LMFDB - Embedded newform 136.4.a.d.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.550476.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 15x^{2} + 19x + 20 $ x^4 - x^3 - 15x^2 + 19x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.30977$ 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)

$f(q)$	$=$	$q+7.43996 q^{3} +15.1251 q^{5} +5.90519 q^{7} +28.3531 q^{9} +O(q^{10})$	$q+7.43996 q^{3} +15.1251 q^{5} +5.90519 q^{7} +28.3531 q^{9} -15.8676 q^{11} -46.4205 q^{13} +112.530 q^{15} -17.0000 q^{17} +53.2914 q^{19} +43.9344 q^{21} -200.431 q^{23} +103.769 q^{25} +10.0667 q^{27} +273.100 q^{29} +103.505 q^{31} -118.054 q^{33} +89.3166 q^{35} +259.333 q^{37} -345.367 q^{39} -21.9929 q^{41} -217.364 q^{43} +428.843 q^{45} -425.334 q^{47} -308.129 q^{49} -126.479 q^{51} +456.461 q^{53} -239.999 q^{55} +396.486 q^{57} -312.570 q^{59} -134.202 q^{61} +167.430 q^{63} -702.114 q^{65} -897.083 q^{67} -1491.20 q^{69} -29.6329 q^{71} -260.307 q^{73} +772.034 q^{75} -93.7011 q^{77} +857.241 q^{79} -690.637 q^{81} -606.355 q^{83} -257.127 q^{85} +2031.85 q^{87} +102.538 q^{89} -274.122 q^{91} +770.070 q^{93} +806.038 q^{95} -881.343 q^{97} -449.895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 8 q^{5} - 22 q^{7} + 100 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 8 * q^5 - 22 * q^7 + 100 * q^9	$ 4 q - 2 q^{3} + 8 q^{5} - 22 q^{7} + 100 q^{9} + 70 q^{11} + 120 q^{13} + 140 q^{15} - 68 q^{17} - 44 q^{19} + 488 q^{21} + 158 q^{23} + 548 q^{25} - 392 q^{27} + 264 q^{29} + 122 q^{31} + 136 q^{33} - 44 q^{35} + 256 q^{37} - 528 q^{39} + 240 q^{41} - 1100 q^{43} - 880 q^{45} - 800 q^{47} + 12 q^{49} + 34 q^{51} + 432 q^{53} - 532 q^{55} + 472 q^{57} - 148 q^{59} - 728 q^{61} - 1450 q^{63} - 72 q^{65} - 1032 q^{67} - 1024 q^{69} + 798 q^{71} - 1544 q^{73} - 2974 q^{75} + 656 q^{77} + 758 q^{79} - 20 q^{81} + 244 q^{83} - 136 q^{85} + 1524 q^{87} + 1440 q^{89} - 1104 q^{91} - 2464 q^{93} + 7016 q^{95} - 1344 q^{97} - 86 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 8 * q^5 - 22 * q^7 + 100 * q^9 + 70 * q^11 + 120 * q^13 + 140 * q^15 - 68 * q^17 - 44 * q^19 + 488 * q^21 + 158 * q^23 + 548 * q^25 - 392 * q^27 + 264 * q^29 + 122 * q^31 + 136 * q^33 - 44 * q^35 + 256 * q^37 - 528 * q^39 + 240 * q^41 - 1100 * q^43 - 880 * q^45 - 800 * q^47 + 12 * q^49 + 34 * q^51 + 432 * q^53 - 532 * q^55 + 472 * q^57 - 148 * q^59 - 728 * q^61 - 1450 * q^63 - 72 * q^65 - 1032 * q^67 - 1024 * q^69 + 798 * q^71 - 1544 * q^73 - 2974 * q^75 + 656 * q^77 + 758 * q^79 - 20 * q^81 + 244 * q^83 - 136 * q^85 + 1524 * q^87 + 1440 * q^89 - 1104 * q^91 - 2464 * q^93 + 7016 * q^95 - 1344 * q^97 - 86 * 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
$3$	7.43996	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$3$	7.43996	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$4$	0	0
$5$	15.1251	1.35283	0.676415	−	0.736521i	$-0.263533\pi$
$5$	15.1251	1.35283	0.676415	+	0.736521i	$0.263533\pi$
$6$	0	0
$7$	5.90519	0.318850	0.159425	−	0.987210i	$-0.449036\pi$
$7$	5.90519	0.318850	0.159425	+	0.987210i	$0.449036\pi$
$8$	0	0
$9$	28.3531	1.05011
$10$	0	0
$11$	−15.8676	−0.434933	−0.217466	−	0.976068i	$-0.569779\pi$
$11$	−15.8676	−0.434933	−0.217466	+	0.976068i	$0.569779\pi$
$12$	0	0
$13$	−46.4205	−0.990363	−0.495182	−	0.868790i	$-0.664898\pi$
$13$	−46.4205	−0.990363	−0.495182	+	0.868790i	$0.664898\pi$
$14$	0	0
$15$	112.530	1.93701
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	53.2914	0.643468	0.321734	−	0.946830i	$-0.395734\pi$
$19$	53.2914	0.643468	0.321734	+	0.946830i	$0.395734\pi$
$20$	0	0
$21$	43.9344	0.456537
$22$	0	0
$23$	−200.431	−1.81707	−0.908537	−	0.417804i	$-0.862800\pi$
$23$	−200.431	−1.81707	−0.908537	+	0.417804i	$0.862800\pi$
$24$	0	0
$25$	103.769	0.830148
$26$	0	0
$27$	10.0667	0.0717532
$28$	0	0
$29$	273.100	1.74874	0.874369	−	0.485261i	$-0.161276\pi$
$29$	273.100	1.74874	0.874369	+	0.485261i	$0.161276\pi$
$30$	0	0
$31$	103.505	0.599677	0.299838	−	0.953990i	$-0.403067\pi$
$31$	103.505	0.599677	0.299838	+	0.953990i	$0.403067\pi$
$32$	0	0
$33$	−118.054	−0.622746
$34$	0	0
$35$	89.3166	0.431350
$36$	0	0
$37$	259.333	1.15227	0.576136	−	0.817354i	$-0.304560\pi$
$37$	259.333	1.15227	0.576136	+	0.817354i	$0.304560\pi$
$38$	0	0
$39$	−345.367	−1.41802
$40$	0	0
$41$	−21.9929	−0.0837736	−0.0418868	−	0.999122i	$-0.513337\pi$
$41$	−21.9929	−0.0837736	−0.0418868	+	0.999122i	$0.513337\pi$
$42$	0	0
$43$	−217.364	−0.770876	−0.385438	−	0.922734i	$-0.625950\pi$
$43$	−217.364	−0.770876	−0.385438	+	0.922734i	$0.625950\pi$
$44$	0	0
$45$	428.843	1.42062
$46$	0	0
$47$	−425.334	−1.32003	−0.660014	−	0.751253i	$-0.729450\pi$
$47$	−425.334	−1.32003	−0.660014	+	0.751253i	$0.729450\pi$
$48$	0	0
$49$	−308.129	−0.898334
$50$	0	0
$51$	−126.479	−0.347268
$52$	0	0
$53$	456.461	1.18301	0.591507	−	0.806300i	$-0.298533\pi$
$53$	456.461	1.18301	0.591507	+	0.806300i	$0.298533\pi$
$54$	0	0
$55$	−239.999	−0.588390
$56$	0	0
$57$	396.486	0.921332
$58$	0	0
$59$	−312.570	−0.689715	−0.344857	−	0.938655i	$-0.612073\pi$
$59$	−312.570	−0.689715	−0.344857	+	0.938655i	$0.612073\pi$
$60$	0	0
$61$	−134.202	−0.281685	−0.140842	−	0.990032i	$-0.544981\pi$
$61$	−134.202	−0.281685	−0.140842	+	0.990032i	$0.544981\pi$
$62$	0	0
$63$	167.430	0.334829
$64$	0	0
$65$	−702.114	−1.33979
$66$	0	0
$67$	−897.083	−1.63576	−0.817881	−	0.575387i	$-0.804851\pi$
$67$	−897.083	−1.63576	−0.817881	+	0.575387i	$0.804851\pi$
$68$	0	0
$69$	−1491.20	−2.60173
$70$	0	0
$71$	−29.6329	−0.0495320	−0.0247660	−	0.999693i	$-0.507884\pi$
$71$	−29.6329	−0.0495320	−0.0247660	+	0.999693i	$0.507884\pi$
$72$	0	0
$73$	−260.307	−0.417351	−0.208675	−	0.977985i	$-0.566915\pi$
$73$	−260.307	−0.417351	−0.208675	+	0.977985i	$0.566915\pi$
$74$	0	0
$75$	772.034	1.18862
$76$	0	0
$77$	−93.7011	−0.138678
$78$	0	0
$79$	857.241	1.22085	0.610425	−	0.792074i	$-0.290999\pi$
$79$	857.241	1.22085	0.610425	+	0.792074i	$0.290999\pi$
$80$	0	0
$81$	−690.637	−0.947375
$82$	0	0
$83$	−606.355	−0.801880	−0.400940	−	0.916104i	$-0.631316\pi$
$83$	−606.355	−0.801880	−0.400940	+	0.916104i	$0.631316\pi$
$84$	0	0
$85$	−257.127	−0.328109
$86$	0	0
$87$	2031.85	2.50388
$88$	0	0
$89$	102.538	0.122124	0.0610620	−	0.998134i	$-0.480551\pi$
$89$	102.538	0.122124	0.0610620	+	0.998134i	$0.480551\pi$
$90$	0	0
$91$	−274.122	−0.315778
$92$	0	0
$93$	770.070	0.858630
$94$	0	0
$95$	806.038	0.870503
$96$	0	0
$97$	−881.343	−0.922545	−0.461273	−	0.887258i	$-0.652607\pi$
$97$	−881.343	−0.922545	−0.461273	+	0.887258i	$0.652607\pi$
$98$	0	0
$99$	−449.895	−0.456728
$100$	0	0
$101$	1688.07	1.66307	0.831533	−	0.555475i	$-0.187463\pi$
$101$	1688.07	1.66307	0.831533	+	0.555475i	$0.187463\pi$
$102$	0	0
$103$	876.474	0.838461	0.419231	−	0.907880i	$-0.362300\pi$
$103$	876.474	0.838461	0.419231	+	0.907880i	$0.362300\pi$
$104$	0	0
$105$	664.512	0.617617
$106$	0	0
$107$	880.796	0.795792	0.397896	−	0.917431i	$-0.369741\pi$
$107$	880.796	0.795792	0.397896	+	0.917431i	$0.369741\pi$
$108$	0	0
$109$	1028.91	0.904142	0.452071	−	0.891982i	$-0.350685\pi$
$109$	1028.91	0.904142	0.452071	+	0.891982i	$0.350685\pi$
$110$	0	0
$111$	1929.43	1.64985
$112$	0	0
$113$	2104.89	1.75231	0.876156	−	0.482028i	$-0.160099\pi$
$113$	2104.89	1.75231	0.876156	+	0.482028i	$0.160099\pi$
$114$	0	0
$115$	−3031.53	−2.45819
$116$	0	0
$117$	−1316.16	−1.03999
$118$	0	0
$119$	−100.388	−0.0773326
$120$	0	0
$121$	−1079.22	−0.810834
$122$	0	0
$123$	−163.626	−0.119949
$124$	0	0
$125$	−321.128	−0.229781
$126$	0	0
$127$	−1593.41	−1.11333	−0.556663	−	0.830738i	$-0.687919\pi$
$127$	−1593.41	−1.11333	−0.556663	+	0.830738i	$0.687919\pi$
$128$	0	0
$129$	−1617.18	−1.10376
$130$	0	0
$131$	2613.49	1.74306	0.871532	−	0.490339i	$-0.163127\pi$
$131$	2613.49	1.74306	0.871532	+	0.490339i	$0.163127\pi$
$132$	0	0
$133$	314.696	0.205170
$134$	0	0
$135$	152.260	0.0970698
$136$	0	0
$137$	2417.12	1.50736	0.753680	−	0.657242i	$-0.228277\pi$
$137$	2417.12	1.50736	0.753680	+	0.657242i	$0.228277\pi$
$138$	0	0
$139$	−1235.20	−0.753727	−0.376864	−	0.926269i	$-0.622997\pi$
$139$	−1235.20	−0.753727	−0.376864	+	0.926269i	$0.622997\pi$
$140$	0	0
$141$	−3164.47	−1.89005
$142$	0	0
$143$	736.581	0.430741
$144$	0	0
$145$	4130.66	2.36574
$146$	0	0
$147$	−2292.47	−1.28625
$148$	0	0
$149$	−2900.22	−1.59460	−0.797300	−	0.603583i	$-0.793739\pi$
$149$	−2900.22	−1.59460	−0.797300	+	0.603583i	$0.793739\pi$
$150$	0	0
$151$	228.597	0.123198	0.0615991	−	0.998101i	$-0.480380\pi$
$151$	228.597	0.123198	0.0615991	+	0.998101i	$0.480380\pi$
$152$	0	0
$153$	−482.002	−0.254690
$154$	0	0
$155$	1565.52	0.811260
$156$	0	0
$157$	−1644.67	−0.836046	−0.418023	−	0.908437i	$-0.637277\pi$
$157$	−1644.67	−0.836046	−0.418023	+	0.908437i	$0.637277\pi$
$158$	0	0
$159$	3396.05	1.69387
$160$	0	0
$161$	−1183.58	−0.579375
$162$	0	0
$163$	1668.91	0.801959	0.400979	−	0.916087i	$-0.368670\pi$
$163$	1668.91	0.801959	0.400979	+	0.916087i	$0.368670\pi$
$164$	0	0
$165$	−1785.58	−0.842469
$166$	0	0
$167$	910.745	0.422009	0.211005	−	0.977485i	$-0.432326\pi$
$167$	910.745	0.422009	0.211005	+	0.977485i	$0.432326\pi$
$168$	0	0
$169$	−42.1407	−0.0191810
$170$	0	0
$171$	1510.98	0.675714
$172$	0	0
$173$	4316.99	1.89720	0.948598	−	0.316482i	$-0.102502\pi$
$173$	4316.99	1.89720	0.948598	+	0.316482i	$0.102502\pi$
$174$	0	0
$175$	612.773	0.264693
$176$	0	0
$177$	−2325.51	−0.987548
$178$	0	0
$179$	−2758.10	−1.15168	−0.575838	−	0.817564i	$-0.695324\pi$
$179$	−2758.10	−1.15168	−0.575838	+	0.817564i	$0.695324\pi$
$180$	0	0
$181$	607.939	0.249656	0.124828	−	0.992178i	$-0.460162\pi$
$181$	607.939	0.249656	0.124828	+	0.992178i	$0.460162\pi$
$182$	0	0
$183$	−998.455	−0.403322
$184$	0	0
$185$	3922.44	1.55883
$186$	0	0
$187$	269.749	0.105487
$188$	0	0
$189$	59.4457	0.0228785
$190$	0	0
$191$	1510.35	0.572172	0.286086	−	0.958204i	$-0.407646\pi$
$191$	1510.35	0.572172	0.286086	+	0.958204i	$0.407646\pi$
$192$	0	0
$193$	1253.98	0.467686	0.233843	−	0.972274i	$-0.424870\pi$
$193$	1253.98	0.467686	0.233843	+	0.972274i	$0.424870\pi$
$194$	0	0
$195$	−5223.70	−1.91834
$196$	0	0
$197$	−5096.38	−1.84316	−0.921578	−	0.388192i	$-0.873100\pi$
$197$	−5096.38	−1.84316	−0.921578	+	0.388192i	$0.873100\pi$
$198$	0	0
$199$	−4554.81	−1.62252	−0.811262	−	0.584683i	$-0.801219\pi$
$199$	−4554.81	−1.62252	−0.811262	+	0.584683i	$0.801219\pi$
$200$	0	0
$201$	−6674.26	−2.34212
$202$	0	0
$203$	1612.71	0.557586
$204$	0	0
$205$	−332.645	−0.113331
$206$	0	0
$207$	−5682.82	−1.90813
$208$	0	0
$209$	−845.607	−0.279865
$210$	0	0
$211$	−1172.10	−0.382420	−0.191210	−	0.981549i	$-0.561241\pi$
$211$	−1172.10	−0.382420	−0.191210	+	0.981549i	$0.561241\pi$
$212$	0	0
$213$	−220.467	−0.0709210
$214$	0	0
$215$	−3287.65	−1.04286
$216$	0	0
$217$	611.214	0.191207
$218$	0	0
$219$	−1936.67	−0.597571
$220$	0	0
$221$	789.148	0.240198
$222$	0	0
$223$	3518.81	1.05667	0.528334	−	0.849037i	$-0.322817\pi$
$223$	3518.81	1.05667	0.528334	+	0.849037i	$0.322817\pi$
$224$	0	0
$225$	2942.15	0.871750
$226$	0	0
$227$	5494.07	1.60640	0.803202	−	0.595706i	$-0.203128\pi$
$227$	5494.07	1.60640	0.803202	+	0.595706i	$0.203128\pi$
$228$	0	0
$229$	1244.01	0.358979	0.179490	−	0.983760i	$-0.442555\pi$
$229$	1244.01	0.358979	0.179490	+	0.983760i	$0.442555\pi$
$230$	0	0
$231$	−697.133	−0.198563
$232$	0	0
$233$	−288.643	−0.0811572	−0.0405786	−	0.999176i	$-0.512920\pi$
$233$	−288.643	−0.0811572	−0.0405786	+	0.999176i	$0.512920\pi$
$234$	0	0
$235$	−6433.22	−1.78577
$236$	0	0
$237$	6377.84	1.74804
$238$	0	0
$239$	6795.34	1.83914	0.919569	−	0.392928i	$-0.128538\pi$
$239$	6795.34	1.83914	0.919569	+	0.392928i	$0.128538\pi$
$240$	0	0
$241$	5750.98	1.53715	0.768575	−	0.639759i	$-0.220966\pi$
$241$	5750.98	1.53715	0.768575	+	0.639759i	$0.220966\pi$
$242$	0	0
$243$	−5410.11	−1.42823
$244$	0	0
$245$	−4660.48	−1.21529
$246$	0	0
$247$	−2473.81	−0.637267
$248$	0	0
$249$	−4511.26	−1.14815
$250$	0	0
$251$	1231.52	0.309692	0.154846	−	0.987939i	$-0.450512\pi$
$251$	1231.52	0.309692	0.154846	+	0.987939i	$0.450512\pi$
$252$	0	0
$253$	3180.35	0.790305
$254$	0	0
$255$	−1913.01	−0.469794
$256$	0	0
$257$	703.053	0.170643	0.0853215	−	0.996353i	$-0.472808\pi$
$257$	703.053	0.170643	0.0853215	+	0.996353i	$0.472808\pi$
$258$	0	0
$259$	1531.41	0.367402
$260$	0	0
$261$	7743.22	1.83637
$262$	0	0
$263$	202.414	0.0474577	0.0237289	−	0.999718i	$-0.492446\pi$
$263$	202.414	0.0474577	0.0237289	+	0.999718i	$0.492446\pi$
$264$	0	0
$265$	6904.02	1.60042
$266$	0	0
$267$	762.881	0.174860
$268$	0	0
$269$	2524.94	0.572297	0.286149	−	0.958185i	$-0.407625\pi$
$269$	2524.94	0.572297	0.286149	+	0.958185i	$0.407625\pi$
$270$	0	0
$271$	7188.99	1.61144	0.805719	−	0.592298i	$-0.201779\pi$
$271$	7188.99	1.61144	0.805719	+	0.592298i	$0.201779\pi$
$272$	0	0
$273$	−2039.46	−0.452137
$274$	0	0
$275$	−1646.56	−0.361058
$276$	0	0
$277$	−5723.98	−1.24159	−0.620795	−	0.783973i	$-0.713190\pi$
$277$	−5723.98	−1.24159	−0.620795	+	0.783973i	$0.713190\pi$
$278$	0	0
$279$	2934.67	0.629728
$280$	0	0
$281$	1341.79	0.284857	0.142428	−	0.989805i	$-0.454509\pi$
$281$	1341.79	0.284857	0.142428	+	0.989805i	$0.454509\pi$
$282$	0	0
$283$	−3800.22	−0.798232	−0.399116	−	0.916901i	$-0.630683\pi$
$283$	−3800.22	−0.798232	−0.399116	+	0.916901i	$0.630683\pi$
$284$	0	0
$285$	5996.89	1.24640
$286$	0	0
$287$	−129.872	−0.0267112
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−6557.16	−1.32092
$292$	0	0
$293$	6479.98	1.29203	0.646015	−	0.763325i	$-0.276434\pi$
$293$	6479.98	1.29203	0.646015	+	0.763325i	$0.276434\pi$
$294$	0	0
$295$	−4727.65	−0.933066
$296$	0	0
$297$	−159.734	−0.0312078
$298$	0	0
$299$	9304.09	1.79956
$300$	0	0
$301$	−1283.57	−0.245794
$302$	0	0
$303$	12559.2	2.38121
$304$	0	0
$305$	−2029.81	−0.381071
$306$	0	0
$307$	−4065.59	−0.755816	−0.377908	−	0.925843i	$-0.623356\pi$
$307$	−4065.59	−0.755816	−0.377908	+	0.925843i	$0.623356\pi$
$308$	0	0
$309$	6520.93	1.20053
$310$	0	0
$311$	2253.42	0.410867	0.205434	−	0.978671i	$-0.434140\pi$
$311$	2253.42	0.410867	0.205434	+	0.978671i	$0.434140\pi$
$312$	0	0
$313$	−5504.09	−0.993960	−0.496980	−	0.867762i	$-0.665558\pi$
$313$	−5504.09	−0.993960	−0.496980	+	0.867762i	$0.665558\pi$
$314$	0	0
$315$	2532.40	0.452967
$316$	0	0
$317$	−8460.40	−1.49900	−0.749501	−	0.662004i	$-0.769706\pi$
$317$	−8460.40	−1.49900	−0.749501	+	0.662004i	$0.769706\pi$
$318$	0	0
$319$	−4333.44	−0.760583
$320$	0	0
$321$	6553.09	1.13943
$322$	0	0
$323$	−905.954	−0.156064
$324$	0	0
$325$	−4816.98	−0.822148
$326$	0	0
$327$	7655.03	1.29457
$328$	0	0
$329$	−2511.68	−0.420892
$330$	0	0
$331$	−6259.86	−1.03950	−0.519748	−	0.854320i	$-0.673974\pi$
$331$	−6259.86	−1.03950	−0.519748	+	0.854320i	$0.673974\pi$
$332$	0	0
$333$	7352.88	1.21002
$334$	0	0
$335$	−13568.5	−2.21291
$336$	0	0
$337$	1418.40	0.229274	0.114637	−	0.993407i	$-0.463429\pi$
$337$	1418.40	0.229274	0.114637	+	0.993407i	$0.463429\pi$
$338$	0	0
$339$	15660.3	2.50900
$340$	0	0
$341$	−1642.37	−0.260819
$342$	0	0
$343$	−3845.04	−0.605285
$344$	0	0
$345$	−22554.5	−3.51969
$346$	0	0
$347$	5833.48	0.902472	0.451236	−	0.892405i	$-0.350983\pi$
$347$	5833.48	0.902472	0.451236	+	0.892405i	$0.350983\pi$
$348$	0	0
$349$	−300.780	−0.0461330	−0.0230665	−	0.999734i	$-0.507343\pi$
$349$	−300.780	−0.0461330	−0.0230665	+	0.999734i	$0.507343\pi$
$350$	0	0
$351$	−467.300	−0.0710617
$352$	0	0
$353$	−4160.99	−0.627386	−0.313693	−	0.949524i	$-0.601566\pi$
$353$	−4160.99	−0.627386	−0.313693	+	0.949524i	$0.601566\pi$
$354$	0	0
$355$	−448.200	−0.0670084
$356$	0	0
$357$	−746.885	−0.110726
$358$	0	0
$359$	−319.563	−0.0469801	−0.0234901	−	0.999724i	$-0.507478\pi$
$359$	−319.563	−0.0469801	−0.0234901	+	0.999724i	$0.507478\pi$
$360$	0	0
$361$	−4019.02	−0.585949
$362$	0	0
$363$	−8029.35	−1.16097
$364$	0	0
$365$	−3937.16	−0.564604
$366$	0	0
$367$	10153.2	1.44413	0.722063	−	0.691827i	$-0.243194\pi$
$367$	10153.2	1.44413	0.722063	+	0.691827i	$0.243194\pi$
$368$	0	0
$369$	−623.566	−0.0879717
$370$	0	0
$371$	2695.49	0.377204
$372$	0	0
$373$	4962.39	0.688855	0.344427	−	0.938813i	$-0.388073\pi$
$373$	4962.39	0.688855	0.344427	+	0.938813i	$0.388073\pi$
$374$	0	0
$375$	−2389.18	−0.329005
$376$	0	0
$377$	−12677.4	−1.73189
$378$	0	0
$379$	5071.05	0.687289	0.343644	−	0.939100i	$-0.388338\pi$
$379$	5071.05	0.687289	0.343644	+	0.939100i	$0.388338\pi$
$380$	0	0
$381$	−11854.9	−1.59408
$382$	0	0
$383$	−7076.82	−0.944148	−0.472074	−	0.881559i	$-0.656494\pi$
$383$	−7076.82	−0.944148	−0.472074	+	0.881559i	$0.656494\pi$
$384$	0	0
$385$	−1417.24	−0.187608
$386$	0	0
$387$	−6162.93	−0.809507
$388$	0	0
$389$	7424.62	0.967721	0.483860	−	0.875145i	$-0.339234\pi$
$389$	7424.62	0.967721	0.483860	+	0.875145i	$0.339234\pi$
$390$	0	0
$391$	3407.32	0.440705
$392$	0	0
$393$	19444.2	2.49576
$394$	0	0
$395$	12965.8	1.65160
$396$	0	0
$397$	−4587.78	−0.579985	−0.289993	−	0.957029i	$-0.593653\pi$
$397$	−4587.78	−0.579985	−0.289993	+	0.957029i	$0.593653\pi$
$398$	0	0
$399$	2341.33	0.293767
$400$	0	0
$401$	6484.38	0.807518	0.403759	−	0.914865i	$-0.367703\pi$
$401$	6484.38	0.807518	0.403759	+	0.914865i	$0.367703\pi$
$402$	0	0
$403$	−4804.73	−0.593897
$404$	0	0
$405$	−10445.9	−1.28164
$406$	0	0
$407$	−4114.99	−0.501161
$408$	0	0
$409$	−6939.89	−0.839011	−0.419506	−	0.907753i	$-0.637797\pi$
$409$	−6939.89	−0.839011	−0.419506	+	0.907753i	$0.637797\pi$
$410$	0	0
$411$	17983.3	2.15827
$412$	0	0
$413$	−1845.79	−0.219916
$414$	0	0
$415$	−9171.17	−1.08481
$416$	0	0
$417$	−9189.83	−1.07920
$418$	0	0
$419$	−2784.10	−0.324611	−0.162306	−	0.986741i	$-0.551893\pi$
$419$	−2784.10	−0.324611	−0.162306	+	0.986741i	$0.551893\pi$
$420$	0	0
$421$	−679.178	−0.0786250	−0.0393125	−	0.999227i	$-0.512517\pi$
$421$	−679.178	−0.0786250	−0.0393125	+	0.999227i	$0.512517\pi$
$422$	0	0
$423$	−12059.5	−1.38618
$424$	0	0
$425$	−1764.06	−0.201340
$426$	0	0
$427$	−792.486	−0.0898152
$428$	0	0
$429$	5480.13	0.616744
$430$	0	0
$431$	7755.99	0.866805	0.433403	−	0.901200i	$-0.357313\pi$
$431$	7755.99	0.866805	0.433403	+	0.901200i	$0.357313\pi$
$432$	0	0
$433$	10849.2	1.20411	0.602057	−	0.798453i	$-0.294348\pi$
$433$	10849.2	1.20411	0.602057	+	0.798453i	$0.294348\pi$
$434$	0	0
$435$	30732.0	3.38732
$436$	0	0
$437$	−10681.2	−1.16923
$438$	0	0
$439$	3926.13	0.426843	0.213422	−	0.976960i	$-0.431539\pi$
$439$	3926.13	0.426843	0.213422	+	0.976960i	$0.431539\pi$
$440$	0	0
$441$	−8736.39	−0.943353
$442$	0	0
$443$	−12772.0	−1.36979	−0.684893	−	0.728644i	$-0.740151\pi$
$443$	−12772.0	−1.36979	−0.684893	+	0.728644i	$0.740151\pi$
$444$	0	0
$445$	1550.90	0.165213
$446$	0	0
$447$	−21577.6	−2.28318
$448$	0	0
$449$	10424.9	1.09573	0.547863	−	0.836568i	$-0.315441\pi$
$449$	10424.9	1.09573	0.547863	+	0.836568i	$0.315441\pi$
$450$	0	0
$451$	348.974	0.0364358
$452$	0	0
$453$	1700.75	0.176398
$454$	0	0
$455$	−4146.12	−0.427193
$456$	0	0
$457$	4352.15	0.445482	0.222741	−	0.974878i	$-0.428500\pi$
$457$	4352.15	0.445482	0.222741	+	0.974878i	$0.428500\pi$
$458$	0	0
$459$	−171.134	−0.0174027
$460$	0	0
$461$	2529.94	0.255599	0.127799	−	0.991800i	$-0.459209\pi$
$461$	2529.94	0.255599	0.127799	+	0.991800i	$0.459209\pi$
$462$	0	0
$463$	−2430.54	−0.243967	−0.121984	−	0.992532i	$-0.538926\pi$
$463$	−2430.54	−0.243967	−0.121984	+	0.992532i	$0.538926\pi$
$464$	0	0
$465$	11647.4	1.16158
$466$	0	0
$467$	−12209.4	−1.20982	−0.604909	−	0.796294i	$-0.706791\pi$
$467$	−12209.4	−1.20982	−0.604909	+	0.796294i	$0.706791\pi$
$468$	0	0
$469$	−5297.44	−0.521563
$470$	0	0
$471$	−12236.3	−1.19707
$472$	0	0
$473$	3449.04	0.335279
$474$	0	0
$475$	5529.97	0.534174
$476$	0	0
$477$	12942.1	1.24230
$478$	0	0
$479$	−3922.90	−0.374200	−0.187100	−	0.982341i	$-0.559909\pi$
$479$	−3922.90	−0.374200	−0.187100	+	0.982341i	$0.559909\pi$
$480$	0	0
$481$	−12038.4	−1.14117
$482$	0	0
$483$	−8805.81	−0.829561
$484$	0	0
$485$	−13330.4	−1.24805
$486$	0	0
$487$	−1143.30	−0.106382	−0.0531909	−	0.998584i	$-0.516939\pi$
$487$	−1143.30	−0.106382	−0.0531909	+	0.998584i	$0.516939\pi$
$488$	0	0
$489$	12416.6	1.14826
$490$	0	0
$491$	−8540.74	−0.785007	−0.392503	−	0.919751i	$-0.628391\pi$
$491$	−8540.74	−0.785007	−0.392503	+	0.919751i	$0.628391\pi$
$492$	0	0
$493$	−4642.70	−0.424131
$494$	0	0
$495$	−6804.70	−0.617876
$496$	0	0
$497$	−174.988	−0.0157933
$498$	0	0
$499$	15478.1	1.38857	0.694284	−	0.719701i	$-0.255721\pi$
$499$	15478.1	1.38857	0.694284	+	0.719701i	$0.255721\pi$
$500$	0	0
$501$	6775.91	0.604242
$502$	0	0
$503$	−7278.03	−0.645151	−0.322576	−	0.946544i	$-0.604549\pi$
$503$	−7278.03	−0.645151	−0.322576	+	0.946544i	$0.604549\pi$
$504$	0	0
$505$	25532.3	2.24985
$506$	0	0
$507$	−313.526	−0.0274638
$508$	0	0
$509$	−523.823	−0.0456151	−0.0228075	−	0.999740i	$-0.507260\pi$
$509$	−523.823	−0.0456151	−0.0228075	+	0.999740i	$0.507260\pi$
$510$	0	0
$511$	−1537.16	−0.133072
$512$	0	0
$513$	536.468	0.0461709
$514$	0	0
$515$	13256.7	1.13430
$516$	0	0
$517$	6749.03	0.574123
$518$	0	0
$519$	32118.3	2.71645
$520$	0	0
$521$	−23618.6	−1.98608	−0.993042	−	0.117762i	$-0.962428\pi$
$521$	−23618.6	−1.98608	−0.993042	+	0.117762i	$0.962428\pi$
$522$	0	0
$523$	8878.45	0.742309	0.371155	−	0.928571i	$-0.378962\pi$
$523$	8878.45	0.742309	0.371155	+	0.928571i	$0.378962\pi$
$524$	0	0
$525$	4559.01	0.378993
$526$	0	0
$527$	−1759.58	−0.145443
$528$	0	0
$529$	28005.5	2.30176
$530$	0	0
$531$	−8862.32	−0.724278
$532$	0	0
$533$	1020.92	0.0829662
$534$	0	0
$535$	13322.1	1.07657
$536$	0	0
$537$	−20520.2	−1.64899
$538$	0	0
$539$	4889.26	0.390715
$540$	0	0
$541$	−240.187	−0.0190877	−0.00954385	−	0.999954i	$-0.503038\pi$
$541$	−240.187	−0.0190877	−0.00954385	+	0.999954i	$0.503038\pi$
$542$	0	0
$543$	4523.04	0.357463
$544$	0	0
$545$	15562.3	1.22315
$546$	0	0
$547$	7598.17	0.593920	0.296960	−	0.954890i	$-0.404027\pi$
$547$	7598.17	0.593920	0.296960	+	0.954890i	$0.404027\pi$
$548$	0	0
$549$	−3805.03	−0.295801
$550$	0	0
$551$	14553.9	1.12526
$552$	0	0
$553$	5062.17	0.389268
$554$	0	0
$555$	29182.8	2.23196
$556$	0	0
$557$	−16801.1	−1.27807	−0.639034	−	0.769179i	$-0.720665\pi$
$557$	−16801.1	−1.27807	−0.639034	+	0.769179i	$0.720665\pi$
$558$	0	0
$559$	10090.1	0.763447
$560$	0	0
$561$	2006.92	0.151038
$562$	0	0
$563$	12514.6	0.936819	0.468409	−	0.883512i	$-0.344827\pi$
$563$	12514.6	0.936819	0.468409	+	0.883512i	$0.344827\pi$
$564$	0	0
$565$	31836.6	2.37058
$566$	0	0
$567$	−4078.34	−0.302071
$568$	0	0
$569$	−14337.0	−1.05631	−0.528154	−	0.849148i	$-0.677116\pi$
$569$	−14337.0	−1.05631	−0.528154	+	0.849148i	$0.677116\pi$
$570$	0	0
$571$	−20865.4	−1.52923	−0.764616	−	0.644486i	$-0.777071\pi$
$571$	−20865.4	−1.52923	−0.764616	+	0.644486i	$0.777071\pi$
$572$	0	0
$573$	11236.9	0.819248
$574$	0	0
$575$	−20798.4	−1.50844
$576$	0	0
$577$	15317.0	1.10512	0.552562	−	0.833472i	$-0.313650\pi$
$577$	15317.0	1.10512	0.552562	+	0.833472i	$0.313650\pi$
$578$	0	0
$579$	9329.55	0.669643
$580$	0	0
$581$	−3580.64	−0.255680
$582$	0	0
$583$	−7242.94	−0.514531
$584$	0	0
$585$	−19907.1	−1.40693
$586$	0	0
$587$	−7104.60	−0.499554	−0.249777	−	0.968303i	$-0.580357\pi$
$587$	−7104.60	−0.499554	−0.249777	+	0.968303i	$0.580357\pi$
$588$	0	0
$589$	5515.91	0.385873
$590$	0	0
$591$	−37916.9	−2.63907
$592$	0	0
$593$	15376.9	1.06485	0.532424	−	0.846478i	$-0.321281\pi$
$593$	15376.9	1.06485	0.532424	+	0.846478i	$0.321281\pi$
$594$	0	0
$595$	−1518.38	−0.104618
$596$	0	0
$597$	−33887.6	−2.32316
$598$	0	0
$599$	−9123.57	−0.622336	−0.311168	−	0.950355i	$-0.600720\pi$
$599$	−9123.57	−0.622336	−0.311168	+	0.950355i	$0.600720\pi$
$600$	0	0
$601$	−87.7144	−0.00595332	−0.00297666	−	0.999996i	$-0.500948\pi$
$601$	−87.7144	−0.00595332	−0.00297666	+	0.999996i	$0.500948\pi$
$602$	0	0
$603$	−25435.0	−1.71774
$604$	0	0
$605$	−16323.3	−1.09692
$606$	0	0
$607$	−24347.7	−1.62808	−0.814039	−	0.580811i	$-0.802736\pi$
$607$	−24347.7	−1.62808	−0.814039	+	0.580811i	$0.802736\pi$
$608$	0	0
$609$	11998.5	0.798363
$610$	0	0
$611$	19744.2	1.30731
$612$	0	0
$613$	−8069.34	−0.531677	−0.265838	−	0.964018i	$-0.585649\pi$
$613$	−8069.34	−0.531677	−0.265838	+	0.964018i	$0.585649\pi$
$614$	0	0
$615$	−2474.87	−0.162270
$616$	0	0
$617$	16233.9	1.05924	0.529622	−	0.848234i	$-0.322334\pi$
$617$	16233.9	1.05924	0.529622	+	0.848234i	$0.322334\pi$
$618$	0	0
$619$	12459.9	0.809055	0.404528	−	0.914526i	$-0.367436\pi$
$619$	12459.9	0.809055	0.404528	+	0.914526i	$0.367436\pi$
$620$	0	0
$621$	−2017.67	−0.130381
$622$	0	0
$623$	605.508	0.0389393
$624$	0	0
$625$	−17828.2	−1.14100
$626$	0	0
$627$	−6291.28	−0.400717
$628$	0	0
$629$	−4408.66	−0.279467
$630$	0	0
$631$	−15291.3	−0.964719	−0.482359	−	0.875973i	$-0.660220\pi$
$631$	−15291.3	−0.964719	−0.482359	+	0.875973i	$0.660220\pi$
$632$	0	0
$633$	−8720.37	−0.547557
$634$	0	0
$635$	−24100.5	−1.50614
$636$	0	0
$637$	14303.5	0.889677
$638$	0	0
$639$	−840.182	−0.0520142
$640$	0	0
$641$	27761.6	1.71063	0.855317	−	0.518106i	$-0.173363\pi$
$641$	27761.6	1.71063	0.855317	+	0.518106i	$0.173363\pi$
$642$	0	0
$643$	−27780.4	−1.70381	−0.851906	−	0.523695i	$-0.824553\pi$
$643$	−27780.4	−1.70381	−0.851906	+	0.523695i	$0.824553\pi$
$644$	0	0
$645$	−24460.0	−1.49320
$646$	0	0
$647$	−2256.36	−0.137105	−0.0685523	−	0.997648i	$-0.521838\pi$
$647$	−2256.36	−0.137105	−0.0685523	+	0.997648i	$0.521838\pi$
$648$	0	0
$649$	4959.73	0.299979
$650$	0	0
$651$	4547.41	0.273774
$652$	0	0
$653$	4902.57	0.293802	0.146901	−	0.989151i	$-0.453070\pi$
$653$	4902.57	0.293802	0.146901	+	0.989151i	$0.453070\pi$
$654$	0	0
$655$	39529.2	2.35807
$656$	0	0
$657$	−7380.49	−0.438265
$658$	0	0
$659$	9040.81	0.534416	0.267208	−	0.963639i	$-0.413899\pi$
$659$	9040.81	0.534416	0.267208	+	0.963639i	$0.413899\pi$
$660$	0	0
$661$	−7339.63	−0.431889	−0.215944	−	0.976406i	$-0.569283\pi$
$661$	−7339.63	−0.431889	−0.215944	+	0.976406i	$0.569283\pi$
$662$	0	0
$663$	5871.23	0.343921
$664$	0	0
$665$	4759.81	0.277560
$666$	0	0
$667$	−54737.7	−3.17759
$668$	0	0
$669$	26179.8	1.51296
$670$	0	0
$671$	2129.46	0.122514
$672$	0	0
$673$	−21193.2	−1.21387	−0.606937	−	0.794750i	$-0.707602\pi$
$673$	−21193.2	−1.21387	−0.606937	+	0.794750i	$0.707602\pi$
$674$	0	0
$675$	1044.61	0.0595658
$676$	0	0
$677$	−16826.8	−0.955252	−0.477626	−	0.878563i	$-0.658503\pi$
$677$	−16826.8	−0.955252	−0.477626	+	0.878563i	$0.658503\pi$
$678$	0	0
$679$	−5204.50	−0.294154
$680$	0	0
$681$	40875.6	2.30009
$682$	0	0
$683$	−26526.3	−1.48609	−0.743046	−	0.669241i	$-0.766619\pi$
$683$	−26526.3	−1.48609	−0.743046	+	0.669241i	$0.766619\pi$
$684$	0	0
$685$	36559.1	2.03920
$686$	0	0
$687$	9255.36	0.513994
$688$	0	0
$689$	−21189.1	−1.17161
$690$	0	0
$691$	−588.600	−0.0324043	−0.0162022	−	0.999869i	$-0.505158\pi$
$691$	−588.600	−0.0324043	−0.0162022	+	0.999869i	$0.505158\pi$
$692$	0	0
$693$	−2656.71	−0.145628
$694$	0	0
$695$	−18682.5	−1.01966
$696$	0	0
$697$	373.880	0.0203181
$698$	0	0
$699$	−2147.49	−0.116203
$700$	0	0
$701$	19693.3	1.06106	0.530531	−	0.847665i	$-0.321992\pi$
$701$	19693.3	1.06106	0.530531	+	0.847665i	$0.321992\pi$
$702$	0	0
$703$	13820.2	0.741451
$704$	0	0
$705$	−47862.9	−2.55691
$706$	0	0
$707$	9968.40	0.530269
$708$	0	0
$709$	−27678.8	−1.46615	−0.733075	−	0.680148i	$-0.761915\pi$
$709$	−27678.8	−1.46615	−0.733075	+	0.680148i	$0.761915\pi$
$710$	0	0
$711$	24305.4	1.28203
$712$	0	0
$713$	−20745.5	−1.08966
$714$	0	0
$715$	11140.9	0.582719
$716$	0	0
$717$	50557.1	2.63332
$718$	0	0
$719$	1729.03	0.0896830	0.0448415	−	0.998994i	$-0.485722\pi$
$719$	1729.03	0.0896830	0.0448415	+	0.998994i	$0.485722\pi$
$720$	0	0
$721$	5175.74	0.267344
$722$	0	0
$723$	42787.1	2.20093
$724$	0	0
$725$	28339.2	1.45171
$726$	0	0
$727$	18235.5	0.930285	0.465142	−	0.885236i	$-0.346003\pi$
$727$	18235.5	0.930285	0.465142	+	0.885236i	$0.346003\pi$
$728$	0	0
$729$	−21603.8	−1.09759
$730$	0	0
$731$	3695.19	0.186965
$732$	0	0
$733$	14309.4	0.721051	0.360526	−	0.932749i	$-0.382597\pi$
$733$	14309.4	0.721051	0.360526	+	0.932749i	$0.382597\pi$
$734$	0	0
$735$	−34673.8	−1.74008
$736$	0	0
$737$	14234.5	0.711446
$738$	0	0
$739$	−4196.16	−0.208874	−0.104437	−	0.994531i	$-0.533304\pi$
$739$	−4196.16	−0.208874	−0.104437	+	0.994531i	$0.533304\pi$
$740$	0	0
$741$	−18405.1	−0.912453
$742$	0	0
$743$	18336.1	0.905367	0.452684	−	0.891671i	$-0.350467\pi$
$743$	18336.1	0.905367	0.452684	+	0.891671i	$0.350467\pi$
$744$	0	0
$745$	−43866.1	−2.15722
$746$	0	0
$747$	−17192.0	−0.842065
$748$	0	0
$749$	5201.27	0.253739
$750$	0	0
$751$	142.880	0.00694242	0.00347121	−	0.999994i	$-0.498895\pi$
$751$	142.880	0.00694242	0.00347121	+	0.999994i	$0.498895\pi$
$752$	0	0
$753$	9162.45	0.443424
$754$	0	0
$755$	3457.55	0.166666
$756$	0	0
$757$	−31096.7	−1.49304	−0.746519	−	0.665364i	$-0.768276\pi$
$757$	−31096.7	−1.49304	−0.746519	+	0.665364i	$0.768276\pi$
$758$	0	0
$759$	23661.7	1.13158
$760$	0	0
$761$	−30621.4	−1.45864	−0.729319	−	0.684173i	$-0.760163\pi$
$761$	−30621.4	−1.45864	−0.729319	+	0.684173i	$0.760163\pi$
$762$	0	0
$763$	6075.89	0.288286
$764$	0	0
$765$	−7290.33	−0.344552
$766$	0	0
$767$	14509.6	0.683068
$768$	0	0
$769$	−23099.3	−1.08320	−0.541600	−	0.840636i	$-0.682181\pi$
$769$	−23099.3	−1.08320	−0.541600	+	0.840636i	$0.682181\pi$
$770$	0	0
$771$	5230.69	0.244330
$772$	0	0
$773$	−5960.54	−0.277343	−0.138671	−	0.990338i	$-0.544283\pi$
$773$	−5960.54	−0.277343	−0.138671	+	0.990338i	$0.544283\pi$
$774$	0	0
$775$	10740.5	0.497820
$776$	0	0
$777$	11393.6	0.526055
$778$	0	0
$779$	−1172.03	−0.0539056
$780$	0	0
$781$	470.202	0.0215431
$782$	0	0
$783$	2749.21	0.125478
$784$	0	0
$785$	−24875.8	−1.13103
$786$	0	0
$787$	−9989.23	−0.452449	−0.226225	−	0.974075i	$-0.572638\pi$
$787$	−9989.23	−0.452449	−0.226225	+	0.974075i	$0.572638\pi$
$788$	0	0
$789$	1505.95	0.0679510
$790$	0	0
$791$	12429.8	0.558725
$792$	0	0
$793$	6229.70	0.278970
$794$	0	0
$795$	51365.6	2.29151
$796$	0	0
$797$	−30885.6	−1.37268	−0.686339	−	0.727282i	$-0.740783\pi$
$797$	−30885.6	−1.37268	−0.686339	+	0.727282i	$0.740783\pi$
$798$	0	0
$799$	7230.68	0.320154
$800$	0	0
$801$	2907.27	0.128244
$802$	0	0
$803$	4130.44	0.181519
$804$	0	0
$805$	−17901.8	−0.783795
$806$	0	0
$807$	18785.4	0.819428
$808$	0	0
$809$	−5112.83	−0.222197	−0.111099	−	0.993809i	$-0.535437\pi$
$809$	−5112.83	−0.222197	−0.111099	+	0.993809i	$0.535437\pi$
$810$	0	0
$811$	10184.8	0.440984	0.220492	−	0.975389i	$-0.429234\pi$
$811$	10184.8	0.440984	0.220492	+	0.975389i	$0.429234\pi$
$812$	0	0
$813$	53485.8	2.30729
$814$	0	0
$815$	25242.4	1.08491
$816$	0	0
$817$	−11583.6	−0.496034
$818$	0	0
$819$	−7772.19	−0.331602
$820$	0	0
$821$	39049.6	1.65997	0.829987	−	0.557783i	$-0.188348\pi$
$821$	39049.6	1.65997	0.829987	+	0.557783i	$0.188348\pi$
$822$	0	0
$823$	−3059.28	−0.129575	−0.0647873	−	0.997899i	$-0.520637\pi$
$823$	−3059.28	−0.129575	−0.0647873	+	0.997899i	$0.520637\pi$
$824$	0	0
$825$	−12250.3	−0.516971
$826$	0	0
$827$	31185.3	1.31127	0.655634	−	0.755078i	$-0.272401\pi$
$827$	31185.3	1.31127	0.655634	+	0.755078i	$0.272401\pi$
$828$	0	0
$829$	−6843.29	−0.286704	−0.143352	−	0.989672i	$-0.545788\pi$
$829$	−6843.29	−0.286704	−0.143352	+	0.989672i	$0.545788\pi$
$830$	0	0
$831$	−42586.2	−1.77774
$832$	0	0
$833$	5238.19	0.217878
$834$	0	0
$835$	13775.1	0.570907
$836$	0	0
$837$	1041.95	0.0430287
$838$	0	0
$839$	−23157.7	−0.952909	−0.476455	−	0.879199i	$-0.658078\pi$
$839$	−23157.7	−0.952909	−0.476455	+	0.879199i	$0.658078\pi$
$840$	0	0
$841$	50194.6	2.05809
$842$	0	0
$843$	9982.90	0.407864
$844$	0	0
$845$	−637.383	−0.0259487
$846$	0	0
$847$	−6373.00	−0.258535
$848$	0	0
$849$	−28273.5	−1.14293
$850$	0	0
$851$	−51978.3	−2.09376
$852$	0	0
$853$	1500.27	0.0602208	0.0301104	−	0.999547i	$-0.490414\pi$
$853$	1500.27	0.0602208	0.0301104	+	0.999547i	$0.490414\pi$
$854$	0	0
$855$	22853.6	0.914126
$856$	0	0
$857$	14227.1	0.567080	0.283540	−	0.958960i	$-0.408491\pi$
$857$	14227.1	0.567080	0.283540	+	0.958960i	$0.408491\pi$
$858$	0	0
$859$	9400.73	0.373398	0.186699	−	0.982417i	$-0.440221\pi$
$859$	9400.73	0.373398	0.186699	+	0.982417i	$0.440221\pi$
$860$	0	0
$861$	−966.246	−0.0382457
$862$	0	0
$863$	−21211.0	−0.836654	−0.418327	−	0.908296i	$-0.637383\pi$
$863$	−21211.0	−0.836654	−0.418327	+	0.908296i	$0.637383\pi$
$864$	0	0
$865$	65295.0	2.56658
$866$	0	0
$867$	2150.15	0.0842248
$868$	0	0
$869$	−13602.3	−0.530987
$870$	0	0
$871$	41643.0	1.62000
$872$	0	0
$873$	−24988.8	−0.968777
$874$	0	0
$875$	−1896.32	−0.0732656
$876$	0	0
$877$	−39474.7	−1.51992	−0.759958	−	0.649972i	$-0.774780\pi$
$877$	−39474.7	−1.51992	−0.759958	+	0.649972i	$0.774780\pi$
$878$	0	0
$879$	48210.8	1.84996
$880$	0	0
$881$	14821.0	0.566777	0.283389	−	0.959005i	$-0.408541\pi$
$881$	14821.0	0.566777	0.283389	+	0.959005i	$0.408541\pi$
$882$	0	0
$883$	−2197.66	−0.0837567	−0.0418784	−	0.999123i	$-0.513334\pi$
$883$	−2197.66	−0.0837567	−0.0418784	+	0.999123i	$0.513334\pi$
$884$	0	0
$885$	−35173.6	−1.33598
$886$	0	0
$887$	−43430.3	−1.64402	−0.822010	−	0.569472i	$-0.807148\pi$
$887$	−43430.3	−1.64402	−0.822010	+	0.569472i	$0.807148\pi$
$888$	0	0
$889$	−9409.40	−0.354984
$890$	0	0
$891$	10958.7	0.412044
$892$	0	0
$893$	−22666.7	−0.849396
$894$	0	0
$895$	−41716.5	−1.55802
$896$	0	0
$897$	69222.1	2.57665
$898$	0	0
$899$	28267.1	1.04868
$900$	0	0
$901$	−7759.84	−0.286923
$902$	0	0
$903$	−9549.75	−0.351933
$904$	0	0
$905$	9195.13	0.337742
$906$	0	0
$907$	−31645.5	−1.15851	−0.579256	−	0.815145i	$-0.696657\pi$
$907$	−31645.5	−1.15851	−0.579256	+	0.815145i	$0.696657\pi$
$908$	0	0
$909$	47862.1	1.74641
$910$	0	0
$911$	19345.1	0.703546	0.351773	−	0.936085i	$-0.385579\pi$
$911$	19345.1	0.703546	0.351773	+	0.936085i	$0.385579\pi$
$912$	0	0
$913$	9621.39	0.348764
$914$	0	0
$915$	−15101.7	−0.545626
$916$	0	0
$917$	15433.1	0.555776
$918$	0	0
$919$	−5449.53	−0.195607	−0.0978037	−	0.995206i	$-0.531182\pi$
$919$	−5449.53	−0.195607	−0.0978037	+	0.995206i	$0.531182\pi$
$920$	0	0
$921$	−30247.8	−1.08219
$922$	0	0
$923$	1375.57	0.0490547
$924$	0	0
$925$	26910.6	0.956557
$926$	0	0
$927$	24850.7	0.880479
$928$	0	0
$929$	−6262.79	−0.221179	−0.110590	−	0.993866i	$-0.535274\pi$
$929$	−6262.79	−0.221179	−0.110590	+	0.993866i	$0.535274\pi$
$930$	0	0
$931$	−16420.6	−0.578050
$932$	0	0
$933$	16765.4	0.588289
$934$	0	0
$935$	4079.98	0.142705
$936$	0	0
$937$	−11037.6	−0.384828	−0.192414	−	0.981314i	$-0.561632\pi$
$937$	−11037.6	−0.384828	−0.192414	+	0.981314i	$0.561632\pi$
$938$	0	0
$939$	−40950.2	−1.42317
$940$	0	0
$941$	12963.7	0.449100	0.224550	−	0.974463i	$-0.427909\pi$
$941$	12963.7	0.449100	0.224550	+	0.974463i	$0.427909\pi$
$942$	0	0
$943$	4408.06	0.152223
$944$	0	0
$945$	899.122	0.0309507
$946$	0	0
$947$	−44086.6	−1.51280	−0.756400	−	0.654109i	$-0.773044\pi$
$947$	−44086.6	−1.51280	−0.756400	+	0.654109i	$0.773044\pi$
$948$	0	0
$949$	12083.6	0.413329
$950$	0	0
$951$	−62945.1	−2.14630
$952$	0	0
$953$	31988.0	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$953$	31988.0	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$954$	0	0
$955$	22844.1	0.774051
$956$	0	0
$957$	−32240.6	−1.08902
$958$	0	0
$959$	14273.5	0.480622
$960$	0	0
$961$	−19077.8	−0.640388
$962$	0	0
$963$	24973.2	0.835672
$964$	0	0
$965$	18966.5	0.632699
$966$	0	0
$967$	15691.4	0.521822	0.260911	−	0.965363i	$-0.415977\pi$
$967$	15691.4	0.521822	0.260911	+	0.965363i	$0.415977\pi$
$968$	0	0
$969$	−6740.27	−0.223456
$970$	0	0
$971$	−18786.9	−0.620907	−0.310453	−	0.950589i	$-0.600481\pi$
$971$	−18786.9	−0.620907	−0.310453	+	0.950589i	$0.600481\pi$
$972$	0	0
$973$	−7294.08	−0.240326
$974$	0	0
$975$	−35838.2	−1.17717
$976$	0	0
$977$	49002.0	1.60462	0.802309	−	0.596909i	$-0.203605\pi$
$977$	49002.0	1.60462	0.802309	+	0.596909i	$0.203605\pi$
$978$	0	0
$979$	−1627.04	−0.0531157
$980$	0	0
$981$	29172.7	0.949451
$982$	0	0
$983$	51257.4	1.66313	0.831565	−	0.555428i	$-0.187446\pi$
$983$	51257.4	1.66313	0.831565	+	0.555428i	$0.187446\pi$
$984$	0	0
$985$	−77083.2	−2.49348
$986$	0	0
$987$	−18686.8	−0.602642
$988$	0	0
$989$	43566.4	1.40074
$990$	0	0
$991$	36297.4	1.16350	0.581748	−	0.813369i	$-0.302369\pi$
$991$	36297.4	1.16350	0.581748	+	0.813369i	$0.302369\pi$
$992$	0	0
$993$	−46573.1	−1.48837
$994$	0	0
$995$	−68892.0	−2.19500
$996$	0	0
$997$	−7578.52	−0.240736	−0.120368	−	0.992729i	$-0.538408\pi$
$997$	−7578.52	−0.240736	−0.120368	+	0.992729i	$0.538408\pi$
$998$	0	0
$999$	2610.62	0.0826792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.a.d.1.4	✓	4
3.2	odd	2		1224.4.a.l.1.2		4
4.3	odd	2		272.4.a.k.1.1		4
8.3	odd	2		1088.4.a.bb.1.4		4
8.5	even	2		1088.4.a.be.1.1		4
12.11	even	2		2448.4.a.bq.1.2		4
17.16	even	2		2312.4.a.e.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.d.1.4	✓	4	1.1	even	1	trivial
272.4.a.k.1.1		4	4.3	odd	2
1088.4.a.bb.1.4		4	8.3	odd	2
1088.4.a.be.1.1		4	8.5	even	2
1224.4.a.l.1.2		4	3.2	odd	2
2312.4.a.e.1.1		4	17.16	even	2
2448.4.a.bq.1.2		4	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)

\(f(q)\)	\(=\)	\(q+7.43996 q^{3} +15.1251 q^{5} +5.90519 q^{7} +28.3531 q^{9} +O(q^{10})\)	\(q+7.43996 q^{3} +15.1251 q^{5} +5.90519 q^{7} +28.3531 q^{9} -15.8676 q^{11} -46.4205 q^{13} +112.530 q^{15} -17.0000 q^{17} +53.2914 q^{19} +43.9344 q^{21} -200.431 q^{23} +103.769 q^{25} +10.0667 q^{27} +273.100 q^{29} +103.505 q^{31} -118.054 q^{33} +89.3166 q^{35} +259.333 q^{37} -345.367 q^{39} -21.9929 q^{41} -217.364 q^{43} +428.843 q^{45} -425.334 q^{47} -308.129 q^{49} -126.479 q^{51} +456.461 q^{53} -239.999 q^{55} +396.486 q^{57} -312.570 q^{59} -134.202 q^{61} +167.430 q^{63} -702.114 q^{65} -897.083 q^{67} -1491.20 q^{69} -29.6329 q^{71} -260.307 q^{73} +772.034 q^{75} -93.7011 q^{77} +857.241 q^{79} -690.637 q^{81} -606.355 q^{83} -257.127 q^{85} +2031.85 q^{87} +102.538 q^{89} -274.122 q^{91} +770.070 q^{93} +806.038 q^{95} -881.343 q^{97} -449.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 8 q^{5} - 22 q^{7} + 100 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 8 * q^5 - 22 * q^7 + 100 * q^9	\( 4 q - 2 q^{3} + 8 q^{5} - 22 q^{7} + 100 q^{9} + 70 q^{11} + 120 q^{13} + 140 q^{15} - 68 q^{17} - 44 q^{19} + 488 q^{21} + 158 q^{23} + 548 q^{25} - 392 q^{27} + 264 q^{29} + 122 q^{31} + 136 q^{33} - 44 q^{35} + 256 q^{37} - 528 q^{39} + 240 q^{41} - 1100 q^{43} - 880 q^{45} - 800 q^{47} + 12 q^{49} + 34 q^{51} + 432 q^{53} - 532 q^{55} + 472 q^{57} - 148 q^{59} - 728 q^{61} - 1450 q^{63} - 72 q^{65} - 1032 q^{67} - 1024 q^{69} + 798 q^{71} - 1544 q^{73} - 2974 q^{75} + 656 q^{77} + 758 q^{79} - 20 q^{81} + 244 q^{83} - 136 q^{85} + 1524 q^{87} + 1440 q^{89} - 1104 q^{91} - 2464 q^{93} + 7016 q^{95} - 1344 q^{97} - 86 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 8 * q^5 - 22 * q^7 + 100 * q^9 + 70 * q^11 + 120 * q^13 + 140 * q^15 - 68 * q^17 - 44 * q^19 + 488 * q^21 + 158 * q^23 + 548 * q^25 - 392 * q^27 + 264 * q^29 + 122 * q^31 + 136 * q^33 - 44 * q^35 + 256 * q^37 - 528 * q^39 + 240 * q^41 - 1100 * q^43 - 880 * q^45 - 800 * q^47 + 12 * q^49 + 34 * q^51 + 432 * q^53 - 532 * q^55 + 472 * q^57 - 148 * q^59 - 728 * q^61 - 1450 * q^63 - 72 * q^65 - 1032 * q^67 - 1024 * q^69 + 798 * q^71 - 1544 * q^73 - 2974 * q^75 + 656 * q^77 + 758 * q^79 - 20 * q^81 + 244 * q^83 - 136 * q^85 + 1524 * q^87 + 1440 * q^89 - 1104 * q^91 - 2464 * q^93 + 7016 * q^95 - 1344 * q^97 - 86 * q^99

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