LMFDB - Embedded newform 136.4.a.d.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:	$0$
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			1.2
Root			$-0.698199$ 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)

$f(q)$	$=$	$q-5.12911 q^{3} +20.1066 q^{5} -21.0725 q^{7} -0.692234 q^{9} +O(q^{10})$	$q-5.12911 q^{3} +20.1066 q^{5} -21.0725 q^{7} -0.692234 q^{9} +17.0861 q^{11} +78.4238 q^{13} -103.129 q^{15} -17.0000 q^{17} +139.741 q^{19} +108.083 q^{21} +138.464 q^{23} +279.277 q^{25} +142.037 q^{27} -203.057 q^{29} +42.6601 q^{31} -87.6363 q^{33} -423.698 q^{35} -39.9480 q^{37} -402.244 q^{39} +313.527 q^{41} -229.478 q^{43} -13.9185 q^{45} -322.172 q^{47} +101.052 q^{49} +87.1949 q^{51} -213.671 q^{53} +343.543 q^{55} -716.747 q^{57} +488.327 q^{59} +382.584 q^{61} +14.5871 q^{63} +1576.84 q^{65} -426.513 q^{67} -710.196 q^{69} -483.945 q^{71} -637.051 q^{73} -1432.44 q^{75} -360.047 q^{77} -120.822 q^{79} -709.830 q^{81} +465.209 q^{83} -341.813 q^{85} +1041.50 q^{87} +1346.34 q^{89} -1652.59 q^{91} -218.808 q^{93} +2809.72 q^{95} -1117.14 q^{97} -11.8276 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.12911	−0.987098	−0.493549	−	0.869718i	$-0.664301\pi$
$3$	−5.12911	−0.987098	−0.493549	+	0.869718i	$0.664301\pi$
$4$	0	0
$5$	20.1066	1.79839	0.899196	−	0.437545i	$-0.144152\pi$
$5$	20.1066	1.79839	0.899196	+	0.437545i	$0.144152\pi$
$6$	0	0
$7$	−21.0725	−1.13781	−0.568905	−	0.822403i	$-0.692633\pi$
$7$	−21.0725	−1.13781	−0.568905	+	0.822403i	$0.692633\pi$
$8$	0	0
$9$	−0.692234	−0.0256383
$10$	0	0
$11$	17.0861	0.468331	0.234165	−	0.972197i	$-0.424764\pi$
$11$	17.0861	0.468331	0.234165	+	0.972197i	$0.424764\pi$
$12$	0	0
$13$	78.4238	1.67314	0.836572	−	0.547858i	$-0.184556\pi$
$13$	78.4238	1.67314	0.836572	+	0.547858i	$0.184556\pi$
$14$	0	0
$15$	−103.129	−1.77519
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	139.741	1.68730	0.843652	−	0.536890i	$-0.180401\pi$
$19$	139.741	1.68730	0.843652	+	0.536890i	$0.180401\pi$
$20$	0	0
$21$	108.083	1.12313
$22$	0	0
$23$	138.464	1.25529	0.627645	−	0.778500i	$-0.284019\pi$
$23$	138.464	1.25529	0.627645	+	0.778500i	$0.284019\pi$
$24$	0	0
$25$	279.277	2.23422
$26$	0	0
$27$	142.037	1.01241
$28$	0	0
$29$	−203.057	−1.30024	−0.650118	−	0.759834i	$-0.725280\pi$
$29$	−203.057	−1.30024	−0.650118	+	0.759834i	$0.725280\pi$
$30$	0	0
$31$	42.6601	0.247161	0.123580	−	0.992335i	$-0.460562\pi$
$31$	42.6601	0.247161	0.123580	+	0.992335i	$0.460562\pi$
$32$	0	0
$33$	−87.6363	−0.462288
$34$	0	0
$35$	−423.698	−2.04623
$36$	0	0
$37$	−39.9480	−0.177498	−0.0887488	−	0.996054i	$-0.528287\pi$
$37$	−39.9480	−0.177498	−0.0887488	+	0.996054i	$0.528287\pi$
$38$	0	0
$39$	−402.244	−1.65156
$40$	0	0
$41$	313.527	1.19426	0.597130	−	0.802145i	$-0.296308\pi$
$41$	313.527	1.19426	0.597130	+	0.802145i	$0.296308\pi$
$42$	0	0
$43$	−229.478	−0.813839	−0.406920	−	0.913464i	$-0.633397\pi$
$43$	−229.478	−0.813839	−0.406920	+	0.913464i	$0.633397\pi$
$44$	0	0
$45$	−13.9185	−0.0461078
$46$	0	0
$47$	−322.172	−0.999864	−0.499932	−	0.866065i	$-0.666642\pi$
$47$	−322.172	−0.999864	−0.499932	+	0.866065i	$0.666642\pi$
$48$	0	0
$49$	101.052	0.294612
$50$	0	0
$51$	87.1949	0.239406
$52$	0	0
$53$	−213.671	−0.553774	−0.276887	−	0.960903i	$-0.589303\pi$
$53$	−213.671	−0.553774	−0.276887	+	0.960903i	$0.589303\pi$
$54$	0	0
$55$	343.543	0.842243
$56$	0	0
$57$	−716.747	−1.66553
$58$	0	0
$59$	488.327	1.07754	0.538769	−	0.842454i	$-0.318890\pi$
$59$	488.327	1.07754	0.538769	+	0.842454i	$0.318890\pi$
$60$	0	0
$61$	382.584	0.803031	0.401516	−	0.915852i	$-0.368484\pi$
$61$	382.584	0.803031	0.401516	+	0.915852i	$0.368484\pi$
$62$	0	0
$63$	14.5871	0.0291715
$64$	0	0
$65$	1576.84	3.00897
$66$	0	0
$67$	−426.513	−0.777714	−0.388857	−	0.921298i	$-0.627130\pi$
$67$	−426.513	−0.777714	−0.388857	+	0.921298i	$0.627130\pi$
$68$	0	0
$69$	−710.196	−1.23909
$70$	0	0
$71$	−483.945	−0.808925	−0.404462	−	0.914555i	$-0.632541\pi$
$71$	−483.945	−0.808925	−0.404462	+	0.914555i	$0.632541\pi$
$72$	0	0
$73$	−637.051	−1.02139	−0.510693	−	0.859763i	$-0.670611\pi$
$73$	−637.051	−1.02139	−0.510693	+	0.859763i	$0.670611\pi$
$74$	0	0
$75$	−1432.44	−2.20539
$76$	0	0
$77$	−360.047	−0.532872
$78$	0	0
$79$	−120.822	−0.172071	−0.0860354	−	0.996292i	$-0.527420\pi$
$79$	−120.822	−0.172071	−0.0860354	+	0.996292i	$0.527420\pi$
$80$	0	0
$81$	−709.830	−0.973704
$82$	0	0
$83$	465.209	0.615221	0.307611	−	0.951512i	$-0.400471\pi$
$83$	465.209	0.615221	0.307611	+	0.951512i	$0.400471\pi$
$84$	0	0
$85$	−341.813	−0.436174
$86$	0	0
$87$	1041.50	1.28346
$88$	0	0
$89$	1346.34	1.60350	0.801751	−	0.597658i	$-0.203902\pi$
$89$	1346.34	1.60350	0.801751	+	0.597658i	$0.203902\pi$
$90$	0	0
$91$	−1652.59	−1.90372
$92$	0	0
$93$	−218.808	−0.243972
$94$	0	0
$95$	2809.72	3.03444
$96$	0	0
$97$	−1117.14	−1.16936	−0.584680	−	0.811264i	$-0.698780\pi$
$97$	−1117.14	−1.16936	−0.584680	+	0.811264i	$0.698780\pi$
$98$	0	0
$99$	−11.8276	−0.0120072
$100$	0	0
$101$	24.1892	0.0238308	0.0119154	−	0.999929i	$-0.496207\pi$
$101$	24.1892	0.0238308	0.0119154	+	0.999929i	$0.496207\pi$
$102$	0	0
$103$	797.676	0.763081	0.381541	−	0.924352i	$-0.375394\pi$
$103$	797.676	0.763081	0.381541	+	0.924352i	$0.375394\pi$
$104$	0	0
$105$	2173.19	2.01983
$106$	0	0
$107$	−1202.56	−1.08651	−0.543254	−	0.839568i	$-0.682808\pi$
$107$	−1202.56	−1.08651	−0.543254	+	0.839568i	$0.682808\pi$
$108$	0	0
$109$	214.814	0.188766	0.0943829	−	0.995536i	$-0.469912\pi$
$109$	214.814	0.188766	0.0943829	+	0.995536i	$0.469912\pi$
$110$	0	0
$111$	204.898	0.175207
$112$	0	0
$113$	−1535.38	−1.27820	−0.639100	−	0.769124i	$-0.720693\pi$
$113$	−1535.38	−1.27820	−0.639100	+	0.769124i	$0.720693\pi$
$114$	0	0
$115$	2784.04	2.25751
$116$	0	0
$117$	−54.2877	−0.0428966
$118$	0	0
$119$	358.233	0.275959
$120$	0	0
$121$	−1039.07	−0.780666
$122$	0	0
$123$	−1608.11	−1.17885
$124$	0	0
$125$	3102.00	2.21961
$126$	0	0
$127$	659.153	0.460554	0.230277	−	0.973125i	$-0.426037\pi$
$127$	659.153	0.460554	0.230277	+	0.973125i	$0.426037\pi$
$128$	0	0
$129$	1177.02	0.803339
$130$	0	0
$131$	1087.80	0.725509	0.362754	−	0.931885i	$-0.381836\pi$
$131$	1087.80	0.725509	0.362754	+	0.931885i	$0.381836\pi$
$132$	0	0
$133$	−2944.70	−1.91983
$134$	0	0
$135$	2855.88	1.82070
$136$	0	0
$137$	−1762.20	−1.09894	−0.549472	−	0.835512i	$-0.685171\pi$
$137$	−1762.20	−1.09894	−0.549472	+	0.835512i	$0.685171\pi$
$138$	0	0
$139$	−992.512	−0.605639	−0.302819	−	0.953048i	$-0.597928\pi$
$139$	−992.512	−0.605639	−0.302819	+	0.953048i	$0.597928\pi$
$140$	0	0
$141$	1652.45	0.986963
$142$	0	0
$143$	1339.95	0.783585
$144$	0	0
$145$	−4082.80	−2.33833
$146$	0	0
$147$	−518.306	−0.290811
$148$	0	0
$149$	1305.14	0.717590	0.358795	−	0.933416i	$-0.383188\pi$
$149$	1305.14	0.717590	0.358795	+	0.933416i	$0.383188\pi$
$150$	0	0
$151$	1566.45	0.844210	0.422105	−	0.906547i	$-0.361291\pi$
$151$	1566.45	0.844210	0.422105	+	0.906547i	$0.361291\pi$
$152$	0	0
$153$	11.7680	0.00621820
$154$	0	0
$155$	857.751	0.444492
$156$	0	0
$157$	3409.68	1.73326	0.866630	−	0.498951i	$-0.166281\pi$
$157$	3409.68	1.73326	0.866630	+	0.498951i	$0.166281\pi$
$158$	0	0
$159$	1095.94	0.546629
$160$	0	0
$161$	−2917.78	−1.42828
$162$	0	0
$163$	−1775.65	−0.853251	−0.426625	−	0.904428i	$-0.640298\pi$
$163$	−1775.65	−0.853251	−0.426625	+	0.904428i	$0.640298\pi$
$164$	0	0
$165$	−1762.07	−0.831376
$166$	0	0
$167$	−651.188	−0.301739	−0.150870	−	0.988554i	$-0.548207\pi$
$167$	−651.188	−0.301739	−0.150870	+	0.988554i	$0.548207\pi$
$168$	0	0
$169$	3953.30	1.79941
$170$	0	0
$171$	−96.7335	−0.0432596
$172$	0	0
$173$	−1737.90	−0.763758	−0.381879	−	0.924212i	$-0.624723\pi$
$173$	−1737.90	−0.763758	−0.381879	+	0.924212i	$0.624723\pi$
$174$	0	0
$175$	−5885.08	−2.54212
$176$	0	0
$177$	−2504.68	−1.06363
$178$	0	0
$179$	2890.53	1.20698	0.603488	−	0.797372i	$-0.293777\pi$
$179$	2890.53	1.20698	0.603488	+	0.797372i	$0.293777\pi$
$180$	0	0
$181$	−1617.76	−0.664349	−0.332175	−	0.943218i	$-0.607782\pi$
$181$	−1617.76	−0.664349	−0.332175	+	0.943218i	$0.607782\pi$
$182$	0	0
$183$	−1962.32	−0.792670
$184$	0	0
$185$	−803.220	−0.319210
$186$	0	0
$187$	−290.463	−0.113587
$188$	0	0
$189$	−2993.07	−1.15192
$190$	0	0
$191$	−209.705	−0.0794436	−0.0397218	−	0.999211i	$-0.512647\pi$
$191$	−209.705	−0.0794436	−0.0397218	+	0.999211i	$0.512647\pi$
$192$	0	0
$193$	2714.19	1.01229	0.506144	−	0.862449i	$-0.331070\pi$
$193$	2714.19	1.01229	0.506144	+	0.862449i	$0.331070\pi$
$194$	0	0
$195$	−8087.79	−2.97015
$196$	0	0
$197$	−1932.09	−0.698760	−0.349380	−	0.936981i	$-0.613608\pi$
$197$	−1932.09	−0.698760	−0.349380	+	0.936981i	$0.613608\pi$
$198$	0	0
$199$	−2114.22	−0.753132	−0.376566	−	0.926390i	$-0.622895\pi$
$199$	−2114.22	−0.753132	−0.376566	+	0.926390i	$0.622895\pi$
$200$	0	0
$201$	2187.63	0.767680
$202$	0	0
$203$	4278.94	1.47942
$204$	0	0
$205$	6303.97	2.14775
$206$	0	0
$207$	−95.8493	−0.0321835
$208$	0	0
$209$	2387.62	0.790217
$210$	0	0
$211$	−3828.43	−1.24910	−0.624550	−	0.780985i	$-0.714718\pi$
$211$	−3828.43	−1.24910	−0.624550	+	0.780985i	$0.714718\pi$
$212$	0	0
$213$	2482.20	0.798488
$214$	0	0
$215$	−4614.04	−1.46360
$216$	0	0
$217$	−898.956	−0.281222
$218$	0	0
$219$	3267.51	1.00821
$220$	0	0
$221$	−1333.21	−0.405797
$222$	0	0
$223$	−3318.07	−0.996387	−0.498194	−	0.867066i	$-0.666003\pi$
$223$	−3318.07	−0.996387	−0.498194	+	0.867066i	$0.666003\pi$
$224$	0	0
$225$	−193.325	−0.0572816
$226$	0	0
$227$	1265.77	0.370097	0.185048	−	0.982729i	$-0.440756\pi$
$227$	1265.77	0.370097	0.185048	+	0.982729i	$0.440756\pi$
$228$	0	0
$229$	470.289	0.135710	0.0678550	−	0.997695i	$-0.478384\pi$
$229$	470.289	0.135710	0.0678550	+	0.997695i	$0.478384\pi$
$230$	0	0
$231$	1846.72	0.525996
$232$	0	0
$233$	−598.736	−0.168345	−0.0841727	−	0.996451i	$-0.526825\pi$
$233$	−598.736	−0.168345	−0.0841727	+	0.996451i	$0.526825\pi$
$234$	0	0
$235$	−6477.80	−1.79815
$236$	0	0
$237$	619.712	0.169851
$238$	0	0
$239$	5431.34	1.46997	0.734987	−	0.678081i	$-0.237188\pi$
$239$	5431.34	1.46997	0.734987	+	0.678081i	$0.237188\pi$
$240$	0	0
$241$	3929.62	1.05033	0.525164	−	0.851001i	$-0.324004\pi$
$241$	3929.62	1.05033	0.525164	+	0.851001i	$0.324004\pi$
$242$	0	0
$243$	−194.187	−0.0512639
$244$	0	0
$245$	2031.81	0.529828
$246$	0	0
$247$	10959.0	2.82310
$248$	0	0
$249$	−2386.11	−0.607283
$250$	0	0
$251$	−5275.18	−1.32656	−0.663280	−	0.748371i	$-0.730836\pi$
$251$	−5275.18	−1.32656	−0.663280	+	0.748371i	$0.730836\pi$
$252$	0	0
$253$	2365.80	0.587891
$254$	0	0
$255$	1753.20	0.430547
$256$	0	0
$257$	3811.20	0.925043	0.462522	−	0.886608i	$-0.346945\pi$
$257$	3811.20	0.925043	0.462522	+	0.886608i	$0.346945\pi$
$258$	0	0
$259$	841.806	0.201959
$260$	0	0
$261$	140.563	0.0333358
$262$	0	0
$263$	5112.71	1.19872	0.599360	−	0.800479i	$-0.295422\pi$
$263$	5112.71	1.19872	0.599360	+	0.800479i	$0.295422\pi$
$264$	0	0
$265$	−4296.21	−0.995903
$266$	0	0
$267$	−6905.52	−1.58281
$268$	0	0
$269$	975.150	0.221026	0.110513	−	0.993875i	$-0.464751\pi$
$269$	975.150	0.221026	0.110513	+	0.993875i	$0.464751\pi$
$270$	0	0
$271$	−4216.45	−0.945133	−0.472567	−	0.881295i	$-0.656672\pi$
$271$	−4216.45	−0.945133	−0.472567	+	0.881295i	$0.656672\pi$
$272$	0	0
$273$	8476.31	1.87916
$274$	0	0
$275$	4771.75	1.04635
$276$	0	0
$277$	−8128.34	−1.76312	−0.881560	−	0.472072i	$-0.843506\pi$
$277$	−8128.34	−1.76312	−0.881560	+	0.472072i	$0.843506\pi$
$278$	0	0
$279$	−29.5308	−0.00633678
$280$	0	0
$281$	−7222.06	−1.53321	−0.766605	−	0.642119i	$-0.778056\pi$
$281$	−7222.06	−1.53321	−0.766605	+	0.642119i	$0.778056\pi$
$282$	0	0
$283$	393.419	0.0826373	0.0413186	−	0.999146i	$-0.486844\pi$
$283$	393.419	0.0826373	0.0413186	+	0.999146i	$0.486844\pi$
$284$	0	0
$285$	−14411.4	−2.99528
$286$	0	0
$287$	−6606.80	−1.35884
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	5729.91	1.15427
$292$	0	0
$293$	2276.65	0.453937	0.226968	−	0.973902i	$-0.427119\pi$
$293$	2276.65	0.453937	0.226968	+	0.973902i	$0.427119\pi$
$294$	0	0
$295$	9818.61	1.93784
$296$	0	0
$297$	2426.84	0.474141
$298$	0	0
$299$	10858.9	2.10028
$300$	0	0
$301$	4835.69	0.925995
$302$	0	0
$303$	−124.069	−0.0235233
$304$	0	0
$305$	7692.49	1.44417
$306$	0	0
$307$	−8281.24	−1.53953	−0.769764	−	0.638328i	$-0.779626\pi$
$307$	−8281.24	−1.53953	−0.769764	+	0.638328i	$0.779626\pi$
$308$	0	0
$309$	−4091.37	−0.753235
$310$	0	0
$311$	846.944	0.154424	0.0772119	−	0.997015i	$-0.475398\pi$
$311$	846.944	0.154424	0.0772119	+	0.997015i	$0.475398\pi$
$312$	0	0
$313$	−1769.07	−0.319469	−0.159734	−	0.987160i	$-0.551064\pi$
$313$	−1769.07	−0.319469	−0.159734	+	0.987160i	$0.551064\pi$
$314$	0	0
$315$	293.298	0.0524619
$316$	0	0
$317$	4119.16	0.729826	0.364913	−	0.931042i	$-0.381099\pi$
$317$	4119.16	0.729826	0.364913	+	0.931042i	$0.381099\pi$
$318$	0	0
$319$	−3469.45	−0.608940
$320$	0	0
$321$	6168.09	1.07249
$322$	0	0
$323$	−2375.60	−0.409231
$324$	0	0
$325$	21902.0	3.73817
$326$	0	0
$327$	−1101.81	−0.186330
$328$	0	0
$329$	6788.98	1.13766
$330$	0	0
$331$	−10126.2	−1.68152	−0.840761	−	0.541407i	$-0.817892\pi$
$331$	−10126.2	−1.68152	−0.840761	+	0.541407i	$0.817892\pi$
$332$	0	0
$333$	27.6534	0.00455074
$334$	0	0
$335$	−8575.75	−1.39864
$336$	0	0
$337$	−2869.47	−0.463828	−0.231914	−	0.972736i	$-0.574499\pi$
$337$	−2869.47	−0.463828	−0.231914	+	0.972736i	$0.574499\pi$
$338$	0	0
$339$	7875.14	1.26171
$340$	0	0
$341$	728.893	0.115753
$342$	0	0
$343$	5098.46	0.802598
$344$	0	0
$345$	−14279.6	−2.22838
$346$	0	0
$347$	6322.40	0.978110	0.489055	−	0.872253i	$-0.337342\pi$
$347$	6322.40	0.978110	0.489055	+	0.872253i	$0.337342\pi$
$348$	0	0
$349$	4459.97	0.684060	0.342030	−	0.939689i	$-0.388886\pi$
$349$	4459.97	0.684060	0.342030	+	0.939689i	$0.388886\pi$
$350$	0	0
$351$	11139.0	1.69390
$352$	0	0
$353$	−2585.23	−0.389795	−0.194898	−	0.980824i	$-0.562437\pi$
$353$	−2585.23	−0.389795	−0.194898	+	0.980824i	$0.562437\pi$
$354$	0	0
$355$	−9730.50	−1.45476
$356$	0	0
$357$	−1837.42	−0.272399
$358$	0	0
$359$	3840.82	0.564654	0.282327	−	0.959318i	$-0.408894\pi$
$359$	3840.82	0.564654	0.282327	+	0.959318i	$0.408894\pi$
$360$	0	0
$361$	12668.5	1.84699
$362$	0	0
$363$	5329.49	0.770594
$364$	0	0
$365$	−12809.0	−1.83685
$366$	0	0
$367$	−9773.07	−1.39006	−0.695028	−	0.718983i	$-0.744608\pi$
$367$	−9773.07	−1.39006	−0.695028	+	0.718983i	$0.744608\pi$
$368$	0	0
$369$	−217.034	−0.0306188
$370$	0	0
$371$	4502.60	0.630089
$372$	0	0
$373$	−11983.1	−1.66344	−0.831719	−	0.555196i	$-0.812643\pi$
$373$	−11983.1	−1.66344	−0.831719	+	0.555196i	$0.812643\pi$
$374$	0	0
$375$	−15910.5	−2.19097
$376$	0	0
$377$	−15924.5	−2.17548
$378$	0	0
$379$	−7730.36	−1.04771	−0.523855	−	0.851808i	$-0.675507\pi$
$379$	−7730.36	−1.04771	−0.523855	+	0.851808i	$0.675507\pi$
$380$	0	0
$381$	−3380.87	−0.454612
$382$	0	0
$383$	−2206.48	−0.294375	−0.147188	−	0.989109i	$-0.547022\pi$
$383$	−2206.48	−0.294375	−0.147188	+	0.989109i	$0.547022\pi$
$384$	0	0
$385$	−7239.33	−0.958313
$386$	0	0
$387$	158.853	0.0208655
$388$	0	0
$389$	−2565.02	−0.334324	−0.167162	−	0.985929i	$-0.553460\pi$
$389$	−2565.02	−0.334324	−0.167162	+	0.985929i	$0.553460\pi$
$390$	0	0
$391$	−2353.88	−0.304453
$392$	0	0
$393$	−5579.45	−0.716148
$394$	0	0
$395$	−2429.33	−0.309451
$396$	0	0
$397$	−7974.54	−1.00814	−0.504069	−	0.863664i	$-0.668164\pi$
$397$	−7974.54	−1.00814	−0.504069	+	0.863664i	$0.668164\pi$
$398$	0	0
$399$	15103.7	1.89506
$400$	0	0
$401$	9035.21	1.12518	0.562590	−	0.826736i	$-0.309805\pi$
$401$	9035.21	1.12518	0.562590	+	0.826736i	$0.309805\pi$
$402$	0	0
$403$	3345.57	0.413535
$404$	0	0
$405$	−14272.3	−1.75110
$406$	0	0
$407$	−682.554	−0.0831276
$408$	0	0
$409$	−10355.1	−1.25190	−0.625950	−	0.779863i	$-0.715289\pi$
$409$	−10355.1	−1.25190	−0.625950	+	0.779863i	$0.715289\pi$
$410$	0	0
$411$	9038.54	1.08476
$412$	0	0
$413$	−10290.3	−1.22603
$414$	0	0
$415$	9353.80	1.10641
$416$	0	0
$417$	5090.70	0.597824
$418$	0	0
$419$	11297.2	1.31720	0.658598	−	0.752495i	$-0.271150\pi$
$419$	11297.2	1.31720	0.658598	+	0.752495i	$0.271150\pi$
$420$	0	0
$421$	−6109.03	−0.707211	−0.353605	−	0.935395i	$-0.615044\pi$
$421$	−6109.03	−0.707211	−0.353605	+	0.935395i	$0.615044\pi$
$422$	0	0
$423$	223.018	0.0256348
$424$	0	0
$425$	−4747.71	−0.541877
$426$	0	0
$427$	−8062.02	−0.913697
$428$	0	0
$429$	−6872.77	−0.773475
$430$	0	0
$431$	−12011.6	−1.34240	−0.671202	−	0.741274i	$-0.734222\pi$
$431$	−12011.6	−1.34240	−0.671202	+	0.741274i	$0.734222\pi$
$432$	0	0
$433$	10677.2	1.18501	0.592507	−	0.805565i	$-0.298138\pi$
$433$	10677.2	1.18501	0.592507	+	0.805565i	$0.298138\pi$
$434$	0	0
$435$	20941.1	2.30816
$436$	0	0
$437$	19349.1	2.11806
$438$	0	0
$439$	6934.54	0.753912	0.376956	−	0.926231i	$-0.376971\pi$
$439$	6934.54	0.753912	0.376956	+	0.926231i	$0.376971\pi$
$440$	0	0
$441$	−69.9516	−0.00755335
$442$	0	0
$443$	9431.64	1.01154	0.505768	−	0.862669i	$-0.331209\pi$
$443$	9431.64	1.01154	0.505768	+	0.862669i	$0.331209\pi$
$444$	0	0
$445$	27070.4	2.88373
$446$	0	0
$447$	−6694.18	−0.708331
$448$	0	0
$449$	−7280.77	−0.765258	−0.382629	−	0.923902i	$-0.624981\pi$
$449$	−7280.77	−0.765258	−0.382629	+	0.923902i	$0.624981\pi$
$450$	0	0
$451$	5356.94	0.559309
$452$	0	0
$453$	−8034.48	−0.833318
$454$	0	0
$455$	−33228.0	−3.42364
$456$	0	0
$457$	−391.607	−0.0400845	−0.0200422	−	0.999799i	$-0.506380\pi$
$457$	−391.607	−0.0400845	−0.0200422	+	0.999799i	$0.506380\pi$
$458$	0	0
$459$	−2414.62	−0.245544
$460$	0	0
$461$	16052.7	1.62179	0.810897	−	0.585190i	$-0.198980\pi$
$461$	16052.7	1.62179	0.810897	+	0.585190i	$0.198980\pi$
$462$	0	0
$463$	9109.91	0.914413	0.457207	−	0.889360i	$-0.348850\pi$
$463$	9109.91	0.914413	0.457207	+	0.889360i	$0.348850\pi$
$464$	0	0
$465$	−4399.50	−0.438757
$466$	0	0
$467$	−15478.7	−1.53376	−0.766882	−	0.641788i	$-0.778193\pi$
$467$	−15478.7	−1.53376	−0.766882	+	0.641788i	$0.778193\pi$
$468$	0	0
$469$	8987.71	0.884891
$470$	0	0
$471$	−17488.6	−1.71090
$472$	0	0
$473$	−3920.88	−0.381146
$474$	0	0
$475$	39026.5	3.76980
$476$	0	0
$477$	147.911	0.0141978
$478$	0	0
$479$	−16337.4	−1.55840	−0.779201	−	0.626774i	$-0.784375\pi$
$479$	−16337.4	−1.55840	−0.779201	+	0.626774i	$0.784375\pi$
$480$	0	0
$481$	−3132.88	−0.296979
$482$	0	0
$483$	14965.6	1.40985
$484$	0	0
$485$	−22461.8	−2.10297
$486$	0	0
$487$	−5914.12	−0.550296	−0.275148	−	0.961402i	$-0.588727\pi$
$487$	−5914.12	−0.550296	−0.275148	+	0.961402i	$0.588727\pi$
$488$	0	0
$489$	9107.52	0.842242
$490$	0	0
$491$	2100.85	0.193096	0.0965478	−	0.995328i	$-0.469220\pi$
$491$	2100.85	0.193096	0.0965478	+	0.995328i	$0.469220\pi$
$492$	0	0
$493$	3451.98	0.315353
$494$	0	0
$495$	−237.813	−0.0215937
$496$	0	0
$497$	10197.9	0.920403
$498$	0	0
$499$	−17306.7	−1.55261	−0.776305	−	0.630357i	$-0.782909\pi$
$499$	−17306.7	−1.55261	−0.776305	+	0.630357i	$0.782909\pi$
$500$	0	0
$501$	3340.01	0.297846
$502$	0	0
$503$	7147.42	0.633574	0.316787	−	0.948497i	$-0.397396\pi$
$503$	7147.42	0.633574	0.316787	+	0.948497i	$0.397396\pi$
$504$	0	0
$505$	486.363	0.0428572
$506$	0	0
$507$	−20276.9	−1.77619
$508$	0	0
$509$	−7892.80	−0.687313	−0.343656	−	0.939095i	$-0.611666\pi$
$509$	−7892.80	−0.687313	−0.343656	+	0.939095i	$0.611666\pi$
$510$	0	0
$511$	13424.3	1.16214
$512$	0	0
$513$	19848.3	1.70824
$514$	0	0
$515$	16038.6	1.37232
$516$	0	0
$517$	−5504.65	−0.468267
$518$	0	0
$519$	8913.88	0.753904
$520$	0	0
$521$	−4246.71	−0.357105	−0.178552	−	0.983930i	$-0.557141\pi$
$521$	−4246.71	−0.357105	−0.178552	+	0.983930i	$0.557141\pi$
$522$	0	0
$523$	9208.75	0.769925	0.384962	−	0.922932i	$-0.374214\pi$
$523$	9208.75	0.769925	0.384962	+	0.922932i	$0.374214\pi$
$524$	0	0
$525$	30185.2	2.50932
$526$	0	0
$527$	−725.222	−0.0599452
$528$	0	0
$529$	7005.20	0.575754
$530$	0	0
$531$	−338.037	−0.0276262
$532$	0	0
$533$	24588.0	1.99817
$534$	0	0
$535$	−24179.5	−1.95397
$536$	0	0
$537$	−14825.9	−1.19140
$538$	0	0
$539$	1726.58	0.137976
$540$	0	0
$541$	19505.1	1.55007	0.775036	−	0.631917i	$-0.217731\pi$
$541$	19505.1	1.55007	0.775036	+	0.631917i	$0.217731\pi$
$542$	0	0
$543$	8297.67	0.655778
$544$	0	0
$545$	4319.20	0.339475
$546$	0	0
$547$	10605.7	0.829005	0.414503	−	0.910048i	$-0.363956\pi$
$547$	10605.7	0.829005	0.414503	+	0.910048i	$0.363956\pi$
$548$	0	0
$549$	−264.838	−0.0205884
$550$	0	0
$551$	−28375.4	−2.19389
$552$	0	0
$553$	2546.04	0.195784
$554$	0	0
$555$	4119.80	0.315092
$556$	0	0
$557$	5623.77	0.427804	0.213902	−	0.976855i	$-0.431383\pi$
$557$	5623.77	0.427804	0.213902	+	0.976855i	$0.431383\pi$
$558$	0	0
$559$	−17996.6	−1.36167
$560$	0	0
$561$	1489.82	0.112121
$562$	0	0
$563$	−16606.1	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$563$	−16606.1	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$564$	0	0
$565$	−30871.4	−2.29871
$566$	0	0
$567$	14957.9	1.10789
$568$	0	0
$569$	−12583.8	−0.927135	−0.463568	−	0.886062i	$-0.653431\pi$
$569$	−12583.8	−0.927135	−0.463568	+	0.886062i	$0.653431\pi$
$570$	0	0
$571$	13239.1	0.970299	0.485149	−	0.874431i	$-0.338765\pi$
$571$	13239.1	0.970299	0.485149	+	0.874431i	$0.338765\pi$
$572$	0	0
$573$	1075.60	0.0784186
$574$	0	0
$575$	38669.8	2.80459
$576$	0	0
$577$	−13563.1	−0.978580	−0.489290	−	0.872121i	$-0.662744\pi$
$577$	−13563.1	−0.978580	−0.489290	+	0.872121i	$0.662744\pi$
$578$	0	0
$579$	−13921.4	−0.999227
$580$	0	0
$581$	−9803.14	−0.700005
$582$	0	0
$583$	−3650.80	−0.259349
$584$	0	0
$585$	−1091.54	−0.0771449
$586$	0	0
$587$	−22754.1	−1.59993	−0.799967	−	0.600044i	$-0.795150\pi$
$587$	−22754.1	−1.59993	−0.799967	+	0.600044i	$0.795150\pi$
$588$	0	0
$589$	5961.36	0.417035
$590$	0	0
$591$	9909.90	0.689744
$592$	0	0
$593$	14368.2	0.994991	0.497496	−	0.867466i	$-0.334253\pi$
$593$	14368.2	0.994991	0.497496	+	0.867466i	$0.334253\pi$
$594$	0	0
$595$	7202.87	0.496284
$596$	0	0
$597$	10844.1	0.743414
$598$	0	0
$599$	26564.5	1.81201	0.906007	−	0.423262i	$-0.139115\pi$
$599$	26564.5	1.81201	0.906007	+	0.423262i	$0.139115\pi$
$600$	0	0
$601$	10021.8	0.680194	0.340097	−	0.940390i	$-0.389540\pi$
$601$	10021.8	0.680194	0.340097	+	0.940390i	$0.389540\pi$
$602$	0	0
$603$	295.247	0.0199393
$604$	0	0
$605$	−20892.1	−1.40394
$606$	0	0
$607$	−25176.7	−1.68351	−0.841756	−	0.539857i	$-0.818478\pi$
$607$	−25176.7	−1.68351	−0.841756	+	0.539857i	$0.818478\pi$
$608$	0	0
$609$	−21947.1	−1.46033
$610$	0	0
$611$	−25266.0	−1.67292
$612$	0	0
$613$	8018.82	0.528347	0.264174	−	0.964475i	$-0.414901\pi$
$613$	8018.82	0.528347	0.264174	+	0.964475i	$0.414901\pi$
$614$	0	0
$615$	−32333.8	−2.12004
$616$	0	0
$617$	−3141.96	−0.205009	−0.102504	−	0.994733i	$-0.532686\pi$
$617$	−3141.96	−0.205009	−0.102504	+	0.994733i	$0.532686\pi$
$618$	0	0
$619$	−103.440	−0.00671663	−0.00335832	−	0.999994i	$-0.501069\pi$
$619$	−103.440	−0.00671663	−0.00335832	+	0.999994i	$0.501069\pi$
$620$	0	0
$621$	19666.9	1.27086
$622$	0	0
$623$	−28370.8	−1.82448
$624$	0	0
$625$	27461.1	1.75751
$626$	0	0
$627$	−12246.4	−0.780021
$628$	0	0
$629$	679.116	0.0430495
$630$	0	0
$631$	1125.52	0.0710085	0.0355043	−	0.999370i	$-0.488696\pi$
$631$	1125.52	0.0710085	0.0355043	+	0.999370i	$0.488696\pi$
$632$	0	0
$633$	19636.4	1.23298
$634$	0	0
$635$	13253.4	0.828257
$636$	0	0
$637$	7924.88	0.492928
$638$	0	0
$639$	335.003	0.0207395
$640$	0	0
$641$	−27474.5	−1.69295	−0.846473	−	0.532432i	$-0.821278\pi$
$641$	−27474.5	−1.69295	−0.846473	+	0.532432i	$0.821278\pi$
$642$	0	0
$643$	−27502.5	−1.68677	−0.843385	−	0.537310i	$-0.819441\pi$
$643$	−27502.5	−1.68677	−0.843385	+	0.537310i	$0.819441\pi$
$644$	0	0
$645$	23665.9	1.44472
$646$	0	0
$647$	19119.2	1.16175	0.580876	−	0.813992i	$-0.302710\pi$
$647$	19119.2	1.16175	0.580876	+	0.813992i	$0.302710\pi$
$648$	0	0
$649$	8343.58	0.504644
$650$	0	0
$651$	4610.85	0.277593
$652$	0	0
$653$	−323.964	−0.0194145	−0.00970726	−	0.999953i	$-0.503090\pi$
$653$	−323.964	−0.0194145	−0.00970726	+	0.999953i	$0.503090\pi$
$654$	0	0
$655$	21872.0	1.30475
$656$	0	0
$657$	440.989	0.0261866
$658$	0	0
$659$	9944.76	0.587850	0.293925	−	0.955829i	$-0.405038\pi$
$659$	9944.76	0.587850	0.293925	+	0.955829i	$0.405038\pi$
$660$	0	0
$661$	7481.89	0.440260	0.220130	−	0.975471i	$-0.429352\pi$
$661$	7481.89	0.440260	0.220130	+	0.975471i	$0.429352\pi$
$662$	0	0
$663$	6838.16	0.400561
$664$	0	0
$665$	−59208.0	−3.45261
$666$	0	0
$667$	−28116.1	−1.63217
$668$	0	0
$669$	17018.7	0.983532
$670$	0	0
$671$	6536.86	0.376084
$672$	0	0
$673$	196.143	0.0112344	0.00561719	−	0.999984i	$-0.498212\pi$
$673$	196.143	0.0112344	0.00561719	+	0.999984i	$0.498212\pi$
$674$	0	0
$675$	39667.6	2.26193
$676$	0	0
$677$	−19213.7	−1.09076	−0.545379	−	0.838190i	$-0.683614\pi$
$677$	−19213.7	−1.09076	−0.545379	+	0.838190i	$0.683614\pi$
$678$	0	0
$679$	23540.9	1.33051
$680$	0	0
$681$	−6492.26	−0.365322
$682$	0	0
$683$	−26093.2	−1.46183	−0.730913	−	0.682471i	$-0.760905\pi$
$683$	−26093.2	−1.46183	−0.730913	+	0.682471i	$0.760905\pi$
$684$	0	0
$685$	−35432.0	−1.97633
$686$	0	0
$687$	−2412.16	−0.133959
$688$	0	0
$689$	−16756.9	−0.926543
$690$	0	0
$691$	−15862.5	−0.873283	−0.436642	−	0.899636i	$-0.643832\pi$
$691$	−15862.5	−0.873283	−0.436642	+	0.899636i	$0.643832\pi$
$692$	0	0
$693$	249.237	0.0136619
$694$	0	0
$695$	−19956.1	−1.08918
$696$	0	0
$697$	−5329.95	−0.289651
$698$	0	0
$699$	3070.98	0.166173
$700$	0	0
$701$	15929.2	0.858256	0.429128	−	0.903244i	$-0.358821\pi$
$701$	15929.2	0.858256	0.429128	+	0.903244i	$0.358821\pi$
$702$	0	0
$703$	−5582.37	−0.299492
$704$	0	0
$705$	33225.3	1.77495
$706$	0	0
$707$	−509.727	−0.0271150
$708$	0	0
$709$	−21143.4	−1.11997	−0.559984	−	0.828504i	$-0.689193\pi$
$709$	−21143.4	−1.11997	−0.559984	+	0.828504i	$0.689193\pi$
$710$	0	0
$711$	83.6375	0.00441160
$712$	0	0
$713$	5906.87	0.310258
$714$	0	0
$715$	26942.0	1.40919
$716$	0	0
$717$	−27857.9	−1.45101
$718$	0	0
$719$	33525.4	1.73892	0.869462	−	0.493999i	$-0.164465\pi$
$719$	33525.4	1.73892	0.869462	+	0.493999i	$0.164465\pi$
$720$	0	0
$721$	−16809.1	−0.868241
$722$	0	0
$723$	−20155.5	−1.03678
$724$	0	0
$725$	−56709.3	−2.90501
$726$	0	0
$727$	−2772.68	−0.141448	−0.0707242	−	0.997496i	$-0.522531\pi$
$727$	−2772.68	−0.141448	−0.0707242	+	0.997496i	$0.522531\pi$
$728$	0	0
$729$	20161.4	1.02431
$730$	0	0
$731$	3901.13	0.197385
$732$	0	0
$733$	−31779.0	−1.60134	−0.800671	−	0.599105i	$-0.795523\pi$
$733$	−31779.0	−1.60134	−0.800671	+	0.599105i	$0.795523\pi$
$734$	0	0
$735$	−10421.4	−0.522992
$736$	0	0
$737$	−7287.43	−0.364228
$738$	0	0
$739$	779.798	0.0388164	0.0194082	−	0.999812i	$-0.493822\pi$
$739$	779.798	0.0388164	0.0194082	+	0.999812i	$0.493822\pi$
$740$	0	0
$741$	−56210.0	−2.78668
$742$	0	0
$743$	−9130.73	−0.450840	−0.225420	−	0.974262i	$-0.572375\pi$
$743$	−9130.73	−0.450840	−0.225420	+	0.974262i	$0.572375\pi$
$744$	0	0
$745$	26241.9	1.29051
$746$	0	0
$747$	−322.034	−0.0157732
$748$	0	0
$749$	25341.1	1.23624
$750$	0	0
$751$	−30497.8	−1.48186	−0.740932	−	0.671580i	$-0.765616\pi$
$751$	−30497.8	−1.48186	−0.740932	+	0.671580i	$0.765616\pi$
$752$	0	0
$753$	27057.0	1.30944
$754$	0	0
$755$	31496.0	1.51822
$756$	0	0
$757$	22888.3	1.09893	0.549466	−	0.835516i	$-0.314831\pi$
$757$	22888.3	1.09893	0.549466	+	0.835516i	$0.314831\pi$
$758$	0	0
$759$	−12134.4	−0.580306
$760$	0	0
$761$	1765.87	0.0841164	0.0420582	−	0.999115i	$-0.486609\pi$
$761$	1765.87	0.0841164	0.0420582	+	0.999115i	$0.486609\pi$
$762$	0	0
$763$	−4526.68	−0.214780
$764$	0	0
$765$	236.615	0.0111828
$766$	0	0
$767$	38296.5	1.80287
$768$	0	0
$769$	19453.3	0.912228	0.456114	−	0.889921i	$-0.349241\pi$
$769$	19453.3	0.912228	0.456114	+	0.889921i	$0.349241\pi$
$770$	0	0
$771$	−19548.1	−0.913108
$772$	0	0
$773$	24322.6	1.13173	0.565863	−	0.824499i	$-0.308543\pi$
$773$	24322.6	1.13173	0.565863	+	0.824499i	$0.308543\pi$
$774$	0	0
$775$	11914.0	0.552210
$776$	0	0
$777$	−4317.71	−0.199353
$778$	0	0
$779$	43812.5	2.01508
$780$	0	0
$781$	−8268.71	−0.378844
$782$	0	0
$783$	−28841.6	−1.31636
$784$	0	0
$785$	68557.2	3.11708
$786$	0	0
$787$	38992.7	1.76612	0.883062	−	0.469256i	$-0.155478\pi$
$787$	38992.7	1.76612	0.883062	+	0.469256i	$0.155478\pi$
$788$	0	0
$789$	−26223.7	−1.18325
$790$	0	0
$791$	32354.4	1.45435
$792$	0	0
$793$	30003.7	1.34359
$794$	0	0
$795$	22035.7	0.983053
$796$	0	0
$797$	1275.33	0.0566806	0.0283403	−	0.999598i	$-0.490978\pi$
$797$	1275.33	0.0566806	0.0283403	+	0.999598i	$0.490978\pi$
$798$	0	0
$799$	5476.92	0.242503
$800$	0	0
$801$	−931.982	−0.0411111
$802$	0	0
$803$	−10884.7	−0.478347
$804$	0	0
$805$	−58666.8	−2.56861
$806$	0	0
$807$	−5001.65	−0.218174
$808$	0	0
$809$	−21147.8	−0.919056	−0.459528	−	0.888163i	$-0.651981\pi$
$809$	−21147.8	−0.919056	−0.459528	+	0.888163i	$0.651981\pi$
$810$	0	0
$811$	−14387.6	−0.622955	−0.311477	−	0.950254i	$-0.600824\pi$
$811$	−14387.6	−0.622955	−0.311477	+	0.950254i	$0.600824\pi$
$812$	0	0
$813$	21626.6	0.932939
$814$	0	0
$815$	−35702.4	−1.53448
$816$	0	0
$817$	−32067.5	−1.37319
$818$	0	0
$819$	1143.98	0.0488081
$820$	0	0
$821$	18679.9	0.794070	0.397035	−	0.917803i	$-0.370039\pi$
$821$	18679.9	0.794070	0.397035	+	0.917803i	$0.370039\pi$
$822$	0	0
$823$	19564.0	0.828623	0.414311	−	0.910135i	$-0.364022\pi$
$823$	19564.0	0.828623	0.414311	+	0.910135i	$0.364022\pi$
$824$	0	0
$825$	−24474.8	−1.03285
$826$	0	0
$827$	23623.7	0.993323	0.496661	−	0.867944i	$-0.334559\pi$
$827$	23623.7	0.993323	0.496661	+	0.867944i	$0.334559\pi$
$828$	0	0
$829$	34397.8	1.44111	0.720557	−	0.693396i	$-0.243886\pi$
$829$	34397.8	1.44111	0.720557	+	0.693396i	$0.243886\pi$
$830$	0	0
$831$	41691.1	1.74037
$832$	0	0
$833$	−1717.88	−0.0714539
$834$	0	0
$835$	−13093.2	−0.542645
$836$	0	0
$837$	6059.29	0.250227
$838$	0	0
$839$	11412.1	0.469593	0.234797	−	0.972045i	$-0.424558\pi$
$839$	11412.1	0.469593	0.234797	+	0.972045i	$0.424558\pi$
$840$	0	0
$841$	16843.3	0.690611
$842$	0	0
$843$	37042.7	1.51343
$844$	0	0
$845$	79487.6	3.23604
$846$	0	0
$847$	21895.8	0.888250
$848$	0	0
$849$	−2017.89	−0.0815711
$850$	0	0
$851$	−5531.35	−0.222811
$852$	0	0
$853$	15920.1	0.639030	0.319515	−	0.947581i	$-0.396480\pi$
$853$	15920.1	0.639030	0.319515	+	0.947581i	$0.396480\pi$
$854$	0	0
$855$	−1944.99	−0.0777978
$856$	0	0
$857$	173.324	0.00690854	0.00345427	−	0.999994i	$-0.498900\pi$
$857$	173.324	0.00690854	0.00345427	+	0.999994i	$0.498900\pi$
$858$	0	0
$859$	−9092.59	−0.361159	−0.180579	−	0.983560i	$-0.557797\pi$
$859$	−9092.59	−0.361159	−0.180579	+	0.983560i	$0.557797\pi$
$860$	0	0
$861$	33887.0	1.34131
$862$	0	0
$863$	1032.20	0.0407145	0.0203572	−	0.999793i	$-0.493520\pi$
$863$	1032.20	0.0407145	0.0203572	+	0.999793i	$0.493520\pi$
$864$	0	0
$865$	−34943.4	−1.37354
$866$	0	0
$867$	−1482.31	−0.0580646
$868$	0	0
$869$	−2064.38	−0.0805861
$870$	0	0
$871$	−33448.8	−1.30123
$872$	0	0
$873$	773.319	0.0299804
$874$	0	0
$875$	−65366.9	−2.52549
$876$	0	0
$877$	26483.1	1.01969	0.509847	−	0.860265i	$-0.329702\pi$
$877$	26483.1	1.01969	0.509847	+	0.860265i	$0.329702\pi$
$878$	0	0
$879$	−11677.2	−0.448080
$880$	0	0
$881$	11437.8	0.437402	0.218701	−	0.975792i	$-0.429818\pi$
$881$	11437.8	0.437402	0.218701	+	0.975792i	$0.429818\pi$
$882$	0	0
$883$	−31024.5	−1.18240	−0.591200	−	0.806525i	$-0.701345\pi$
$883$	−31024.5	−1.18240	−0.591200	+	0.806525i	$0.701345\pi$
$884$	0	0
$885$	−50360.7	−1.91283
$886$	0	0
$887$	−8871.64	−0.335829	−0.167915	−	0.985802i	$-0.553703\pi$
$887$	−8871.64	−0.335829	−0.167915	+	0.985802i	$0.553703\pi$
$888$	0	0
$889$	−13890.0	−0.524023
$890$	0	0
$891$	−12128.2	−0.456016
$892$	0	0
$893$	−45020.6	−1.68707
$894$	0	0
$895$	58118.9	2.17062
$896$	0	0
$897$	−55696.3	−2.07318
$898$	0	0
$899$	−8662.45	−0.321367
$900$	0	0
$901$	3632.41	0.134310
$902$	0	0
$903$	−24802.8	−0.914047
$904$	0	0
$905$	−32527.7	−1.19476
$906$	0	0
$907$	19665.1	0.719921	0.359961	−	0.932967i	$-0.382790\pi$
$907$	19665.1	0.719921	0.359961	+	0.932967i	$0.382790\pi$
$908$	0	0
$909$	−16.7446	−0.000610982 0
$910$	0	0
$911$	33849.7	1.23105	0.615527	−	0.788116i	$-0.288943\pi$
$911$	33849.7	1.23105	0.615527	+	0.788116i	$0.288943\pi$
$912$	0	0
$913$	7948.60	0.288127
$914$	0	0
$915$	−39455.6	−1.42553
$916$	0	0
$917$	−22922.7	−0.825491
$918$	0	0
$919$	39175.1	1.40617	0.703083	−	0.711108i	$-0.251806\pi$
$919$	39175.1	1.40617	0.703083	+	0.711108i	$0.251806\pi$
$920$	0	0
$921$	42475.4	1.51967
$922$	0	0
$923$	−37952.8	−1.35345
$924$	0	0
$925$	−11156.6	−0.396568
$926$	0	0
$927$	−552.179	−0.0195641
$928$	0	0
$929$	25575.4	0.903230	0.451615	−	0.892213i	$-0.350848\pi$
$929$	25575.4	0.903230	0.451615	+	0.892213i	$0.350848\pi$
$930$	0	0
$931$	14121.1	0.497100
$932$	0	0
$933$	−4344.07	−0.152431
$934$	0	0
$935$	−5840.24	−0.204274
$936$	0	0
$937$	−2451.58	−0.0854745	−0.0427373	−	0.999086i	$-0.513608\pi$
$937$	−2451.58	−0.0854745	−0.0427373	+	0.999086i	$0.513608\pi$
$938$	0	0
$939$	9073.76	0.315347
$940$	0	0
$941$	3901.72	0.135167	0.0675837	−	0.997714i	$-0.478471\pi$
$941$	3901.72	0.135167	0.0675837	+	0.997714i	$0.478471\pi$
$942$	0	0
$943$	43412.1	1.49914
$944$	0	0
$945$	−60180.6	−2.07161
$946$	0	0
$947$	14785.0	0.507335	0.253668	−	0.967291i	$-0.418363\pi$
$947$	14785.0	0.507335	0.253668	+	0.967291i	$0.418363\pi$
$948$	0	0
$949$	−49960.0	−1.70893
$950$	0	0
$951$	−21127.6	−0.720410
$952$	0	0
$953$	7120.38	0.242027	0.121014	−	0.992651i	$-0.461386\pi$
$953$	7120.38	0.242027	0.121014	+	0.992651i	$0.461386\pi$
$954$	0	0
$955$	−4216.46	−0.142871
$956$	0	0
$957$	17795.2	0.601084
$958$	0	0
$959$	37134.1	1.25039
$960$	0	0
$961$	−27971.1	−0.938912
$962$	0	0
$963$	832.457	0.0278562
$964$	0	0
$965$	54573.2	1.82049
$966$	0	0
$967$	−18022.9	−0.599355	−0.299677	−	0.954041i	$-0.596879\pi$
$967$	−18022.9	−0.599355	−0.299677	+	0.954041i	$0.596879\pi$
$968$	0	0
$969$	12184.7	0.403951
$970$	0	0
$971$	30934.5	1.02238	0.511192	−	0.859467i	$-0.329204\pi$
$971$	30934.5	1.02238	0.511192	+	0.859467i	$0.329204\pi$
$972$	0	0
$973$	20914.7	0.689102
$974$	0	0
$975$	−112338.	−3.68993
$976$	0	0
$977$	9224.43	0.302063	0.151032	−	0.988529i	$-0.451740\pi$
$977$	9224.43	0.302063	0.151032	+	0.988529i	$0.451740\pi$
$978$	0	0
$979$	23003.6	0.750970
$980$	0	0
$981$	−148.702	−0.00483964
$982$	0	0
$983$	13784.1	0.447248	0.223624	−	0.974675i	$-0.428211\pi$
$983$	13784.1	0.447248	0.223624	+	0.974675i	$0.428211\pi$
$984$	0	0
$985$	−38847.9	−1.25665
$986$	0	0
$987$	−34821.4	−1.12298
$988$	0	0
$989$	−31774.4	−1.02160
$990$	0	0
$991$	22391.2	0.717740	0.358870	−	0.933388i	$-0.383162\pi$
$991$	22391.2	0.717740	0.358870	+	0.933388i	$0.383162\pi$
$992$	0	0
$993$	51938.1	1.65983
$994$	0	0
$995$	−42509.9	−1.35443
$996$	0	0
$997$	36575.1	1.16183	0.580916	−	0.813963i	$-0.302694\pi$
$997$	36575.1	1.16183	0.580916	+	0.813963i	$0.302694\pi$
$998$	0	0
$999$	−5674.07	−0.179699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.a.d.1.2	✓	4
3.2	odd	2		1224.4.a.l.1.1		4
4.3	odd	2		272.4.a.k.1.3		4
8.3	odd	2		1088.4.a.bb.1.2		4
8.5	even	2		1088.4.a.be.1.3		4
12.11	even	2		2448.4.a.bq.1.1		4
17.16	even	2		2312.4.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.d.1.2	✓	4	1.1	even	1	trivial
272.4.a.k.1.3		4	4.3	odd	2
1088.4.a.bb.1.2		4	8.3	odd	2
1088.4.a.be.1.3		4	8.5	even	2
1224.4.a.l.1.1		4	3.2	odd	2
2312.4.a.e.1.3		4	17.16	even	2
2448.4.a.bq.1.1		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)

\(f(q)\)	\(=\)	\(q-5.12911 q^{3} +20.1066 q^{5} -21.0725 q^{7} -0.692234 q^{9} +O(q^{10})\)	\(q-5.12911 q^{3} +20.1066 q^{5} -21.0725 q^{7} -0.692234 q^{9} +17.0861 q^{11} +78.4238 q^{13} -103.129 q^{15} -17.0000 q^{17} +139.741 q^{19} +108.083 q^{21} +138.464 q^{23} +279.277 q^{25} +142.037 q^{27} -203.057 q^{29} +42.6601 q^{31} -87.6363 q^{33} -423.698 q^{35} -39.9480 q^{37} -402.244 q^{39} +313.527 q^{41} -229.478 q^{43} -13.9185 q^{45} -322.172 q^{47} +101.052 q^{49} +87.1949 q^{51} -213.671 q^{53} +343.543 q^{55} -716.747 q^{57} +488.327 q^{59} +382.584 q^{61} +14.5871 q^{63} +1576.84 q^{65} -426.513 q^{67} -710.196 q^{69} -483.945 q^{71} -637.051 q^{73} -1432.44 q^{75} -360.047 q^{77} -120.822 q^{79} -709.830 q^{81} +465.209 q^{83} -341.813 q^{85} +1041.50 q^{87} +1346.34 q^{89} -1652.59 q^{91} -218.808 q^{93} +2809.72 q^{95} -1117.14 q^{97} -11.8276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 8 q^{5} - 22 q^{7} + 100 q^{9}+O(q^{10}) \) 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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