0.877432
$474$	0	0
$475$	14354.9	1.38662
$476$	0	0
$477$	−5344.00	−0.512966
$478$	0	0
$479$	−16888.5	−1.61097	−0.805484	−	0.592617i	$-0.798095\pi$
$479$	−16888.5	−1.61097	−0.805484	+	0.592617i	$0.798095\pi$
$480$	0	0
$481$	−20204.5	−1.91527
$482$	0	0
$483$	−719.571	−0.0677881
$484$	0	0
$485$	196.770	0.0184224
$486$	0	0
$487$	5325.78	0.495552	0.247776	−	0.968817i	$-0.420300\pi$
$487$	5325.78	0.495552	0.247776	+	0.968817i	$0.420300\pi$
$488$	0	0
$489$	−5141.77	−0.475499
$490$	0	0
$491$	−18743.1	−1.72274	−0.861368	−	0.507982i	$-0.830392\pi$
$491$	−18743.1	−1.72274	−0.861368	+	0.507982i	$0.830392\pi$
$492$	0	0
$493$	−4897.13	−0.447375
$494$	0	0
$495$	−808.282	−0.0733930
$496$	0	0
$497$	7311.60	0.659899
$498$	0	0
$499$	−557.533	−0.0500172	−0.0250086	−	0.999687i	$-0.507961\pi$
$499$	−557.533	−0.0500172	−0.0250086	+	0.999687i	$0.507961\pi$
$500$	0	0
$501$	633.742	0.0565140
$502$	0	0
$503$	9888.07	0.876515	0.438257	−	0.898849i	$-0.355596\pi$
$503$	9888.07	0.876515	0.438257	+	0.898849i	$0.355596\pi$
$504$	0	0
$505$	−803.395	−0.0707933
$506$	0	0
$507$	−7008.38	−0.613912
$508$	0	0
$509$	8219.25	0.715740	0.357870	−	0.933771i	$-0.383503\pi$
$509$	8219.25	0.715740	0.357870	+	0.933771i	$0.383503\pi$
$510$	0	0
$511$	2605.03	0.225518
$512$	0	0
$513$	−8779.43	−0.755597
$514$	0	0
$515$	−716.801	−0.0613321
$516$	0	0
$517$	−1406.72	−0.119667
$518$	0	0
$519$	−3423.29	−0.289529
$520$	0	0
$521$	−12967.4	−1.09042	−0.545212	−	0.838298i	$-0.683551\pi$
$521$	−12967.4	−1.09042	−0.545212	+	0.838298i	$0.683551\pi$
$522$	0	0
$523$	12391.5	1.03603	0.518014	−	0.855372i	$-0.326671\pi$
$523$	12391.5	1.03603	0.518014	+	0.855372i	$0.326671\pi$
$524$	0	0
$525$	−1382.71	−0.114945
$526$	0	0
$527$	3164.92	0.261605
$528$	0	0
$529$	−7993.51	−0.656983
$530$	0	0
$531$	−14563.7	−1.19023
$532$	0	0
$533$	3665.05	0.297844
$534$	0	0
$535$	−2042.26	−0.165037
$536$	0	0
$537$	1481.30	0.119037
$538$	0	0
$539$	−9988.80	−0.798234
$540$	0	0
$541$	−20467.6	−1.62657	−0.813283	−	0.581868i	$-0.802322\pi$
$541$	−20467.6	−1.62657	−0.813283	+	0.581868i	$0.802322\pi$
$542$	0	0
$543$	2904.24	0.229527
$544$	0	0
$545$	−37.0471	−0.00291179
$546$	0	0
$547$	2410.39	0.188411	0.0942056	−	0.995553i	$-0.469969\pi$
$547$	2410.39	0.188411	0.0942056	+	0.995553i	$0.469969\pi$
$548$	0	0
$549$	−8246.33	−0.641065
$550$	0	0
$551$	33310.8	2.57548
$552$	0	0
$553$	3904.95	0.300281
$554$	0	0
$555$	−328.560	−0.0251290
$556$	0	0
$557$	−8165.50	−0.621155	−0.310577	−	0.950548i	$-0.600522\pi$
$557$	−8165.50	−0.621155	−0.310577	+	0.950548i	$0.600522\pi$
$558$	0	0
$559$	−21531.3	−1.62912
$560$	0	0
$561$	−871.970	−0.0656232
$562$	0	0
$563$	19714.5	1.47579	0.737893	−	0.674918i	$-0.235821\pi$
$563$	19714.5	1.47579	0.737893	+	0.674918i	$0.235821\pi$
$564$	0	0
$565$	−764.582	−0.0569314
$566$	0	0
$567$	−4260.03	−0.315528
$568$	0	0
$569$	6808.46	0.501627	0.250813	−	0.968035i	$-0.419302\pi$
$569$	6808.46	0.501627	0.250813	+	0.968035i	$0.419302\pi$
$570$	0	0
$571$	11660.8	0.854621	0.427311	−	0.904105i	$-0.359461\pi$
$571$	11660.8	0.854621	0.427311	+	0.904105i	$0.359461\pi$
$572$	0	0
$573$	−3156.68	−0.230143
$574$	0	0
$575$	8019.66	0.581640
$576$	0	0
$577$	−17508.0	−1.26320	−0.631600	−	0.775294i	$-0.717602\pi$
$577$	−17508.0	−1.26320	−0.631600	+	0.775294i	$0.717602\pi$
$578$	0	0
$579$	−2865.66	−0.205687
$580$	0	0
$581$	−2060.83	−0.147156
$582$	0	0
$583$	7531.98	0.535065
$584$	0	0
$585$	1928.09	0.136268
$586$	0	0
$587$	19549.2	1.37459	0.687293	−	0.726380i	$-0.258799\pi$
$587$	19549.2	1.37459	0.687293	+	0.726380i	$0.258799\pi$
$588$	0	0
$589$	−21528.1	−1.50603
$590$	0	0
$591$	−2667.78	−0.185682
$592$	0	0
$593$	22738.3	1.57462	0.787310	−	0.616558i	$-0.211473\pi$
$593$	22738.3	1.57462	0.787310	+	0.616558i	$0.211473\pi$
$594$	0	0
$595$	−120.045	−0.00827122
$596$	0	0
$597$	5842.56	0.400536
$598$	0	0
$599$	−3679.68	−0.250998	−0.125499	−	0.992094i	$-0.540053\pi$
$599$	−3679.68	−0.250998	−0.125499	+	0.992094i	$0.540053\pi$
$600$	0	0
$601$	2834.52	0.192384	0.0961918	−	0.995363i	$-0.469334\pi$
$601$	2834.52	0.192384	0.0961918	+	0.995363i	$0.469334\pi$
$602$	0	0
$603$	−20368.0	−1.37554
$604$	0	0
$605$	−96.2235	−0.00646618
$606$	0	0
$607$	7728.36	0.516778	0.258389	−	0.966041i	$-0.416808\pi$
$607$	7728.36	0.516778	0.258389	+	0.966041i	$0.416808\pi$
$608$	0	0
$609$	−3208.61	−0.213497
$610$	0	0
$611$	3355.63	0.222184
$612$	0	0
$613$	−3729.75	−0.245748	−0.122874	−	0.992422i	$-0.539211\pi$
$613$	−3729.75	−0.245748	−0.122874	+	0.992422i	$0.539211\pi$
$614$	0	0
$615$	59.6001	0.00390782
$616$	0	0
$617$	10148.2	0.662156	0.331078	−	0.943603i	$-0.392588\pi$
$617$	10148.2	0.662156	0.331078	+	0.943603i	$0.392588\pi$
$618$	0	0
$619$	−24975.6	−1.62174	−0.810868	−	0.585229i	$-0.801005\pi$
$619$	−24975.6	−1.62174	−0.810868	+	0.585229i	$0.801005\pi$
$620$	0	0
$621$	−4904.82	−0.316946
$622$	0	0
$623$	−5539.02	−0.356206
$624$	0	0
$625$	15302.7	0.979370
$626$	0	0
$627$	5931.23	0.377784
$628$	0	0
$629$	4110.08	0.260540
$630$	0	0
$631$	−10577.2	−0.667310	−0.333655	−	0.942695i	$-0.608282\pi$
$631$	−10577.2	−0.667310	−0.333655	+	0.942695i	$0.608282\pi$
$632$	0	0
$633$	−8742.92	−0.548973
$634$	0	0
$635$	−1961.09	−0.122557
$636$	0	0
$637$	23827.5	1.48207
$638$	0	0
$639$	23889.0	1.47893
$640$	0	0
$641$	−4744.60	−0.292357	−0.146178	−	0.989258i	$-0.546697\pi$
$641$	−4744.60	−0.292357	−0.146178	+	0.989258i	$0.546697\pi$
$642$	0	0
$643$	13420.1	0.823075	0.411537	−	0.911393i	$-0.364992\pi$
$643$	13420.1	0.823075	0.411537	+	0.911393i	$0.364992\pi$
$644$	0	0
$645$	−350.137	−0.0213746
$646$	0	0
$647$	30769.7	1.86968	0.934839	−	0.355072i	$-0.115544\pi$
$647$	30769.7	1.86968	0.934839	+	0.355072i	$0.115544\pi$
$648$	0	0
$649$	20526.5	1.24150
$650$	0	0
$651$	2073.66	0.124844
$652$	0	0
$653$	−30697.6	−1.83965	−0.919825	−	0.392328i	$-0.871670\pi$
$653$	−30697.6	−1.83965	−0.919825	+	0.392328i	$0.871670\pi$
$654$	0	0
$655$	−855.163	−0.0510137
$656$	0	0
$657$	8511.34	0.505417
$658$	0	0
$659$	−6532.13	−0.386124	−0.193062	−	0.981187i	$-0.561842\pi$
$659$	−6532.13	−0.386124	−0.193062	+	0.981187i	$0.561842\pi$
$660$	0	0
$661$	25064.2	1.47486	0.737432	−	0.675421i	$-0.236038\pi$
$661$	25064.2	1.47486	0.737432	+	0.675421i	$0.236038\pi$
$662$	0	0
$663$	2080.02	0.121842
$664$	0	0
$665$	816.561	0.0476164
$666$	0	0
$667$	18609.8	1.08032
$668$	0	0
$669$	−2714.83	−0.156893
$670$	0	0
$671$	11622.6	0.668683
$672$	0	0
$673$	−28852.3	−1.65256	−0.826280	−	0.563260i	$-0.809547\pi$
$673$	−28852.3	−1.65256	−0.826280	+	0.563260i	$0.809547\pi$
$674$	0	0
$675$	−9424.97	−0.537433
$676$	0	0
$677$	17281.2	0.981050	0.490525	−	0.871427i	$-0.336805\pi$
$677$	17281.2	0.981050	0.490525	+	0.871427i	$0.336805\pi$
$678$	0	0
$679$	−1612.75	−0.0911514
$680$	0	0
$681$	−2867.31	−0.161344
$682$	0	0
$683$	−11782.9	−0.660117	−0.330058	−	0.943961i	$-0.607068\pi$
$683$	−11782.9	−0.660117	−0.330058	+	0.943961i	$0.607068\pi$
$684$	0	0
$685$	639.213	0.0356541
$686$	0	0
$687$	4987.69	0.276990
$688$	0	0
$689$	−17967.0	−0.993449
$690$	0	0
$691$	8292.73	0.456541	0.228271	−	0.973598i	$-0.426693\pi$
$691$	8292.73	0.456541	0.228271	+	0.973598i	$0.426693\pi$
$692$	0	0
$693$	6624.80	0.363139
$694$	0	0
$695$	1978.97	0.108009
$696$	0	0
$697$	−745.559	−0.0405166
$698$	0	0
$699$	1639.67	0.0887241
$700$	0	0
$701$	−1920.82	−0.103493	−0.0517464	−	0.998660i	$-0.516479\pi$
$701$	−1920.82	−0.103493	−0.0517464	+	0.998660i	$0.516479\pi$
$702$	0	0
$703$	−27957.2	−1.49989
$704$	0	0
$705$	54.5685	0.00291513
$706$	0	0
$707$	6584.75	0.350276
$708$	0	0
$709$	34241.4	1.81377	0.906886	−	0.421377i	$-0.138453\pi$
$709$	34241.4	1.81377	0.906886	+	0.421377i	$0.138453\pi$
$710$	0	0
$711$	12758.5	0.672971
$712$	0	0
$713$	−12027.2	−0.631727
$714$	0	0
$715$	−2717.51	−0.142138
$716$	0	0
$717$	7085.74	0.369068
$718$	0	0
$719$	−1945.22	−0.100896	−0.0504482	−	0.998727i	$-0.516065\pi$
$719$	−1945.22	−0.100896	−0.0504482	+	0.998727i	$0.516065\pi$
$720$	0	0
$721$	5875.01	0.303463
$722$	0	0
$723$	8882.98	0.456932
$724$	0	0
$725$	35760.1	1.83186
$726$	0	0
$727$	−27193.9	−1.38730	−0.693650	−	0.720313i	$-0.743998\pi$
$727$	−27193.9	−1.38730	−0.693650	+	0.720313i	$0.743998\pi$
$728$	0	0
$729$	−10917.4	−0.554661
$730$	0	0
$731$	4379.98	0.221614
$732$	0	0
$733$	15351.2	0.773545	0.386773	−	0.922175i	$-0.373590\pi$
$733$	15351.2	0.773545	0.386773	+	0.922175i	$0.373590\pi$
$734$	0	0
$735$	387.478	0.0194453
$736$	0	0
$737$	28707.2	1.43479
$738$	0	0
$739$	−10027.3	−0.499133	−0.249566	−	0.968358i	$-0.580288\pi$
$739$	−10027.3	−0.499133	−0.249566	+	0.968358i	$0.580288\pi$
$740$	0	0
$741$	−14148.5	−0.701428
$742$	0	0
$743$	−17396.2	−0.858955	−0.429477	−	0.903078i	$-0.641302\pi$
$743$	−17396.2	−0.858955	−0.429477	+	0.903078i	$0.641302\pi$
$744$	0	0
$745$	−2325.75	−0.114375
$746$	0	0
$747$	−6733.28	−0.329796
$748$	0	0
$749$	16738.7	0.816581
$750$	0	0
$751$	3286.42	0.159685	0.0798423	−	0.996808i	$-0.474558\pi$
$751$	3286.42	0.159685	0.0798423	+	0.996808i	$0.474558\pi$
$752$	0	0
$753$	4339.49	0.210013
$754$	0	0
$755$	−1180.73	−0.0569153
$756$	0	0
$757$	−33578.7	−1.61220	−0.806101	−	0.591777i	$-0.798426\pi$
$757$	−33578.7	−1.61220	−0.806101	+	0.591777i	$0.798426\pi$
$758$	0	0
$759$	3313.62	0.158467
$760$	0	0
$761$	31225.7	1.48742	0.743712	−	0.668500i	$-0.233063\pi$
$761$	31225.7	1.48742	0.743712	+	0.668500i	$0.233063\pi$
$762$	0	0
$763$	303.644	0.0144071
$764$	0	0
$765$	−392.221	−0.0185369
$766$	0	0
$767$	−48964.3	−2.30508
$768$	0	0
$769$	17386.4	0.815304	0.407652	−	0.913137i	$-0.366348\pi$
$769$	17386.4	0.815304	0.407652	+	0.913137i	$0.366348\pi$
$770$	0	0
$771$	−5500.43	−0.256930
$772$	0	0
$773$	−9764.68	−0.454348	−0.227174	−	0.973854i	$-0.572949\pi$
$773$	−9764.68	−0.454348	−0.227174	+	0.973854i	$0.572949\pi$
$774$	0	0
$775$	−23111.1	−1.07119
$776$	0	0
$777$	2692.93	0.124335
$778$	0	0
$779$	5071.37	0.233249
$780$	0	0
$781$	−33669.8	−1.54264
$782$	0	0
$783$	−21870.9	−0.998215
$784$	0	0
$785$	165.525	0.00752591
$786$	0	0
$787$	27285.8	1.23588	0.617939	−	0.786226i	$-0.287968\pi$
$787$	27285.8	1.23588	0.617939	+	0.786226i	$0.287968\pi$
$788$	0	0
$789$	7149.60	0.322601
$790$	0	0
$791$	6266.63	0.281689
$792$	0	0
$793$	−27724.8	−1.24154
$794$	0	0
$795$	−292.175	−0.0130344
$796$	0	0
$797$	−18419.4	−0.818632	−0.409316	−	0.912393i	$-0.634233\pi$
$797$	−18419.4	−0.818632	−0.409316	+	0.912393i	$0.634233\pi$
$798$	0	0
$799$	−682.616	−0.0302243
$800$	0	0
$801$	−18097.5	−0.798307
$802$	0	0
$803$	−11996.1	−0.527191
$804$	0	0
$805$	456.190	0.0199734
$806$	0	0
$807$	6265.28	0.273294
$808$	0	0
$809$	−34203.0	−1.48642	−0.743211	−	0.669057i	$-0.766698\pi$
$809$	−34203.0	−1.48642	−0.743211	+	0.669057i	$0.766698\pi$
$810$	0	0
$811$	4605.05	0.199390	0.0996950	−	0.995018i	$-0.468213\pi$
$811$	4605.05	0.199390	0.0996950	+	0.995018i	$0.468213\pi$
$812$	0	0
$813$	−9343.22	−0.403052
$814$	0	0
$815$	3259.75	0.140103
$816$	0	0
$817$	−29793.1	−1.27580
$818$	0	0
$819$	−15802.9	−0.674236
$820$	0	0
$821$	−11557.4	−0.491298	−0.245649	−	0.969359i	$-0.579001\pi$
$821$	−11557.4	−0.491298	−0.245649	+	0.969359i	$0.579001\pi$
$822$	0	0
$823$	16810.8	0.712016	0.356008	−	0.934483i	$-0.384138\pi$
$823$	16810.8	0.712016	0.356008	+	0.934483i	$0.384138\pi$
$824$	0	0
$825$	6367.35	0.268706
$826$	0	0
$827$	−15832.7	−0.665726	−0.332863	−	0.942975i	$-0.608015\pi$
$827$	−15832.7	−0.665726	−0.332863	+	0.942975i	$0.608015\pi$
$828$	0	0
$829$	28451.0	1.19197	0.595986	−	0.802995i	$-0.296761\pi$
$829$	28451.0	1.19197	0.595986	+	0.802995i	$0.296761\pi$
$830$	0	0
$831$	11013.6	0.459757
$832$	0	0
$833$	−4847.09	−0.201611
$834$	0	0
$835$	−401.776	−0.0166516
$836$	0	0
$837$	14134.7	0.583713
$838$	0	0
$839$	−20249.1	−0.833226	−0.416613	−	0.909084i	$-0.636783\pi$
$839$	−20249.1	−0.833226	−0.416613	+	0.909084i	$0.636783\pi$
$840$	0	0
$841$	58593.4	2.40245
$842$	0	0
$843$	9604.31	0.392396
$844$	0	0
$845$	4443.14	0.180886
$846$	0	0
$847$	788.663	0.0319938
$848$	0	0
$849$	2575.53	0.104113
$850$	0	0
$851$	−15618.9	−0.629153
$852$	0	0
$853$	16346.3	0.656140	0.328070	−	0.944653i	$-0.393602\pi$
$853$	16346.3	0.656140	0.328070	+	0.944653i	$0.393602\pi$
$854$	0	0
$855$	2667.93	0.106715
$856$	0	0
$857$	15549.0	0.619770	0.309885	−	0.950774i	$-0.399709\pi$
$857$	15549.0	0.619770	0.309885	+	0.950774i	$0.399709\pi$
$858$	0	0
$859$	−17843.1	−0.708728	−0.354364	−	0.935107i	$-0.615303\pi$
$859$	−17843.1	−0.708728	−0.354364	+	0.935107i	$0.615303\pi$
$860$	0	0
$861$	−488.492	−0.0193354
$862$	0	0
$863$	29343.7	1.15744	0.578721	−	0.815526i	$-0.303552\pi$
$863$	29343.7	1.15744	0.578721	+	0.815526i	$0.303552\pi$
$864$	0	0
$865$	2170.28	0.0853082
$866$	0	0
$867$	−423.125	−0.0165745
$868$	0	0
$869$	−17982.2	−0.701963
$870$	0	0
$871$	−68478.8	−2.66397
$872$	0	0
$873$	−5269.30	−0.204283
$874$	0	0
$875$	1759.29	0.0679712
$876$	0	0
$877$	−9538.17	−0.367253	−0.183627	−	0.982996i	$-0.558784\pi$
$877$	−9538.17	−0.367253	−0.183627	+	0.982996i	$0.558784\pi$
$878$	0	0
$879$	−4508.58	−0.173004
$880$	0	0
$881$	−35280.5	−1.34918	−0.674591	−	0.738192i	$-0.735680\pi$
$881$	−35280.5	−1.34918	−0.674591	+	0.738192i	$0.735680\pi$
$882$	0	0
$883$	−8515.92	−0.324557	−0.162278	−	0.986745i	$-0.551884\pi$
$883$	−8515.92	−0.324557	−0.162278	+	0.986745i	$0.551884\pi$
$884$	0	0
$885$	−796.246	−0.0302435
$886$	0	0
$887$	−39734.0	−1.50410	−0.752051	−	0.659105i	$-0.770935\pi$
$887$	−39734.0	−1.50410	−0.752051	+	0.659105i	$0.770935\pi$
$888$	0	0
$889$	16073.4	0.606395
$890$	0	0
$891$	19617.4	0.737607
$892$	0	0
$893$	4643.23	0.173998
$894$	0	0
$895$	−939.108	−0.0350737
$896$	0	0
$897$	−7904.37	−0.294224
$898$	0	0
$899$	−53629.9	−1.98961
$900$	0	0
$901$	3654.91	0.135142
$902$	0	0
$903$	2869.78	0.105759
$904$	0	0
$905$	−1841.22	−0.0676288
$906$	0	0
$907$	22025.4	0.806329	0.403165	−	0.915128i	$-0.367910\pi$
$907$	22025.4	0.806329	0.403165	+	0.915128i	$0.367910\pi$
$908$	0	0
$909$	21514.2	0.785017
$910$	0	0
$911$	4187.78	0.152302	0.0761512	−	0.997096i	$-0.475737\pi$
$911$	4187.78	0.152302	0.0761512	+	0.997096i	$0.475737\pi$
$912$	0	0
$913$	9490.07	0.344004
$914$	0	0
$915$	−450.855	−0.0162894
$916$	0	0
$917$	7009.05	0.252409
$918$	0	0
$919$	−9322.61	−0.334629	−0.167315	−	0.985904i	$-0.553510\pi$
$919$	−9322.61	−0.334629	−0.167315	+	0.985904i	$0.553510\pi$
$920$	0	0
$921$	−10587.7	−0.378801
$922$	0	0
$923$	80316.7	2.86420
$924$	0	0
$925$	−30012.8	−1.06683
$926$	0	0
$927$	19195.3	0.680103
$928$	0	0
$929$	25028.1	0.883901	0.441950	−	0.897040i	$-0.354287\pi$
$929$	25028.1	0.883901	0.441950	+	0.897040i	$0.354287\pi$
$930$	0	0
$931$	32970.4	1.16065
$932$	0	0
$933$	12510.8	0.438998
$934$	0	0
$935$	552.807	0.0193355
$936$	0	0
$937$	36634.8	1.27727	0.638637	−	0.769508i	$-0.279498\pi$
$937$	36634.8	1.27727	0.638637	+	0.769508i	$0.279498\pi$
$938$	0	0
$939$	8503.71	0.295536
$940$	0	0
$941$	−56248.4	−1.94861	−0.974307	−	0.225225i	$-0.927688\pi$
$941$	−56248.4	−1.94861	−0.974307	+	0.225225i	$0.927688\pi$
$942$	0	0
$943$	2833.24	0.0978397
$944$	0	0
$945$	−536.130	−0.0184553
$946$	0	0
$947$	44981.0	1.54349	0.771745	−	0.635932i	$-0.219384\pi$
$947$	44981.0	1.54349	0.771745	+	0.635932i	$0.219384\pi$
$948$	0	0
$949$	28615.8	0.978829
$950$	0	0
$951$	5092.33	0.173638
$952$	0	0
$953$	−5346.14	−0.181719	−0.0908596	−	0.995864i	$-0.528961\pi$
$953$	−5346.14	−0.181719	−0.0908596	+	0.995864i	$0.528961\pi$
$954$	0	0
$955$	2001.25	0.0678105
$956$	0	0
$957$	14775.6	0.499089
$958$	0	0
$959$	−5239.09	−0.176412
$960$	0	0
$961$	4868.92	0.163436
$962$	0	0
$963$	54689.9	1.83007
$964$	0	0
$965$	1816.75	0.0606046
$966$	0	0
$967$	−32790.2	−1.09045	−0.545223	−	0.838291i	$-0.683555\pi$
$967$	−32790.2	−1.09045	−0.545223	+	0.838291i	$0.683555\pi$
$968$	0	0
$969$	2878.15	0.0954173
$970$	0	0
$971$	−3313.57	−0.109513	−0.0547567	−	0.998500i	$-0.517438\pi$
$971$	−3313.57	−0.109513	−0.0547567	+	0.998500i	$0.517438\pi$
$972$	0	0
$973$	−16219.9	−0.534416
$974$	0	0
$975$	−15188.8	−0.498904
$976$	0	0
$977$	−44576.3	−1.45970	−0.729848	−	0.683609i	$-0.760409\pi$
$977$	−44576.3	−1.45970	−0.729848	+	0.683609i	$0.760409\pi$
$978$	0	0
$979$	25507.1	0.832698
$980$	0	0
$981$	992.087	0.0322884
$982$	0	0
$983$	11436.7	0.371081	0.185541	−	0.982637i	$-0.440596\pi$
$983$	11436.7	0.371081	0.185541	+	0.982637i	$0.440596\pi$
$984$	0	0
$985$	1691.30	0.0547101
$986$	0	0
$987$	−447.252	−0.0144237
$988$	0	0
$989$	−16644.6	−0.535154
$990$	0	0
$991$	−1965.50	−0.0630033	−0.0315017	−	0.999504i	$-0.510029\pi$
$991$	−1965.50	−0.0630033	−0.0315017	+	0.999504i	$0.510029\pi$
$992$	0	0
$993$	3996.44	0.127717
$994$	0	0
$995$	−3704.04	−0.118016
$996$	0	0
$997$	−2878.40	−0.0914343	−0.0457171	−	0.998954i	$-0.514557\pi$
$997$	−2878.40	−0.0914343	−0.0457171	+	0.998954i	$0.514557\pi$
$998$	0	0
$999$	18355.9	0.581335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.a.a.1.1	✓	2
3.2	odd	2		1224.4.a.d.1.1		2
4.3	odd	2		272.4.a.f.1.2		2
8.3	odd	2		1088.4.a.p.1.1		2
8.5	even	2		1088.4.a.n.1.2		2
12.11	even	2		2448.4.a.z.1.1		2
17.16	even	2		2312.4.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.a.1.1	✓	2	1.1	even	1	trivial
272.4.a.f.1.2		2	4.3	odd	2
1088.4.a.n.1.2		2	8.5	even	2
1088.4.a.p.1.1		2	8.3	odd	2
1224.4.a.d.1.1		2	3.2	odd	2
2312.4.a.b.1.2		2	17.16	even	2
2448.4.a.z.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46410 q^{3} +0.928203 q^{5} -7.60770 q^{7} -24.8564 q^{9} +O(q^{10})\)	\(q-1.46410 q^{3} +0.928203 q^{5} -7.60770 q^{7} -24.8564 q^{9} +35.0333 q^{11} -83.5692 q^{13} -1.35898 q^{15} +17.0000 q^{17} -115.636 q^{19} +11.1384 q^{21} -64.6025 q^{23} -124.138 q^{25} +75.9230 q^{27} -288.067 q^{29} +186.172 q^{31} -51.2923 q^{33} -7.06149 q^{35} +241.769 q^{37} +122.354 q^{39} -43.8564 q^{41} +257.646 q^{43} -23.0718 q^{45} -40.1539 q^{47} -285.123 q^{49} -24.8897 q^{51} +214.995 q^{53} +32.5180 q^{55} +169.303 q^{57} +585.913 q^{59} +331.759 q^{61} +189.100 q^{63} -77.5692 q^{65} +819.426 q^{67} +94.5847 q^{69} -961.079 q^{71} -342.420 q^{73} +181.751 q^{75} -266.523 q^{77} -513.290 q^{79} +559.964 q^{81} +270.887 q^{83} +15.7795 q^{85} +421.759 q^{87} +728.082 q^{89} +635.769 q^{91} -272.574 q^{93} -107.334 q^{95} +211.990 q^{97} -870.802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 12 * q^5 - 36 * q^7 - 22 * q^9	\( 2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9} - 20 q^{11} - 84 q^{13} - 72 q^{15} + 34 q^{17} + 32 q^{19} - 144 q^{21} + 44 q^{23} - 82 q^{25} - 56 q^{27} - 396 q^{29} + 116 q^{31} - 352 q^{33} + 360 q^{35} - 140 q^{37} + 120 q^{39} - 60 q^{41} + 640 q^{43} - 60 q^{45} - 496 q^{47} + 178 q^{49} + 68 q^{51} + 236 q^{53} + 744 q^{55} + 976 q^{57} + 576 q^{59} - 348 q^{61} + 108 q^{63} - 72 q^{65} + 1528 q^{67} + 688 q^{69} - 876 q^{71} - 380 q^{73} + 412 q^{75} + 1296 q^{77} + 172 q^{79} - 238 q^{81} - 1024 q^{83} - 204 q^{85} - 168 q^{87} - 844 q^{89} + 648 q^{91} - 656 q^{93} - 2016 q^{95} + 36 q^{97} - 1028 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 12 * q^5 - 36 * q^7 - 22 * q^9 - 20 * q^11 - 84 * q^13 - 72 * q^15 + 34 * q^17 + 32 * q^19 - 144 * q^21 + 44 * q^23 - 82 * q^25 - 56 * q^27 - 396 * q^29 + 116 * q^31 - 352 * q^33 + 360 * q^35 - 140 * q^37 + 120 * q^39 - 60 * q^41 + 640 * q^43 - 60 * q^45 - 496 * q^47 + 178 * q^49 + 68 * q^51 + 236 * q^53 + 744 * q^55 + 976 * q^57 + 576 * q^59 - 348 * q^61 + 108 * q^63 - 72 * q^65 + 1528 * q^67 + 688 * q^69 - 876 * q^71 - 380 * q^73 + 412 * q^75 + 1296 * q^77 + 172 * q^79 - 238 * q^81 - 1024 * q^83 - 204 * q^85 - 168 * q^87 - 844 * q^89 + 648 * q^91 - 656 * q^93 - 2016 * q^95 + 36 * q^97 - 1028 * q^99

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