LMFDB - Embedded newform 136.4.a.a.1.2

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			1.2
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)

$f(q)$	$=$	$q+5.46410 q^{3} -12.9282 q^{5} -28.3923 q^{7} +2.85641 q^{9} +O(q^{10})$	$q+5.46410 q^{3} -12.9282 q^{5} -28.3923 q^{7} +2.85641 q^{9} -55.0333 q^{11} -0.430781 q^{13} -70.6410 q^{15} +17.0000 q^{17} +147.636 q^{19} -155.138 q^{21} +108.603 q^{23} +42.1384 q^{25} -131.923 q^{27} -107.933 q^{29} -70.1718 q^{31} -300.708 q^{33} +367.061 q^{35} -381.769 q^{37} -2.35383 q^{39} -16.1436 q^{41} +382.354 q^{43} -36.9282 q^{45} -455.846 q^{47} +463.123 q^{49} +92.8897 q^{51} +21.0052 q^{53} +711.482 q^{55} +806.697 q^{57} -9.91274 q^{59} -679.759 q^{61} -81.1000 q^{63} +5.56922 q^{65} +708.574 q^{67} +593.415 q^{69} +85.0793 q^{71} -37.5795 q^{73} +230.249 q^{75} +1562.52 q^{77} +685.290 q^{79} -797.964 q^{81} -1294.89 q^{83} -219.779 q^{85} -589.759 q^{87} -1572.08 q^{89} +12.2309 q^{91} -383.426 q^{93} -1908.67 q^{95} -175.990 q^{97} -157.198 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	5.46410	1.05157	0.525783	−	0.850618i	$-0.323772\pi$
$3$	5.46410	1.05157	0.525783	+	0.850618i	$0.323772\pi$
$4$	0	0
$5$	−12.9282	−1.15633	−0.578167	−	0.815919i	$-0.696232\pi$
$5$	−12.9282	−1.15633	−0.578167	+	0.815919i	$0.696232\pi$
$6$	0	0
$7$	−28.3923	−1.53304	−0.766520	−	0.642220i	$-0.778013\pi$
$7$	−28.3923	−1.53304	−0.766520	+	0.642220i	$0.778013\pi$
$8$	0	0
$9$	2.85641	0.105793
$10$	0	0
$11$	−55.0333	−1.50847	−0.754235	−	0.656605i	$-0.771992\pi$
$11$	−55.0333	−1.50847	−0.754235	+	0.656605i	$0.771992\pi$
$12$	0	0
$13$	−0.430781	−0.00919054	−0.00459527	−	0.999989i	$-0.501463\pi$
$13$	−0.430781	−0.00919054	−0.00459527	+	0.999989i	$0.501463\pi$
$14$	0	0
$15$	−70.6410	−1.21596
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	147.636	1.78263	0.891316	−	0.453384i	$-0.149783\pi$
$19$	147.636	1.78263	0.891316	+	0.453384i	$0.149783\pi$
$20$	0	0
$21$	−155.138	−1.61209
$22$	0	0
$23$	108.603	0.984574	0.492287	−	0.870433i	$-0.336161\pi$
$23$	108.603	0.984574	0.492287	+	0.870433i	$0.336161\pi$
$24$	0	0
$25$	42.1384	0.337108
$26$	0	0
$27$	−131.923	−0.940319
$28$	0	0
$29$	−107.933	−0.691128	−0.345564	−	0.938395i	$-0.612312\pi$
$29$	−107.933	−0.691128	−0.345564	+	0.938395i	$0.612312\pi$
$30$	0	0
$31$	−70.1718	−0.406555	−0.203278	−	0.979121i	$-0.565159\pi$
$31$	−70.1718	−0.406555	−0.203278	+	0.979121i	$0.565159\pi$
$32$	0	0
$33$	−300.708	−1.58626
$34$	0	0
$35$	367.061	1.77271
$36$	0	0
$37$	−381.769	−1.69628	−0.848141	−	0.529770i	$-0.822278\pi$
$37$	−381.769	−1.69628	−0.848141	+	0.529770i	$0.822278\pi$
$38$	0	0
$39$	−2.35383	−0.00966447
$40$	0	0
$41$	−16.1436	−0.0614928	−0.0307464	−	0.999527i	$-0.509788\pi$
$41$	−16.1436	−0.0614928	−0.0307464	+	0.999527i	$0.509788\pi$
$42$	0	0
$43$	382.354	1.35601	0.678005	−	0.735057i	$-0.262845\pi$
$43$	382.354	1.35601	0.678005	+	0.735057i	$0.262845\pi$
$44$	0	0
$45$	−36.9282	−0.122332
$46$	0	0
$47$	−455.846	−1.41472	−0.707362	−	0.706852i	$-0.750115\pi$
$47$	−455.846	−1.41472	−0.707362	+	0.706852i	$0.750115\pi$
$48$	0	0
$49$	463.123	1.35021
$50$	0	0
$51$	92.8897	0.255042
$52$	0	0
$53$	21.0052	0.0544392	0.0272196	−	0.999629i	$-0.491335\pi$
$53$	21.0052	0.0544392	0.0272196	+	0.999629i	$0.491335\pi$
$54$	0	0
$55$	711.482	1.74429
$56$	0	0
$57$	806.697	1.87456
$58$	0	0
$59$	−9.91274	−0.0218734	−0.0109367	−	0.999940i	$-0.503481\pi$
$59$	−9.91274	−0.0218734	−0.0109367	+	0.999940i	$0.503481\pi$
$60$	0	0
$61$	−679.759	−1.42679	−0.713395	−	0.700762i	$-0.752843\pi$
$61$	−679.759	−1.42679	−0.713395	+	0.700762i	$0.752843\pi$
$62$	0	0
$63$	−81.1000	−0.162185
$64$	0	0
$65$	5.56922	0.0106273
$66$	0	0
$67$	708.574	1.29203	0.646016	−	0.763324i	$-0.276434\pi$
$67$	708.574	1.29203	0.646016	+	0.763324i	$0.276434\pi$
$68$	0	0
$69$	593.415	1.03535
$70$	0	0
$71$	85.0793	0.142212	0.0711061	−	0.997469i	$-0.477347\pi$
$71$	85.0793	0.142212	0.0711061	+	0.997469i	$0.477347\pi$
$72$	0	0
$73$	−37.5795	−0.0602514	−0.0301257	−	0.999546i	$-0.509591\pi$
$73$	−37.5795	−0.0602514	−0.0301257	+	0.999546i	$0.509591\pi$
$74$	0	0
$75$	230.249	0.354491
$76$	0	0
$77$	1562.52	2.31255
$78$	0	0
$79$	685.290	0.975963	0.487982	−	0.872854i	$-0.337733\pi$
$79$	685.290	0.975963	0.487982	+	0.872854i	$0.337733\pi$
$80$	0	0
$81$	−797.964	−1.09460
$82$	0	0
$83$	−1294.89	−1.71244	−0.856219	−	0.516613i	$-0.827192\pi$
$83$	−1294.89	−1.71244	−0.856219	+	0.516613i	$0.827192\pi$
$84$	0	0
$85$	−219.779	−0.280452
$86$	0	0
$87$	−589.759	−0.726768
$88$	0	0
$89$	−1572.08	−1.87236	−0.936182	−	0.351517i	$-0.885666\pi$
$89$	−1572.08	−1.87236	−0.936182	+	0.351517i	$0.885666\pi$
$90$	0	0
$91$	12.2309	0.0140895
$92$	0	0
$93$	−383.426	−0.427520
$94$	0	0
$95$	−1908.67	−2.06132
$96$	0	0
$97$	−175.990	−0.184217	−0.0921085	−	0.995749i	$-0.529361\pi$
$97$	−175.990	−0.184217	−0.0921085	+	0.995749i	$0.529361\pi$
$98$	0	0
$99$	−157.198	−0.159585
$100$	0	0
$101$	381.538	0.375886	0.187943	−	0.982180i	$-0.439818\pi$
$101$	381.538	0.375886	0.187943	+	0.982180i	$0.439818\pi$
$102$	0	0
$103$	724.246	0.692836	0.346418	−	0.938080i	$-0.387398\pi$
$103$	724.246	0.692836	0.346418	+	0.938080i	$0.387398\pi$
$104$	0	0
$105$	2005.66	1.86412
$106$	0	0
$107$	1132.23	1.02296	0.511482	−	0.859294i	$-0.329097\pi$
$107$	1132.23	1.02296	0.511482	+	0.859294i	$0.329097\pi$
$108$	0	0
$109$	555.913	0.488503	0.244251	−	0.969712i	$-0.421458\pi$
$109$	555.913	0.488503	0.244251	+	0.969712i	$0.421458\pi$
$110$	0	0
$111$	−2086.03	−1.78375
$112$	0	0
$113$	−1156.28	−0.962596	−0.481298	−	0.876557i	$-0.659835\pi$
$113$	−1156.28	−0.962596	−0.481298	+	0.876557i	$0.659835\pi$
$114$	0	0
$115$	−1404.04	−1.13850
$116$	0	0
$117$	−1.23048	−0.000972293 0
$118$	0	0
$119$	−482.669	−0.371817
$120$	0	0
$121$	1697.67	1.27548
$122$	0	0
$123$	−88.2102	−0.0646638
$124$	0	0
$125$	1071.25	0.766525
$126$	0	0
$127$	−2071.22	−1.44717	−0.723585	−	0.690235i	$-0.757507\pi$
$127$	−2071.22	−1.44717	−0.723585	+	0.690235i	$0.757507\pi$
$128$	0	0
$129$	2089.22	1.42593
$130$	0	0
$131$	−498.690	−0.332601	−0.166300	−	0.986075i	$-0.553182\pi$
$131$	−498.690	−0.332601	−0.166300	+	0.986075i	$0.553182\pi$
$132$	0	0
$133$	−4191.72	−2.73285
$134$	0	0
$135$	1705.53	1.08732
$136$	0	0
$137$	−1500.66	−0.935837	−0.467919	−	0.883772i	$-0.654996\pi$
$137$	−1500.66	−0.935837	−0.467919	+	0.883772i	$0.654996\pi$
$138$	0	0
$139$	−272.043	−0.166003	−0.0830015	−	0.996549i	$-0.526451\pi$
$139$	−272.043	−0.166003	−0.0830015	+	0.996549i	$0.526451\pi$
$140$	0	0
$141$	−2490.79	−1.48768
$142$	0	0
$143$	23.7073	0.0138637
$144$	0	0
$145$	1395.38	0.799175
$146$	0	0
$147$	2530.55	1.41984
$148$	0	0
$149$	−122.349	−0.0672700	−0.0336350	−	0.999434i	$-0.510708\pi$
$149$	−122.349	−0.0672700	−0.0336350	+	0.999434i	$0.510708\pi$
$150$	0	0
$151$	−703.944	−0.379378	−0.189689	−	0.981844i	$-0.560748\pi$
$151$	−703.944	−0.379378	−0.189689	+	0.981844i	$0.560748\pi$
$152$	0	0
$153$	48.5589	0.0256585
$154$	0	0
$155$	907.195	0.470114
$156$	0	0
$157$	1785.67	0.907720	0.453860	−	0.891073i	$-0.350047\pi$
$157$	1785.67	0.907720	0.453860	+	0.891073i	$0.350047\pi$
$158$	0	0
$159$	114.774	0.0572465
$160$	0	0
$161$	−3083.48	−1.50939
$162$	0	0
$163$	3588.11	1.72418	0.862092	−	0.506751i	$-0.169154\pi$
$163$	3588.11	1.72418	0.862092	+	0.506751i	$0.169154\pi$
$164$	0	0
$165$	3887.61	1.83424
$166$	0	0
$167$	−1659.15	−0.768794	−0.384397	−	0.923168i	$-0.625591\pi$
$167$	−1659.15	−0.768794	−0.384397	+	0.923168i	$0.625591\pi$
$168$	0	0
$169$	−2196.81	−0.999916
$170$	0	0
$171$	421.708	0.188590
$172$	0	0
$173$	−142.148	−0.0624702	−0.0312351	−	0.999512i	$-0.509944\pi$
$173$	−142.148	−0.0624702	−0.0312351	+	0.999512i	$0.509944\pi$
$174$	0	0
$175$	−1196.41	−0.516799
$176$	0	0
$177$	−54.1642	−0.0230013
$178$	0	0
$179$	387.749	0.161909	0.0809544	−	0.996718i	$-0.474203\pi$
$179$	387.749	0.161909	0.0809544	+	0.996718i	$0.474203\pi$
$180$	0	0
$181$	3683.64	1.51272	0.756360	−	0.654155i	$-0.226976\pi$
$181$	3683.64	1.51272	0.756360	+	0.654155i	$0.226976\pi$
$182$	0	0
$183$	−3714.27	−1.50037
$184$	0	0
$185$	4935.59	1.96147
$186$	0	0
$187$	−935.566	−0.365858
$188$	0	0
$189$	3745.60	1.44155
$190$	0	0
$191$	−1308.05	−0.495535	−0.247768	−	0.968819i	$-0.579697\pi$
$191$	−1308.05	−0.495535	−0.247768	+	0.968819i	$0.579697\pi$
$192$	0	0
$193$	−3585.28	−1.33717	−0.668586	−	0.743635i	$-0.733100\pi$
$193$	−3585.28	−1.33717	−0.668586	+	0.743635i	$0.733100\pi$
$194$	0	0
$195$	30.4308	0.0111754
$196$	0	0
$197$	−1434.13	−0.518667	−0.259333	−	0.965788i	$-0.583503\pi$
$197$	−1434.13	−0.518667	−0.259333	+	0.965788i	$0.583503\pi$
$198$	0	0
$199$	1018.55	0.362828	0.181414	−	0.983407i	$-0.441933\pi$
$199$	1018.55	0.362828	0.181414	+	0.983407i	$0.441933\pi$
$200$	0	0
$201$	3871.72	1.35866
$202$	0	0
$203$	3064.48	1.05953
$204$	0	0
$205$	208.708	0.0711062
$206$	0	0
$207$	310.213	0.104161
$208$	0	0
$209$	−8124.89	−2.68905
$210$	0	0
$211$	2888.47	0.942421	0.471210	−	0.882021i	$-0.343817\pi$
$211$	2888.47	0.942421	0.471210	+	0.882021i	$0.343817\pi$
$212$	0	0
$213$	464.882	0.149546
$214$	0	0
$215$	−4943.15	−1.56800
$216$	0	0
$217$	1992.34	0.623266
$218$	0	0
$219$	−205.338	−0.0633584
$220$	0	0
$221$	−7.32327	−0.00222903
$222$	0	0
$223$	3641.74	1.09358	0.546791	−	0.837269i	$-0.315849\pi$
$223$	3641.74	1.09358	0.546791	+	0.837269i	$0.315849\pi$
$224$	0	0
$225$	120.365	0.0356636
$226$	0	0
$227$	5221.59	1.52674	0.763368	−	0.645963i	$-0.223544\pi$
$227$	5221.59	1.52674	0.763368	+	0.645963i	$0.223544\pi$
$228$	0	0
$229$	4186.66	1.20813	0.604065	−	0.796935i	$-0.293547\pi$
$229$	4186.66	1.20813	0.604065	+	0.796935i	$0.293547\pi$
$230$	0	0
$231$	8537.78	2.43180
$232$	0	0
$233$	−3420.08	−0.961618	−0.480809	−	0.876825i	$-0.659657\pi$
$233$	−3420.08	−0.961618	−0.480809	+	0.876825i	$0.659657\pi$
$234$	0	0
$235$	5893.27	1.63589
$236$	0	0
$237$	3744.49	1.02629
$238$	0	0
$239$	−2456.35	−0.664803	−0.332402	−	0.943138i	$-0.607859\pi$
$239$	−2456.35	−0.664803	−0.332402	+	0.943138i	$0.607859\pi$
$240$	0	0
$241$	2967.19	0.793085	0.396542	−	0.918016i	$-0.370210\pi$
$241$	2967.19	0.793085	0.396542	+	0.918016i	$0.370210\pi$
$242$	0	0
$243$	−798.234	−0.210727
$244$	0	0
$245$	−5987.35	−1.56130
$246$	0	0
$247$	−63.5987	−0.0163833
$248$	0	0
$249$	−7075.39	−1.80074
$250$	0	0
$251$	−5652.07	−1.42134	−0.710668	−	0.703527i	$-0.751607\pi$
$251$	−5652.07	−1.42134	−0.710668	+	0.703527i	$0.751607\pi$
$252$	0	0
$253$	−5976.76	−1.48520
$254$	0	0
$255$	−1200.90	−0.294914
$256$	0	0
$257$	−1480.86	−0.359430	−0.179715	−	0.983719i	$-0.557518\pi$
$257$	−1480.86	−0.359430	−0.179715	+	0.983719i	$0.557518\pi$
$258$	0	0
$259$	10839.3	2.60047
$260$	0	0
$261$	−308.302	−0.0731164
$262$	0	0
$263$	−1460.73	−0.342482	−0.171241	−	0.985229i	$-0.554778\pi$
$263$	−1460.73	−0.342482	−0.171241	+	0.985229i	$0.554778\pi$
$264$	0	0
$265$	−271.559	−0.0629499
$266$	0	0
$267$	−8590.01	−1.96892
$268$	0	0
$269$	−6260.73	−1.41905	−0.709523	−	0.704682i	$-0.751090\pi$
$269$	−6260.73	−1.41905	−0.709523	+	0.704682i	$0.751090\pi$
$270$	0	0
$271$	−269.538	−0.0604179	−0.0302089	−	0.999544i	$-0.509617\pi$
$271$	−269.538	−0.0604179	−0.0302089	+	0.999544i	$0.509617\pi$
$272$	0	0
$273$	66.8306	0.0148160
$274$	0	0
$275$	−2319.02	−0.508517
$276$	0	0
$277$	−5097.56	−1.10571	−0.552857	−	0.833276i	$-0.686463\pi$
$277$	−5097.56	−1.10571	−0.552857	+	0.833276i	$0.686463\pi$
$278$	0	0
$279$	−200.439	−0.0430107
$280$	0	0
$281$	−1516.13	−0.321868	−0.160934	−	0.986965i	$-0.551451\pi$
$281$	−1516.13	−0.321868	−0.160934	+	0.986965i	$0.551451\pi$
$282$	0	0
$283$	−7668.88	−1.61084	−0.805420	−	0.592705i	$-0.798060\pi$
$283$	−7668.88	−1.61084	−0.805420	+	0.592705i	$0.798060\pi$
$284$	0	0
$285$	−10429.1	−2.16761
$286$	0	0
$287$	458.354	0.0942710
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−961.626	−0.193716
$292$	0	0
$293$	2580.58	0.514537	0.257269	−	0.966340i	$-0.417178\pi$
$293$	2580.58	0.514537	0.257269	+	0.966340i	$0.417178\pi$
$294$	0	0
$295$	128.154	0.0252929
$296$	0	0
$297$	7260.16	1.41844
$298$	0	0
$299$	−46.7839	−0.00904877
$300$	0	0
$301$	−10855.9	−2.07882
$302$	0	0
$303$	2084.76	0.395269
$304$	0	0
$305$	8788.06	1.64985
$306$	0	0
$307$	−583.507	−0.108477	−0.0542386	−	0.998528i	$-0.517273\pi$
$307$	−583.507	−0.108477	−0.0542386	+	0.998528i	$0.517273\pi$
$308$	0	0
$309$	3957.35	0.728563
$310$	0	0
$311$	−154.973	−0.0282563	−0.0141282	−	0.999900i	$-0.504497\pi$
$311$	−154.973	−0.0282563	−0.0141282	+	0.999900i	$0.504497\pi$
$312$	0	0
$313$	−5835.86	−1.05387	−0.526936	−	0.849905i	$-0.676659\pi$
$313$	−5835.86	−1.05387	−0.526936	+	0.849905i	$0.676659\pi$
$314$	0	0
$315$	1048.48	0.187540
$316$	0	0
$317$	10586.1	1.87563	0.937817	−	0.347130i	$-0.112844\pi$
$317$	10586.1	1.87563	0.937817	+	0.347130i	$0.112844\pi$
$318$	0	0
$319$	5939.93	1.04255
$320$	0	0
$321$	6186.64	1.07571
$322$	0	0
$323$	2509.81	0.432352
$324$	0	0
$325$	−18.1524	−0.00309820
$326$	0	0
$327$	3037.56	0.513693
$328$	0	0
$329$	12942.5	2.16883
$330$	0	0
$331$	−7094.38	−1.17807	−0.589037	−	0.808106i	$-0.700493\pi$
$331$	−7094.38	−1.17807	−0.589037	+	0.808106i	$0.700493\pi$
$332$	0	0
$333$	−1090.49	−0.179455
$334$	0	0
$335$	−9160.59	−1.49402
$336$	0	0
$337$	−1579.48	−0.255310	−0.127655	−	0.991819i	$-0.540745\pi$
$337$	−1579.48	−0.255310	−0.127655	+	0.991819i	$0.540745\pi$
$338$	0	0
$339$	−6318.01	−1.01223
$340$	0	0
$341$	3861.78	0.613277
$342$	0	0
$343$	−3410.57	−0.536890
$344$	0	0
$345$	−7671.79	−1.19720
$346$	0	0
$347$	−1717.38	−0.265688	−0.132844	−	0.991137i	$-0.542411\pi$
$347$	−1717.38	−0.265688	−0.132844	+	0.991137i	$0.542411\pi$
$348$	0	0
$349$	−6237.70	−0.956724	−0.478362	−	0.878163i	$-0.658769\pi$
$349$	−6237.70	−0.956724	−0.478362	+	0.878163i	$0.658769\pi$
$350$	0	0
$351$	56.8299	0.00864204
$352$	0	0
$353$	9900.99	1.49285	0.746426	−	0.665469i	$-0.231768\pi$
$353$	9900.99	1.49285	0.746426	+	0.665469i	$0.231768\pi$
$354$	0	0
$355$	−1099.92	−0.164445
$356$	0	0
$357$	−2637.35	−0.390990
$358$	0	0
$359$	−7524.52	−1.10621	−0.553104	−	0.833112i	$-0.686557\pi$
$359$	−7524.52	−1.10621	−0.553104	+	0.833112i	$0.686557\pi$
$360$	0	0
$361$	14937.3	2.17777
$362$	0	0
$363$	9276.22	1.34125
$364$	0	0
$365$	485.836	0.0696707
$366$	0	0
$367$	10169.2	1.44640	0.723199	−	0.690640i	$-0.242671\pi$
$367$	10169.2	1.44640	0.723199	+	0.690640i	$0.242671\pi$
$368$	0	0
$369$	−46.1127	−0.00650550
$370$	0	0
$371$	−596.385	−0.0834576
$372$	0	0
$373$	14115.4	1.95943	0.979716	−	0.200393i	$-0.0642219\pi$
$373$	14115.4	1.95943	0.979716	+	0.200393i	$0.0642219\pi$
$374$	0	0
$375$	5853.42	0.806052
$376$	0	0
$377$	46.4956	0.00635184
$378$	0	0
$379$	−10480.9	−1.42050	−0.710249	−	0.703951i	$-0.751417\pi$
$379$	−10480.9	−1.42050	−0.710249	+	0.703951i	$0.751417\pi$
$380$	0	0
$381$	−11317.3	−1.52180
$382$	0	0
$383$	9726.14	1.29760	0.648802	−	0.760957i	$-0.275270\pi$
$383$	9726.14	1.29760	0.648802	+	0.760957i	$0.275270\pi$
$384$	0	0
$385$	−20200.6	−2.67407
$386$	0	0
$387$	1092.16	0.143456
$388$	0	0
$389$	13217.9	1.72282	0.861409	−	0.507912i	$-0.169583\pi$
$389$	13217.9	1.72282	0.861409	+	0.507912i	$0.169583\pi$
$390$	0	0
$391$	1846.24	0.238794
$392$	0	0
$393$	−2724.89	−0.349752
$394$	0	0
$395$	−8859.56	−1.12854
$396$	0	0
$397$	5628.14	0.711507	0.355753	−	0.934580i	$-0.384224\pi$
$397$	5628.14	0.711507	0.355753	+	0.934580i	$0.384224\pi$
$398$	0	0
$399$	−22904.0	−2.87377
$400$	0	0
$401$	8890.10	1.10711	0.553554	−	0.832813i	$-0.313271\pi$
$401$	8890.10	1.10711	0.553554	+	0.832813i	$0.313271\pi$
$402$	0	0
$403$	30.2286	0.00373647
$404$	0	0
$405$	10316.2	1.26572
$406$	0	0
$407$	21010.0	2.55879
$408$	0	0
$409$	721.057	0.0871736	0.0435868	−	0.999050i	$-0.486121\pi$
$409$	721.057	0.0871736	0.0435868	+	0.999050i	$0.486121\pi$
$410$	0	0
$411$	−8199.74	−0.984095
$412$	0	0
$413$	281.446	0.0335328
$414$	0	0
$415$	16740.6	1.98015
$416$	0	0
$417$	−1486.47	−0.174563
$418$	0	0
$419$	−16306.9	−1.90129	−0.950647	−	0.310275i	$-0.899579\pi$
$419$	−16306.9	−1.90129	−0.950647	+	0.310275i	$0.899579\pi$
$420$	0	0
$421$	−11568.8	−1.33926	−0.669628	−	0.742697i	$-0.733546\pi$
$421$	−11568.8	−1.33926	−0.669628	+	0.742697i	$0.733546\pi$
$422$	0	0
$423$	−1302.08	−0.149668
$424$	0	0
$425$	716.353	0.0817606
$426$	0	0
$427$	19299.9	2.18733
$428$	0	0
$429$	129.539	0.0145786
$430$	0	0
$431$	−11733.2	−1.31129	−0.655647	−	0.755067i	$-0.727604\pi$
$431$	−11733.2	−1.31129	−0.655647	+	0.755067i	$0.727604\pi$
$432$	0	0
$433$	−8900.96	−0.987882	−0.493941	−	0.869495i	$-0.664444\pi$
$433$	−8900.96	−0.987882	−0.493941	+	0.869495i	$0.664444\pi$
$434$	0	0
$435$	7624.52	0.840386
$436$	0	0
$437$	16033.6	1.75513
$438$	0	0
$439$	−8857.07	−0.962927	−0.481464	−	0.876466i	$-0.659895\pi$
$439$	−8857.07	−0.962927	−0.481464	+	0.876466i	$0.659895\pi$
$440$	0	0
$441$	1322.87	0.142843
$442$	0	0
$443$	13328.1	1.42943	0.714716	−	0.699415i	$-0.246556\pi$
$443$	13328.1	1.42943	0.714716	+	0.699415i	$0.246556\pi$
$444$	0	0
$445$	20324.2	2.16508
$446$	0	0
$447$	−668.528	−0.0707389
$448$	0	0
$449$	−8432.61	−0.886323	−0.443162	−	0.896442i	$-0.646143\pi$
$449$	−8432.61	−0.886323	−0.443162	+	0.896442i	$0.646143\pi$
$450$	0	0
$451$	888.436	0.0927601
$452$	0	0
$453$	−3846.42	−0.398942
$454$	0	0
$455$	−158.123	−0.0162921
$456$	0	0
$457$	6628.94	0.678531	0.339266	−	0.940691i	$-0.389821\pi$
$457$	6628.94	0.678531	0.339266	+	0.940691i	$0.389821\pi$
$458$	0	0
$459$	−2242.69	−0.228061
$460$	0	0
$461$	−7327.69	−0.740314	−0.370157	−	0.928969i	$-0.620696\pi$
$461$	−7327.69	−0.740314	−0.370157	+	0.928969i	$0.620696\pi$
$462$	0	0
$463$	−8996.83	−0.903063	−0.451531	−	0.892255i	$-0.649122\pi$
$463$	−8996.83	−0.903063	−0.451531	+	0.892255i	$0.649122\pi$
$464$	0	0
$465$	4957.00	0.494356
$466$	0	0
$467$	12138.4	1.20278	0.601388	−	0.798957i	$-0.294614\pi$
$467$	12138.4	1.20278	0.601388	+	0.798957i	$0.294614\pi$
$468$	0	0
$469$	−20118.1	−1.98074
$470$	0	0
$471$	9757.09	0.954529
$472$	0	0
$473$	−21042.2	−2.04550
$474$	0	0
$475$	6221.14	0.600938
$476$	0	0
$477$	59.9993	0.00575928
$478$	0	0
$479$	1644.47	0.156864	0.0784320	−	0.996919i	$-0.475009\pi$
$479$	1644.47	0.156864	0.0784320	+	0.996919i	$0.475009\pi$
$480$	0	0
$481$	164.459	0.0155898
$482$	0	0
$483$	−16848.4	−1.58723
$484$	0	0
$485$	2275.23	0.213016
$486$	0	0
$487$	−1761.78	−0.163930	−0.0819648	−	0.996635i	$-0.526119\pi$
$487$	−1761.78	−0.163930	−0.0819648	+	0.996635i	$0.526119\pi$
$488$	0	0
$489$	19605.8	1.81310
$490$	0	0
$491$	2471.08	0.227125	0.113562	−	0.993531i	$-0.463774\pi$
$491$	2471.08	0.227125	0.113562	+	0.993531i	$0.463774\pi$
$492$	0	0
$493$	−1834.87	−0.167623
$494$	0	0
$495$	2032.28	0.184534
$496$	0	0
$497$	−2415.60	−0.218017
$498$	0	0
$499$	−16582.5	−1.48764	−0.743821	−	0.668379i	$-0.766988\pi$
$499$	−16582.5	−1.48764	−0.743821	+	0.668379i	$0.766988\pi$
$500$	0	0
$501$	−9065.74	−0.808438
$502$	0	0
$503$	19067.9	1.69025	0.845126	−	0.534566i	$-0.179525\pi$
$503$	19067.9	1.69025	0.845126	+	0.534566i	$0.179525\pi$
$504$	0	0
$505$	−4932.60	−0.434650
$506$	0	0
$507$	−12003.6	−1.05148
$508$	0	0
$509$	−9295.25	−0.809439	−0.404720	−	0.914441i	$-0.632631\pi$
$509$	−9295.25	−0.809439	−0.404720	+	0.914441i	$0.632631\pi$
$510$	0	0
$511$	1066.97	0.0923678
$512$	0	0
$513$	−19476.6	−1.67624
$514$	0	0
$515$	−9363.20	−0.801149
$516$	0	0
$517$	25086.7	2.13407
$518$	0	0
$519$	−776.713	−0.0656916
$520$	0	0
$521$	4907.38	0.412661	0.206330	−	0.978482i	$-0.433848\pi$
$521$	4907.38	0.412661	0.206330	+	0.978482i	$0.433848\pi$
$522$	0	0
$523$	−11247.5	−0.940381	−0.470191	−	0.882565i	$-0.655815\pi$
$523$	−11247.5	−0.940381	−0.470191	+	0.882565i	$0.655815\pi$
$524$	0	0
$525$	−6537.29	−0.543449
$526$	0	0
$527$	−1192.92	−0.0986042
$528$	0	0
$529$	−372.488	−0.0306146
$530$	0	0
$531$	−28.3148	−0.00231405
$532$	0	0
$533$	6.95435	0.000565152 0
$534$	0	0
$535$	−14637.7	−1.18289
$536$	0	0
$537$	2118.70	0.170258
$538$	0	0
$539$	−25487.2	−2.03676
$540$	0	0
$541$	−3992.37	−0.317274	−0.158637	−	0.987337i	$-0.550710\pi$
$541$	−3992.37	−0.317274	−0.158637	+	0.987337i	$0.550710\pi$
$542$	0	0
$543$	20127.8	1.59073
$544$	0	0
$545$	−7186.95	−0.564872
$546$	0	0
$547$	−3014.39	−0.235624	−0.117812	−	0.993036i	$-0.537588\pi$
$547$	−3014.39	−0.235624	−0.117812	+	0.993036i	$0.537588\pi$
$548$	0	0
$549$	−1941.67	−0.150944
$550$	0	0
$551$	−15934.8	−1.23203
$552$	0	0
$553$	−19457.0	−1.49619
$554$	0	0
$555$	26968.6	2.06262
$556$	0	0
$557$	−16174.5	−1.23041	−0.615203	−	0.788369i	$-0.710926\pi$
$557$	−16174.5	−1.23041	−0.615203	+	0.788369i	$0.710926\pi$
$558$	0	0
$559$	−164.711	−0.0124625
$560$	0	0
$561$	−5112.03	−0.384724
$562$	0	0
$563$	−6626.51	−0.496047	−0.248023	−	0.968754i	$-0.579781\pi$
$563$	−6626.51	−0.496047	−0.248023	+	0.968754i	$0.579781\pi$
$564$	0	0
$565$	14948.6	1.11308
$566$	0	0
$567$	22656.0	1.67807
$568$	0	0
$569$	2651.54	0.195357	0.0976786	−	0.995218i	$-0.468858\pi$
$569$	2651.54	0.195357	0.0976786	+	0.995218i	$0.468858\pi$
$570$	0	0
$571$	7663.21	0.561638	0.280819	−	0.959761i	$-0.409394\pi$
$571$	7663.21	0.561638	0.280819	+	0.959761i	$0.409394\pi$
$572$	0	0
$573$	−7147.32	−0.521088
$574$	0	0
$575$	4576.34	0.331907
$576$	0	0
$577$	−22136.0	−1.59711	−0.798557	−	0.601919i	$-0.794403\pi$
$577$	−22136.0	−1.59711	−0.798557	+	0.601919i	$0.794403\pi$
$578$	0	0
$579$	−19590.3	−1.40613
$580$	0	0
$581$	36764.8	2.62524
$582$	0	0
$583$	−1155.98	−0.0821200
$584$	0	0
$585$	15.9080	0.00112430
$586$	0	0
$587$	24994.8	1.75749	0.878744	−	0.477294i	$-0.158382\pi$
$587$	24994.8	1.75749	0.878744	+	0.477294i	$0.158382\pi$
$588$	0	0
$589$	−10359.9	−0.724738
$590$	0	0
$591$	−7836.22	−0.545413
$592$	0	0
$593$	11985.7	0.830007	0.415004	−	0.909820i	$-0.363780\pi$
$593$	11985.7	0.830007	0.415004	+	0.909820i	$0.363780\pi$
$594$	0	0
$595$	6240.05	0.429944
$596$	0	0
$597$	5565.44	0.381538
$598$	0	0
$599$	2943.68	0.200794	0.100397	−	0.994947i	$-0.467989\pi$
$599$	2943.68	0.200794	0.100397	+	0.994947i	$0.467989\pi$
$600$	0	0
$601$	−4398.52	−0.298535	−0.149267	−	0.988797i	$-0.547692\pi$
$601$	−4398.52	−0.298535	−0.149267	+	0.988797i	$0.547692\pi$
$602$	0	0
$603$	2023.98	0.136688
$604$	0	0
$605$	−21947.8	−1.47488
$606$	0	0
$607$	19859.6	1.32797	0.663985	−	0.747746i	$-0.268864\pi$
$607$	19859.6	1.32797	0.663985	+	0.747746i	$0.268864\pi$
$608$	0	0
$609$	16744.6	1.11416
$610$	0	0
$611$	196.370	0.0130021
$612$	0	0
$613$	5581.75	0.367773	0.183887	−	0.982947i	$-0.441132\pi$
$613$	5581.75	0.367773	0.183887	+	0.982947i	$0.441132\pi$
$614$	0	0
$615$	1140.40	0.0747729
$616$	0	0
$617$	−9888.18	−0.645192	−0.322596	−	0.946537i	$-0.604555\pi$
$617$	−9888.18	−0.645192	−0.322596	+	0.946537i	$0.604555\pi$
$618$	0	0
$619$	9547.62	0.619953	0.309977	−	0.950744i	$-0.399679\pi$
$619$	9547.62	0.619953	0.309977	+	0.950744i	$0.399679\pi$
$620$	0	0
$621$	−14327.2	−0.925813
$622$	0	0
$623$	44635.0	2.87041
$624$	0	0
$625$	−19116.7	−1.22347
$626$	0	0
$627$	−44395.2	−2.82771
$628$	0	0
$629$	−6490.08	−0.411409
$630$	0	0
$631$	−198.776	−0.0125406	−0.00627032	−	0.999980i	$-0.501996\pi$
$631$	−198.776	−0.0125406	−0.00627032	+	0.999980i	$0.501996\pi$
$632$	0	0
$633$	15782.9	0.991019
$634$	0	0
$635$	26777.1	1.67341
$636$	0	0
$637$	−199.504	−0.0124092
$638$	0	0
$639$	243.021	0.0150450
$640$	0	0
$641$	15596.6	0.961043	0.480522	−	0.876983i	$-0.340447\pi$
$641$	15596.6	0.961043	0.480522	+	0.876983i	$0.340447\pi$
$642$	0	0
$643$	18359.9	1.12604	0.563020	−	0.826443i	$-0.309639\pi$
$643$	18359.9	1.12604	0.563020	+	0.826443i	$0.309639\pi$
$644$	0	0
$645$	−27009.9	−1.64886
$646$	0	0
$647$	−23713.7	−1.44093	−0.720465	−	0.693491i	$-0.756072\pi$
$647$	−23713.7	−1.44093	−0.720465	+	0.693491i	$0.756072\pi$
$648$	0	0
$649$	545.531	0.0329953
$650$	0	0
$651$	10886.3	0.655406
$652$	0	0
$653$	22829.6	1.36814	0.684068	−	0.729418i	$-0.260209\pi$
$653$	22829.6	1.36814	0.684068	+	0.729418i	$0.260209\pi$
$654$	0	0
$655$	6447.16	0.384598
$656$	0	0
$657$	−107.342	−0.00637416
$658$	0	0
$659$	−16979.9	−1.00371	−0.501853	−	0.864953i	$-0.667348\pi$
$659$	−16979.9	−1.00371	−0.501853	+	0.864953i	$0.667348\pi$
$660$	0	0
$661$	−15036.2	−0.884782	−0.442391	−	0.896822i	$-0.645870\pi$
$661$	−15036.2	−0.884782	−0.442391	+	0.896822i	$0.645870\pi$
$662$	0	0
$663$	−40.0151	−0.00234398
$664$	0	0
$665$	54191.4	3.16008
$666$	0	0
$667$	−11721.8	−0.680467
$668$	0	0
$669$	19898.8	1.14997
$670$	0	0
$671$	37409.4	2.15227
$672$	0	0
$673$	−16935.7	−0.970022	−0.485011	−	0.874508i	$-0.661184\pi$
$673$	−16935.7	−0.970022	−0.485011	+	0.874508i	$0.661184\pi$
$674$	0	0
$675$	−5559.03	−0.316988
$676$	0	0
$677$	6514.79	0.369843	0.184921	−	0.982753i	$-0.440797\pi$
$677$	6514.79	0.369843	0.184921	+	0.982753i	$0.440797\pi$
$678$	0	0
$679$	4996.75	0.282412
$680$	0	0
$681$	28531.3	1.60547
$682$	0	0
$683$	−6261.11	−0.350768	−0.175384	−	0.984500i	$-0.556117\pi$
$683$	−6261.11	−0.350768	−0.175384	+	0.984500i	$0.556117\pi$
$684$	0	0
$685$	19400.8	1.08214
$686$	0	0
$687$	22876.3	1.27043
$688$	0	0
$689$	−9.04861	−0.000500326 0
$690$	0	0
$691$	12775.3	0.703320	0.351660	−	0.936128i	$-0.385617\pi$
$691$	12775.3	0.703320	0.351660	+	0.936128i	$0.385617\pi$
$692$	0	0
$693$	4463.20	0.244651
$694$	0	0
$695$	3517.03	0.191955
$696$	0	0
$697$	−274.441	−0.0149142
$698$	0	0
$699$	−18687.7	−1.01121
$700$	0	0
$701$	−5939.18	−0.320000	−0.160000	−	0.987117i	$-0.551149\pi$
$701$	−5939.18	−0.320000	−0.160000	+	0.987117i	$0.551149\pi$
$702$	0	0
$703$	−56362.8	−3.02385
$704$	0	0
$705$	32201.4	1.72025
$706$	0	0
$707$	−10832.8	−0.576248
$708$	0	0
$709$	−22029.4	−1.16690	−0.583450	−	0.812149i	$-0.698298\pi$
$709$	−22029.4	−1.16690	−0.583450	+	0.812149i	$0.698298\pi$
$710$	0	0
$711$	1957.47	0.103250
$712$	0	0
$713$	−7620.83	−0.400284
$714$	0	0
$715$	−306.493	−0.0160310
$716$	0	0
$717$	−13421.7	−0.699085
$718$	0	0
$719$	4477.22	0.232228	0.116114	−	0.993236i	$-0.462956\pi$
$719$	4477.22	0.232228	0.116114	+	0.993236i	$0.462956\pi$
$720$	0	0
$721$	−20563.0	−1.06215
$722$	0	0
$723$	16213.0	0.833982
$724$	0	0
$725$	−4548.14	−0.232985
$726$	0	0
$727$	−7878.08	−0.401901	−0.200950	−	0.979601i	$-0.564403\pi$
$727$	−7878.08	−0.401901	−0.200950	+	0.979601i	$0.564403\pi$
$728$	0	0
$729$	17183.4	0.873007
$730$	0	0
$731$	6500.02	0.328881
$732$	0	0
$733$	11332.8	0.571060	0.285530	−	0.958370i	$-0.407830\pi$
$733$	11332.8	0.571060	0.285530	+	0.958370i	$0.407830\pi$
$734$	0	0
$735$	−32715.5	−1.64181
$736$	0	0
$737$	−38995.2	−1.94899
$738$	0	0
$739$	20027.3	0.996908	0.498454	−	0.866916i	$-0.333901\pi$
$739$	20027.3	0.996908	0.498454	+	0.866916i	$0.333901\pi$
$740$	0	0
$741$	−347.510	−0.0172282
$742$	0	0
$743$	−8839.83	−0.436477	−0.218238	−	0.975896i	$-0.570031\pi$
$743$	−8839.83	−0.436477	−0.218238	+	0.975896i	$0.570031\pi$
$744$	0	0
$745$	1581.75	0.0777865
$746$	0	0
$747$	−3698.72	−0.181164
$748$	0	0
$749$	−32146.7	−1.56824
$750$	0	0
$751$	1533.58	0.0745156	0.0372578	−	0.999306i	$-0.488138\pi$
$751$	1533.58	0.0745156	0.0372578	+	0.999306i	$0.488138\pi$
$752$	0	0
$753$	−30883.5	−1.49463
$754$	0	0
$755$	9100.73	0.438688
$756$	0	0
$757$	16470.7	0.790801	0.395401	−	0.918509i	$-0.370606\pi$
$757$	16470.7	0.790801	0.395401	+	0.918509i	$0.370606\pi$
$758$	0	0
$759$	−32657.6	−1.56179
$760$	0	0
$761$	−18629.7	−0.887418	−0.443709	−	0.896171i	$-0.646338\pi$
$761$	−18629.7	−0.887418	−0.443709	+	0.896171i	$0.646338\pi$
$762$	0	0
$763$	−15783.6	−0.748894
$764$	0	0
$765$	−627.779	−0.0296698
$766$	0	0
$767$	4.27022	0.000201028 0
$768$	0	0
$769$	−6086.38	−0.285410	−0.142705	−	0.989765i	$-0.545580\pi$
$769$	−6086.38	−0.285410	−0.142705	+	0.989765i	$0.545580\pi$
$770$	0	0
$771$	−8091.57	−0.377965
$772$	0	0
$773$	7472.68	0.347702	0.173851	−	0.984772i	$-0.444379\pi$
$773$	7472.68	0.347702	0.173851	+	0.984772i	$0.444379\pi$
$774$	0	0
$775$	−2956.93	−0.137053
$776$	0	0
$777$	59227.1	2.73457
$778$	0	0
$779$	−2383.37	−0.109619
$780$	0	0
$781$	−4682.20	−0.214523
$782$	0	0
$783$	14238.9	0.649881
$784$	0	0
$785$	−23085.5	−1.04963
$786$	0	0
$787$	28318.2	1.28263	0.641317	−	0.767276i	$-0.278388\pi$
$787$	28318.2	1.28263	0.641317	+	0.767276i	$0.278388\pi$
$788$	0	0
$789$	−7981.60	−0.360142
$790$	0	0
$791$	32829.4	1.47570
$792$	0	0
$793$	292.827	0.0131130
$794$	0	0
$795$	−1483.83	−0.0661961
$796$	0	0
$797$	29551.4	1.31338	0.656691	−	0.754160i	$-0.271956\pi$
$797$	29551.4	1.31338	0.656691	+	0.754160i	$0.271956\pi$
$798$	0	0
$799$	−7749.38	−0.343121
$800$	0	0
$801$	−4490.50	−0.198083
$802$	0	0
$803$	2068.13	0.0908874
$804$	0	0
$805$	39863.8	1.74536
$806$	0	0
$807$	−34209.3	−1.49222
$808$	0	0
$809$	28511.0	1.23905	0.619527	−	0.784975i	$-0.287324\pi$
$809$	28511.0	1.23905	0.619527	+	0.784975i	$0.287324\pi$
$810$	0	0
$811$	−29641.1	−1.28340	−0.641700	−	0.766955i	$-0.721771\pi$
$811$	−29641.1	−1.28340	−0.641700	+	0.766955i	$0.721771\pi$
$812$	0	0
$813$	−1472.78	−0.0635334
$814$	0	0
$815$	−46387.8	−1.99373
$816$	0	0
$817$	56449.1	2.41726
$818$	0	0
$819$	34.9363	0.00149057
$820$	0	0
$821$	8825.39	0.375162	0.187581	−	0.982249i	$-0.439935\pi$
$821$	8825.39	0.375162	0.187581	+	0.982249i	$0.439935\pi$
$822$	0	0
$823$	−19790.8	−0.838233	−0.419116	−	0.907933i	$-0.637660\pi$
$823$	−19790.8	−0.838233	−0.419116	+	0.907933i	$0.637660\pi$
$824$	0	0
$825$	−12671.4	−0.534739
$826$	0	0
$827$	38948.7	1.63770	0.818850	−	0.574008i	$-0.194612\pi$
$827$	38948.7	1.63770	0.818850	+	0.574008i	$0.194612\pi$
$828$	0	0
$829$	34825.0	1.45901	0.729506	−	0.683974i	$-0.239750\pi$
$829$	34825.0	1.45901	0.729506	+	0.683974i	$0.239750\pi$
$830$	0	0
$831$	−27853.6	−1.16273
$832$	0	0
$833$	7873.09	0.327475
$834$	0	0
$835$	21449.8	0.888982
$836$	0	0
$837$	9257.27	0.382292
$838$	0	0
$839$	−17082.9	−0.702941	−0.351470	−	0.936199i	$-0.614318\pi$
$839$	−17082.9	−0.702941	−0.351470	+	0.936199i	$0.614318\pi$
$840$	0	0
$841$	−12739.4	−0.522342
$842$	0	0
$843$	−8284.31	−0.338466
$844$	0	0
$845$	28400.9	1.15624
$846$	0	0
$847$	−48200.7	−1.95537
$848$	0	0
$849$	−41903.5	−1.69391
$850$	0	0
$851$	−41461.1	−1.67012
$852$	0	0
$853$	−38982.3	−1.56475	−0.782373	−	0.622810i	$-0.785991\pi$
$853$	−38982.3	−1.56475	−0.782373	+	0.622810i	$0.785991\pi$
$854$	0	0
$855$	−5451.93	−0.218072
$856$	0	0
$857$	−6648.98	−0.265023	−0.132512	−	0.991181i	$-0.542304\pi$
$857$	−6648.98	−0.265023	−0.132512	+	0.991181i	$0.542304\pi$
$858$	0	0
$859$	−1644.93	−0.0653368	−0.0326684	−	0.999466i	$-0.510401\pi$
$859$	−1644.93	−0.0653368	−0.0326684	+	0.999466i	$0.510401\pi$
$860$	0	0
$861$	2504.49	0.0991322
$862$	0	0
$863$	24272.3	0.957402	0.478701	−	0.877978i	$-0.341108\pi$
$863$	24272.3	0.957402	0.478701	+	0.877978i	$0.341108\pi$
$864$	0	0
$865$	1837.72	0.0722364
$866$	0	0
$867$	1579.13	0.0618569
$868$	0	0
$869$	−37713.8	−1.47221
$870$	0	0
$871$	−305.240	−0.0118745
$872$	0	0
$873$	−502.698	−0.0194888
$874$	0	0
$875$	−30415.3	−1.17511
$876$	0	0
$877$	−51065.8	−1.96621	−0.983107	−	0.183030i	$-0.941410\pi$
$877$	−51065.8	−1.96621	−0.983107	+	0.183030i	$0.941410\pi$
$878$	0	0
$879$	14100.6	0.541070
$880$	0	0
$881$	−10283.5	−0.393258	−0.196629	−	0.980478i	$-0.563000\pi$
$881$	−10283.5	−0.393258	−0.196629	+	0.980478i	$0.563000\pi$
$882$	0	0
$883$	−5412.08	−0.206264	−0.103132	−	0.994668i	$-0.532886\pi$
$883$	−5412.08	−0.206264	−0.103132	+	0.994668i	$0.532886\pi$
$884$	0	0
$885$	700.246	0.0265972
$886$	0	0
$887$	11306.0	0.427981	0.213991	−	0.976836i	$-0.431354\pi$
$887$	11306.0	0.427981	0.213991	+	0.976836i	$0.431354\pi$
$888$	0	0
$889$	58806.6	2.21857
$890$	0	0
$891$	43914.6	1.65117
$892$	0	0
$893$	−67299.2	−2.52193
$894$	0	0
$895$	−5012.89	−0.187221
$896$	0	0
$897$	−255.632	−0.00951538
$898$	0	0
$899$	7573.87	0.280982
$900$	0	0
$901$	357.088	0.0132035
$902$	0	0
$903$	−59317.8	−2.18602
$904$	0	0
$905$	−47622.8	−1.74921
$906$	0	0
$907$	13122.6	0.480408	0.240204	−	0.970722i	$-0.422786\pi$
$907$	13122.6	0.480408	0.240204	+	0.970722i	$0.422786\pi$
$908$	0	0
$909$	1089.83	0.0397660
$910$	0	0
$911$	−24127.8	−0.877485	−0.438743	−	0.898613i	$-0.644576\pi$
$911$	−24127.8	−0.877485	−0.438743	+	0.898613i	$0.644576\pi$
$912$	0	0
$913$	71261.9	2.58316
$914$	0	0
$915$	48018.9	1.73492
$916$	0	0
$917$	14159.0	0.509891
$918$	0	0
$919$	−10597.4	−0.380387	−0.190194	−	0.981747i	$-0.560912\pi$
$919$	−10597.4	−0.380387	−0.190194	+	0.981747i	$0.560912\pi$
$920$	0	0
$921$	−3188.34	−0.114071
$922$	0	0
$923$	−36.6505	−0.00130701
$924$	0	0
$925$	−16087.2	−0.571830
$926$	0	0
$927$	2068.74	0.0732970
$928$	0	0
$929$	16159.9	0.570711	0.285356	−	0.958422i	$-0.407888\pi$
$929$	16159.9	0.570711	0.285356	+	0.958422i	$0.407888\pi$
$930$	0	0
$931$	68373.6	2.40693
$932$	0	0
$933$	−846.788	−0.0297134
$934$	0	0
$935$	12095.2	0.423054
$936$	0	0
$937$	12857.2	0.448267	0.224134	−	0.974558i	$-0.428045\pi$
$937$	12857.2	0.448267	0.224134	+	0.974558i	$0.428045\pi$
$938$	0	0
$939$	−31887.7	−1.10822
$940$	0	0
$941$	−31043.6	−1.07544	−0.537722	−	0.843122i	$-0.680715\pi$
$941$	−31043.6	−1.07544	−0.537722	+	0.843122i	$0.680715\pi$
$942$	0	0
$943$	−1753.24	−0.0605442
$944$	0	0
$945$	−48423.9	−1.66691
$946$	0	0
$947$	−39841.0	−1.36712	−0.683558	−	0.729896i	$-0.739568\pi$
$947$	−39841.0	−1.36712	−0.683558	+	0.729896i	$0.739568\pi$
$948$	0	0
$949$	16.1885	0.000553743 0
$950$	0	0
$951$	57843.7	1.97235
$952$	0	0
$953$	−26601.9	−0.904217	−0.452109	−	0.891963i	$-0.649328\pi$
$953$	−26601.9	−0.904217	−0.452109	+	0.891963i	$0.649328\pi$
$954$	0	0
$955$	16910.7	0.573004
$956$	0	0
$957$	32456.4	1.09631
$958$	0	0
$959$	42607.1	1.43468
$960$	0	0
$961$	−24866.9	−0.834713
$962$	0	0
$963$	3234.12	0.108222
$964$	0	0
$965$	46351.2	1.54622
$966$	0	0
$967$	15998.2	0.532024	0.266012	−	0.963970i	$-0.414294\pi$
$967$	15998.2	0.532024	0.266012	+	0.963970i	$0.414294\pi$
$968$	0	0
$969$	13713.9	0.454647
$970$	0	0
$971$	−21950.4	−0.725461	−0.362731	−	0.931894i	$-0.618155\pi$
$971$	−21950.4	−0.725461	−0.362731	+	0.931894i	$0.618155\pi$
$972$	0	0
$973$	7723.94	0.254489
$974$	0	0
$975$	−99.1867	−0.00325797
$976$	0	0
$977$	−24955.7	−0.817198	−0.408599	−	0.912714i	$-0.633983\pi$
$977$	−24955.7	−0.817198	−0.408599	+	0.912714i	$0.633983\pi$
$978$	0	0
$979$	86516.9	2.82440
$980$	0	0
$981$	1587.91	0.0516801
$982$	0	0
$983$	35207.3	1.14236	0.571180	−	0.820825i	$-0.306486\pi$
$983$	35207.3	1.14236	0.571180	+	0.820825i	$0.306486\pi$
$984$	0	0
$985$	18540.7	0.599752
$986$	0	0
$987$	70719.3	2.28067
$988$	0	0
$989$	41524.6	1.33509
$990$	0	0
$991$	47633.5	1.52687	0.763435	−	0.645885i	$-0.223511\pi$
$991$	47633.5	1.52687	0.763435	+	0.645885i	$0.223511\pi$
$992$	0	0
$993$	−38764.4	−1.23882
$994$	0	0
$995$	−13168.0	−0.419550
$996$	0	0
$997$	6114.40	0.194228	0.0971139	−	0.995273i	$-0.469039\pi$
$997$	6114.40	0.194228	0.0971139	+	0.995273i	$0.469039\pi$
$998$	0	0
$999$	50364.1	1.59505

Twists

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

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

\(f(q)\)	\(=\)	\(q+5.46410 q^{3} -12.9282 q^{5} -28.3923 q^{7} +2.85641 q^{9} +O(q^{10})\)	\(q+5.46410 q^{3} -12.9282 q^{5} -28.3923 q^{7} +2.85641 q^{9} -55.0333 q^{11} -0.430781 q^{13} -70.6410 q^{15} +17.0000 q^{17} +147.636 q^{19} -155.138 q^{21} +108.603 q^{23} +42.1384 q^{25} -131.923 q^{27} -107.933 q^{29} -70.1718 q^{31} -300.708 q^{33} +367.061 q^{35} -381.769 q^{37} -2.35383 q^{39} -16.1436 q^{41} +382.354 q^{43} -36.9282 q^{45} -455.846 q^{47} +463.123 q^{49} +92.8897 q^{51} +21.0052 q^{53} +711.482 q^{55} +806.697 q^{57} -9.91274 q^{59} -679.759 q^{61} -81.1000 q^{63} +5.56922 q^{65} +708.574 q^{67} +593.415 q^{69} +85.0793 q^{71} -37.5795 q^{73} +230.249 q^{75} +1562.52 q^{77} +685.290 q^{79} -797.964 q^{81} -1294.89 q^{83} -219.779 q^{85} -589.759 q^{87} -1572.08 q^{89} +12.2309 q^{91} -383.426 q^{93} -1908.67 q^{95} -175.990 q^{97} -157.198 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9}+O(q^{10}) \) 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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