LMFDB - Embedded newform 207.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.23607 q^{2} +9.94427 q^{4} +10.7639 q^{5} +10.6525 q^{7} +8.23607 q^{8} +45.5967 q^{10} +52.3607 q^{11} -79.0820 q^{13} +45.1246 q^{14} -44.6656 q^{16} +77.2361 q^{17} +50.7902 q^{19} +107.039 q^{20} +221.803 q^{22} +23.0000 q^{23} -9.13777 q^{25} -334.997 q^{26} +105.931 q^{28} -12.7477 q^{29} -12.4133 q^{31} -255.095 q^{32} +327.177 q^{34} +114.663 q^{35} -73.1084 q^{37} +215.151 q^{38} +88.6525 q^{40} +38.8916 q^{41} +171.787 q^{43} +520.689 q^{44} +97.4296 q^{46} -614.545 q^{47} -229.525 q^{49} -38.7082 q^{50} -786.413 q^{52} -269.597 q^{53} +563.607 q^{55} +87.7345 q^{56} -54.0000 q^{58} +534.768 q^{59} -838.604 q^{61} -52.5836 q^{62} -723.276 q^{64} -851.234 q^{65} -448.180 q^{67} +768.056 q^{68} +485.718 q^{70} -628.604 q^{71} +925.266 q^{73} -309.692 q^{74} +505.072 q^{76} +557.771 q^{77} -963.479 q^{79} -480.778 q^{80} +164.748 q^{82} -133.358 q^{83} +831.364 q^{85} +727.702 q^{86} +431.246 q^{88} +778.581 q^{89} -842.420 q^{91} +228.718 q^{92} -2603.25 q^{94} +546.703 q^{95} +1603.57 q^{97} -972.282 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{4} + 26 q^{5} - 10 q^{7} + 12 q^{8} + 42 q^{10} + 60 q^{11} - 24 q^{13} + 50 q^{14} + 18 q^{16} + 150 q^{17} - 46 q^{19} - 14 q^{20} + 220 q^{22} + 46 q^{23} + 98 q^{25} - 348 q^{26}+ \cdots - 992 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 2 * q^4 + 26 * q^5 - 10 * q^7 + 12 * q^8 + 42 * q^10 + 60 * q^11 - 24 * q^13 + 50 * q^14 + 18 * q^16 + 150 * q^17 - 46 * q^19 - 14 * q^20 + 220 * q^22 + 46 * q^23 + 98 * q^25 - 348 * q^26 + 270 * q^28 + 216 * q^29 + 324 * q^31 - 300 * q^32 + 310 * q^34 - 200 * q^35 + 140 * q^37 + 238 * q^38 + 146 * q^40 + 364 * q^41 - 126 * q^43 + 460 * q^44 + 92 * q^46 - 120 * q^47 - 146 * q^49 - 64 * q^50 - 1224 * q^52 - 490 * q^53 + 680 * q^55 + 10 * q^56 - 108 * q^58 + 32 * q^59 - 908 * q^61 - 132 * q^62 - 1214 * q^64 - 12 * q^65 - 874 * q^67 + 190 * q^68 + 560 * q^70 - 488 * q^71 + 652 * q^73 - 360 * q^74 + 1274 * q^76 + 400 * q^77 - 1198 * q^79 + 474 * q^80 + 88 * q^82 + 100 * q^83 + 1940 * q^85 + 798 * q^86 + 460 * q^88 - 102 * q^89 - 1980 * q^91 + 46 * q^92 - 2720 * q^94 - 928 * q^95 + 1624 * q^97 - 992 * q^98

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 $

$2$	4.23607	1.49768	0.748838	−	0.662753i	$-0.230612\pi$
$2$	4.23607	1.49768	0.748838	+	0.662753i	$0.230612\pi$
$3$	0	0
$3$	0	0
$4$	9.94427	1.24303
$5$	10.7639	0.962755	0.481378	−	0.876513i	$-0.340137\pi$
$5$	10.7639	0.962755	0.481378	+	0.876513i	$0.340137\pi$
$6$	0	0
$7$	10.6525	0.575180	0.287590	−	0.957754i	$-0.407146\pi$
$7$	10.6525	0.575180	0.287590	+	0.957754i	$0.407146\pi$
$8$	8.23607	0.363986
$9$	0	0
$10$	45.5967	1.44190
$11$	52.3607	1.43521	0.717606	−	0.696449i	$-0.245238\pi$
$11$	52.3607	1.43521	0.717606	+	0.696449i	$0.245238\pi$
$12$	0	0
$13$	−79.0820	−1.68719	−0.843593	−	0.536984i	$-0.819564\pi$
$13$	−79.0820	−1.68719	−0.843593	+	0.536984i	$0.819564\pi$
$14$	45.1246	0.861433
$15$	0	0
$16$	−44.6656	−0.697900
$17$	77.2361	1.10191	0.550956	−	0.834534i	$-0.314263\pi$
$17$	77.2361	1.10191	0.550956	+	0.834534i	$0.314263\pi$
$18$	0	0
$19$	50.7902	0.613267	0.306634	−	0.951828i	$-0.400797\pi$
$19$	50.7902	0.613267	0.306634	+	0.951828i	$0.400797\pi$
$20$	107.039	1.19674
$21$	0	0
$22$	221.803	2.14948
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−9.13777	−0.0731021
$26$	−334.997	−2.52686
$27$	0	0
$28$	105.931	0.714968
$29$	−12.7477	−0.0816270	−0.0408135	−	0.999167i	$-0.512995\pi$
$29$	−12.7477	−0.0816270	−0.0408135	+	0.999167i	$0.512995\pi$
$30$	0	0
$31$	−12.4133	−0.0719192	−0.0359596	−	0.999353i	$-0.511449\pi$
$31$	−12.4133	−0.0719192	−0.0359596	+	0.999353i	$0.511449\pi$
$32$	−255.095	−1.40922
$33$	0	0
$34$	327.177	1.65031
$35$	114.663	0.553757
$36$	0	0
$37$	−73.1084	−0.324836	−0.162418	−	0.986722i	$-0.551929\pi$
$37$	−73.1084	−0.324836	−0.162418	+	0.986722i	$0.551929\pi$
$38$	215.151	0.918476
$39$	0	0
$40$	88.6525	0.350430
$41$	38.8916	0.148143	0.0740714	−	0.997253i	$-0.476401\pi$
$41$	38.8916	0.148143	0.0740714	+	0.997253i	$0.476401\pi$
$42$	0	0
$43$	171.787	0.609239	0.304620	−	0.952474i	$-0.401471\pi$
$43$	171.787	0.609239	0.304620	+	0.952474i	$0.401471\pi$
$44$	520.689	1.78402
$45$	0	0
$46$	97.4296	0.312287
$47$	−614.545	−1.90725	−0.953623	−	0.301003i	$-0.902679\pi$
$47$	−614.545	−1.90725	−0.953623	+	0.301003i	$0.902679\pi$
$48$	0	0
$49$	−229.525	−0.669168
$50$	−38.7082	−0.109483
$51$	0	0
$52$	−786.413	−2.09723
$53$	−269.597	−0.698716	−0.349358	−	0.936989i	$-0.613600\pi$
$53$	−269.597	−0.698716	−0.349358	+	0.936989i	$0.613600\pi$
$54$	0	0
$55$	563.607	1.38176
$56$	87.7345	0.209357
$57$	0	0
$58$	−54.0000	−0.122251
$59$	534.768	1.18001	0.590007	−	0.807398i	$-0.299125\pi$
$59$	534.768	1.18001	0.590007	+	0.807398i	$0.299125\pi$
$60$	0	0
$61$	−838.604	−1.76020	−0.880100	−	0.474788i	$-0.842525\pi$
$61$	−838.604	−1.76020	−0.880100	+	0.474788i	$0.842525\pi$
$62$	−52.5836	−0.107712
$63$	0	0
$64$	−723.276	−1.41265
$65$	−851.234	−1.62435
$66$	0	0
$67$	−448.180	−0.817223	−0.408612	−	0.912708i	$-0.633987\pi$
$67$	−448.180	−0.817223	−0.408612	+	0.912708i	$0.633987\pi$
$68$	768.056	1.36971
$69$	0	0
$70$	485.718	0.829349
$71$	−628.604	−1.05073	−0.525363	−	0.850878i	$-0.676070\pi$
$71$	−628.604	−1.05073	−0.525363	+	0.850878i	$0.676070\pi$
$72$	0	0
$73$	925.266	1.48348	0.741741	−	0.670686i	$-0.234000\pi$
$73$	925.266	1.48348	0.741741	+	0.670686i	$0.234000\pi$
$74$	−309.692	−0.486499
$75$	0	0
$76$	505.072	0.762312
$77$	557.771	0.825505
$78$	0	0
$79$	−963.479	−1.37215	−0.686075	−	0.727531i	$-0.740668\pi$
$79$	−963.479	−1.37215	−0.686075	+	0.727531i	$0.740668\pi$
$80$	−480.778	−0.671907
$81$	0	0
$82$	164.748	0.221870
$83$	−133.358	−0.176360	−0.0881801	−	0.996105i	$-0.528105\pi$
$83$	−133.358	−0.176360	−0.0881801	+	0.996105i	$0.528105\pi$
$84$	0	0
$85$	831.364	1.06087
$86$	727.702	0.912443
$87$	0	0
$88$	431.246	0.522398
$89$	778.581	0.927297	0.463649	−	0.886019i	$-0.346540\pi$
$89$	778.581	0.927297	0.463649	+	0.886019i	$0.346540\pi$
$90$	0	0
$91$	−842.420	−0.970435
$92$	228.718	0.259191
$93$	0	0
$94$	−2603.25	−2.85644
$95$	546.703	0.590426
$96$	0	0
$97$	1603.57	1.67853	0.839266	−	0.543721i	$-0.182985\pi$
$97$	1603.57	1.67853	0.839266	+	0.543721i	$0.182985\pi$
$98$	−972.282	−1.00220
$99$	0	0
$100$	−90.8684	−0.0908684
$101$	−1229.04	−1.21084	−0.605418	−	0.795908i	$-0.706994\pi$
$101$	−1229.04	−1.21084	−0.605418	+	0.795908i	$0.706994\pi$
$102$	0	0
$103$	215.066	0.205738	0.102869	−	0.994695i	$-0.467198\pi$
$103$	215.066	0.205738	0.102869	+	0.994695i	$0.467198\pi$
$104$	−651.325	−0.614112
$105$	0	0
$106$	−1142.03	−1.04645
$107$	1586.58	1.43346	0.716730	−	0.697351i	$-0.245638\pi$
$107$	1586.58	1.43346	0.716730	+	0.697351i	$0.245638\pi$
$108$	0	0
$109$	822.255	0.722549	0.361274	−	0.932460i	$-0.382342\pi$
$109$	822.255	0.722549	0.361274	+	0.932460i	$0.382342\pi$
$110$	2387.48	2.06943
$111$	0	0
$112$	−475.800	−0.401418
$113$	−19.7005	−0.0164006	−0.00820031	−	0.999966i	$-0.502610\pi$
$113$	−19.7005	−0.0164006	−0.00820031	+	0.999966i	$0.502610\pi$
$114$	0	0
$115$	247.570	0.200748
$116$	−126.766	−0.101465
$117$	0	0
$118$	2265.31	1.76728
$119$	822.755	0.633797
$120$	0	0
$121$	1410.64	1.05984
$122$	−3552.38	−2.63621
$123$	0	0
$124$	−123.441	−0.0893980
$125$	−1443.85	−1.03313
$126$	0	0
$127$	1143.23	0.798784	0.399392	−	0.916780i	$-0.369221\pi$
$127$	1143.23	0.798784	0.399392	+	0.916780i	$0.369221\pi$
$128$	−1023.08	−0.706473
$129$	0	0
$130$	−3605.88	−2.43275
$131$	−1154.64	−0.770085	−0.385043	−	0.922899i	$-0.625813\pi$
$131$	−1154.64	−0.770085	−0.385043	+	0.922899i	$0.625813\pi$
$132$	0	0
$133$	541.042	0.352739
$134$	−1898.52	−1.22394
$135$	0	0
$136$	636.122	0.401081
$137$	2153.10	1.34271	0.671356	−	0.741135i	$-0.265712\pi$
$137$	2153.10	1.34271	0.671356	+	0.741135i	$0.265712\pi$
$138$	0	0
$139$	−2392.16	−1.45971	−0.729857	−	0.683600i	$-0.760413\pi$
$139$	−2392.16	−1.45971	−0.729857	+	0.683600i	$0.760413\pi$
$140$	1140.24	0.688339
$141$	0	0
$142$	−2662.81	−1.57365
$143$	−4140.79	−2.42147
$144$	0	0
$145$	−137.215	−0.0785868
$146$	3919.49	2.22178
$147$	0	0
$148$	−727.009	−0.403782
$149$	2259.80	1.24248	0.621242	−	0.783619i	$-0.286628\pi$
$149$	2259.80	1.24248	0.621242	+	0.783619i	$0.286628\pi$
$150$	0	0
$151$	−2410.84	−1.29928	−0.649639	−	0.760243i	$-0.725080\pi$
$151$	−2410.84	−1.29928	−0.649639	+	0.760243i	$0.725080\pi$
$152$	418.312	0.223221
$153$	0	0
$154$	2362.76	1.23634
$155$	−133.616	−0.0692406
$156$	0	0
$157$	−614.960	−0.312606	−0.156303	−	0.987709i	$-0.549958\pi$
$157$	−614.960	−0.312606	−0.156303	+	0.987709i	$0.549958\pi$
$158$	−4081.36	−2.05504
$159$	0	0
$160$	−2745.83	−1.35673
$161$	245.007	0.119933
$162$	0	0
$163$	−50.8127	−0.0244169	−0.0122085	−	0.999925i	$-0.503886\pi$
$163$	−50.8127	−0.0244169	−0.0122085	+	0.999925i	$0.503886\pi$
$164$	386.749	0.184147
$165$	0	0
$166$	−564.912	−0.264130
$167$	2000.84	0.927125	0.463562	−	0.886064i	$-0.346571\pi$
$167$	2000.84	0.927125	0.463562	+	0.886064i	$0.346571\pi$
$168$	0	0
$169$	4056.97	1.84659
$170$	3521.71	1.58884
$171$	0	0
$172$	1708.30	0.757305
$173$	3796.78	1.66858	0.834288	−	0.551330i	$-0.185879\pi$
$173$	3796.78	1.66858	0.834288	+	0.551330i	$0.185879\pi$
$174$	0	0
$175$	−97.3398	−0.0420469
$176$	−2338.72	−1.00164
$177$	0	0
$178$	3298.12	1.38879
$179$	3084.16	1.28783	0.643913	−	0.765099i	$-0.277310\pi$
$179$	3084.16	1.28783	0.643913	+	0.765099i	$0.277310\pi$
$180$	0	0
$181$	4733.12	1.94370	0.971851	−	0.235597i	$-0.0757047\pi$
$181$	4733.12	1.94370	0.971851	+	0.235597i	$0.0757047\pi$
$182$	−3568.55	−1.45340
$183$	0	0
$184$	189.430	0.0758964
$185$	−786.933	−0.312738
$186$	0	0
$187$	4044.13	1.58148
$188$	−6111.20	−2.37077
$189$	0	0
$190$	2315.87	0.884268
$191$	1293.13	0.489885	0.244942	−	0.969538i	$-0.421231\pi$
$191$	1293.13	0.489885	0.244942	+	0.969538i	$0.421231\pi$
$192$	0	0
$193$	−1171.75	−0.437019	−0.218510	−	0.975835i	$-0.570119\pi$
$193$	−1171.75	−0.437019	−0.218510	+	0.975835i	$0.570119\pi$
$194$	6792.82	2.51390
$195$	0	0
$196$	−2282.46	−0.831799
$197$	−4559.19	−1.64888	−0.824438	−	0.565952i	$-0.808509\pi$
$197$	−4559.19	−1.64888	−0.824438	+	0.565952i	$0.808509\pi$
$198$	0	0
$199$	−473.721	−0.168750	−0.0843748	−	0.996434i	$-0.526889\pi$
$199$	−473.721	−0.168750	−0.0843748	+	0.996434i	$0.526889\pi$
$200$	−75.2593	−0.0266082
$201$	0	0
$202$	−5206.31	−1.81344
$203$	−135.794	−0.0469502
$204$	0	0
$205$	418.627	0.142625
$206$	911.033	0.308130
$207$	0	0
$208$	3532.25	1.17749
$209$	2659.41	0.880169
$210$	0	0
$211$	−1423.56	−0.464465	−0.232232	−	0.972660i	$-0.574603\pi$
$211$	−1423.56	−0.464465	−0.232232	+	0.972660i	$0.574603\pi$
$212$	−2680.94	−0.868528
$213$	0	0
$214$	6720.85	2.14686
$215$	1849.11	0.586548
$216$	0	0
$217$	−132.232	−0.0413665
$218$	3483.13	1.08214
$219$	0	0
$220$	5604.66	1.71757
$221$	−6107.99	−1.85913
$222$	0	0
$223$	6088.63	1.82836	0.914181	−	0.405305i	$-0.132835\pi$
$223$	6088.63	1.82836	0.914181	+	0.405305i	$0.132835\pi$
$224$	−2717.40	−0.810552
$225$	0	0
$226$	−83.4528	−0.0245628
$227$	4463.66	1.30513	0.652563	−	0.757734i	$-0.273694\pi$
$227$	4463.66	1.30513	0.652563	+	0.757734i	$0.273694\pi$
$228$	0	0
$229$	1298.87	0.374812	0.187406	−	0.982283i	$-0.439992\pi$
$229$	1298.87	0.374812	0.187406	+	0.982283i	$0.439992\pi$
$230$	1048.73	0.300656
$231$	0	0
$232$	−104.991	−0.0297111
$233$	1257.41	0.353544	0.176772	−	0.984252i	$-0.443434\pi$
$233$	1257.41	0.353544	0.176772	+	0.984252i	$0.443434\pi$
$234$	0	0
$235$	−6614.92	−1.83621
$236$	5317.88	1.46680
$237$	0	0
$238$	3485.25	0.949223
$239$	−3473.96	−0.940217	−0.470108	−	0.882609i	$-0.655785\pi$
$239$	−3473.96	−0.940217	−0.470108	+	0.882609i	$0.655785\pi$
$240$	0	0
$241$	−2140.92	−0.572235	−0.286118	−	0.958195i	$-0.592365\pi$
$241$	−2140.92	−0.572235	−0.286118	+	0.958195i	$0.592365\pi$
$242$	5975.57	1.58729
$243$	0	0
$244$	−8339.30	−2.18799
$245$	−2470.59	−0.644245
$246$	0	0
$247$	−4016.60	−1.03470
$248$	−102.237	−0.0261776
$249$	0	0
$250$	−6116.25	−1.54730
$251$	5613.17	1.41155	0.705777	−	0.708434i	$-0.250598\pi$
$251$	5613.17	1.41155	0.705777	+	0.708434i	$0.250598\pi$
$252$	0	0
$253$	1204.30	0.299263
$254$	4842.82	1.19632
$255$	0	0
$256$	1452.36	0.354579
$257$	−1017.98	−0.247082	−0.123541	−	0.992339i	$-0.539425\pi$
$257$	−1017.98	−0.247082	−0.123541	+	0.992339i	$0.539425\pi$
$258$	0	0
$259$	−778.785	−0.186839
$260$	−8464.90	−2.01912
$261$	0	0
$262$	−4891.12	−1.15334
$263$	995.531	0.233411	0.116705	−	0.993167i	$-0.462767\pi$
$263$	995.531	0.233411	0.116705	+	0.993167i	$0.462767\pi$
$264$	0	0
$265$	−2901.92	−0.672693
$266$	2291.89	0.528289
$267$	0	0
$268$	−4456.83	−1.01584
$269$	3568.95	0.808932	0.404466	−	0.914553i	$-0.367457\pi$
$269$	3568.95	0.808932	0.404466	+	0.914553i	$0.367457\pi$
$270$	0	0
$271$	−6294.26	−1.41088	−0.705441	−	0.708768i	$-0.749251\pi$
$271$	−6294.26	−1.41088	−0.705441	+	0.708768i	$0.749251\pi$
$272$	−3449.80	−0.769025
$273$	0	0
$274$	9120.67	2.01095
$275$	−478.460	−0.104917
$276$	0	0
$277$	−2263.65	−0.491009	−0.245505	−	0.969395i	$-0.578954\pi$
$277$	−2263.65	−0.491009	−0.245505	+	0.969395i	$0.578954\pi$
$278$	−10133.3	−2.18618
$279$	0	0
$280$	944.368	0.201560
$281$	505.900	0.107400	0.0537001	−	0.998557i	$-0.482898\pi$
$281$	505.900	0.107400	0.0537001	+	0.998557i	$0.482898\pi$
$282$	0	0
$283$	−7018.20	−1.47417	−0.737083	−	0.675802i	$-0.763797\pi$
$283$	−7018.20	−1.47417	−0.737083	+	0.675802i	$0.763797\pi$
$284$	−6251.01	−1.30609
$285$	0	0
$286$	−17540.7	−3.62658
$287$	414.292	0.0852087
$288$	0	0
$289$	1052.41	0.214209
$290$	−581.252	−0.117698
$291$	0	0
$292$	9201.10	1.84402
$293$	9488.82	1.89196	0.945978	−	0.324232i	$-0.105106\pi$
$293$	9488.82	1.89196	0.945978	+	0.324232i	$0.105106\pi$
$294$	0	0
$295$	5756.20	1.13606
$296$	−602.125	−0.118236
$297$	0	0
$298$	9572.67	1.86084
$299$	−1818.89	−0.351802
$300$	0	0
$301$	1829.96	0.350422
$302$	−10212.5	−1.94590
$303$	0	0
$304$	−2268.58	−0.428000
$305$	−9026.67	−1.69464
$306$	0	0
$307$	4660.58	0.866427	0.433214	−	0.901291i	$-0.357380\pi$
$307$	4660.58	0.866427	0.433214	+	0.901291i	$0.357380\pi$
$308$	5546.63	1.02613
$309$	0	0
$310$	−566.006	−0.103700
$311$	−4378.51	−0.798336	−0.399168	−	0.916878i	$-0.630701\pi$
$311$	−4378.51	−0.798336	−0.399168	+	0.916878i	$0.630701\pi$
$312$	0	0
$313$	−4444.17	−0.802553	−0.401277	−	0.915957i	$-0.631433\pi$
$313$	−4444.17	−0.802553	−0.401277	+	0.915957i	$0.631433\pi$
$314$	−2605.01	−0.468182
$315$	0	0
$316$	−9581.10	−1.70563
$317$	−7881.61	−1.39645	−0.698226	−	0.715878i	$-0.746027\pi$
$317$	−7881.61	−1.39645	−0.698226	+	0.715878i	$0.746027\pi$
$318$	0	0
$319$	−667.477	−0.117152
$320$	−7785.29	−1.36003
$321$	0	0
$322$	1037.87	0.179621
$323$	3922.84	0.675767
$324$	0	0
$325$	722.633	0.123337
$326$	−215.246	−0.0365686
$327$	0	0
$328$	320.314	0.0539219
$329$	−6546.42	−1.09701
$330$	0	0
$331$	−127.060	−0.0210993	−0.0105497	−	0.999944i	$-0.503358\pi$
$331$	−127.060	−0.0210993	−0.0105497	+	0.999944i	$0.503358\pi$
$332$	−1326.14	−0.219222
$333$	0	0
$334$	8475.70	1.38853
$335$	−4824.18	−0.786786
$336$	0	0
$337$	−1739.07	−0.281107	−0.140554	−	0.990073i	$-0.544888\pi$
$337$	−1739.07	−0.281107	−0.140554	+	0.990073i	$0.544888\pi$
$338$	17185.6	2.76560
$339$	0	0
$340$	8267.31	1.31870
$341$	−649.969	−0.103219
$342$	0	0
$343$	−6098.81	−0.960072
$344$	1414.85	0.221755
$345$	0	0
$346$	16083.4	2.49899
$347$	−7551.30	−1.16823	−0.584114	−	0.811672i	$-0.698558\pi$
$347$	−7551.30	−1.16823	−0.584114	+	0.811672i	$0.698558\pi$
$348$	0	0
$349$	−6237.28	−0.956659	−0.478329	−	0.878180i	$-0.658758\pi$
$349$	−6237.28	−0.956659	−0.478329	+	0.878180i	$0.658758\pi$
$350$	−412.338	−0.0629726
$351$	0	0
$352$	−13357.0	−2.02252
$353$	−1788.57	−0.269678	−0.134839	−	0.990868i	$-0.543052\pi$
$353$	−1788.57	−0.269678	−0.134839	+	0.990868i	$0.543052\pi$
$354$	0	0
$355$	−6766.25	−1.01159
$356$	7742.42	1.15266
$357$	0	0
$358$	13064.7	1.92875
$359$	−2234.70	−0.328531	−0.164266	−	0.986416i	$-0.552525\pi$
$359$	−2234.70	−0.328531	−0.164266	+	0.986416i	$0.552525\pi$
$360$	0	0
$361$	−4279.35	−0.623903
$362$	20049.8	2.91104
$363$	0	0
$364$	−8377.25	−1.20628
$365$	9959.50	1.42823
$366$	0	0
$367$	10551.7	1.50080	0.750399	−	0.660985i	$-0.229861\pi$
$367$	10551.7	1.50080	0.750399	+	0.660985i	$0.229861\pi$
$368$	−1027.31	−0.145522
$369$	0	0
$370$	−3333.50	−0.468380
$371$	−2871.87	−0.401887
$372$	0	0
$373$	−996.068	−0.138269	−0.0691347	−	0.997607i	$-0.522024\pi$
$373$	−996.068	−0.138269	−0.0691347	+	0.997607i	$0.522024\pi$
$374$	17131.2	2.36854
$375$	0	0
$376$	−5061.43	−0.694211
$377$	1008.11	0.137720
$378$	0	0
$379$	−11117.0	−1.50670	−0.753351	−	0.657619i	$-0.771564\pi$
$379$	−11117.0	−1.50670	−0.753351	+	0.657619i	$0.771564\pi$
$380$	5436.56	0.733920
$381$	0	0
$382$	5477.81	0.733688
$383$	−5315.09	−0.709108	−0.354554	−	0.935035i	$-0.615367\pi$
$383$	−5315.09	−0.709108	−0.354554	+	0.935035i	$0.615367\pi$
$384$	0	0
$385$	6003.81	0.794759
$386$	−4963.63	−0.654513
$387$	0	0
$388$	15946.3	2.08647
$389$	8438.06	1.09981	0.549906	−	0.835227i	$-0.314664\pi$
$389$	8438.06	1.09981	0.549906	+	0.835227i	$0.314664\pi$
$390$	0	0
$391$	1776.43	0.229764
$392$	−1890.38	−0.243568
$393$	0	0
$394$	−19313.0	−2.46948
$395$	−10370.8	−1.32104
$396$	0	0
$397$	11582.0	1.46419	0.732094	−	0.681204i	$-0.238543\pi$
$397$	11582.0	1.46419	0.732094	+	0.681204i	$0.238543\pi$
$398$	−2006.71	−0.252732
$399$	0	0
$400$	408.144	0.0510180
$401$	−4267.85	−0.531487	−0.265744	−	0.964044i	$-0.585617\pi$
$401$	−4267.85	−0.531487	−0.265744	+	0.964044i	$0.585617\pi$
$402$	0	0
$403$	981.669	0.121341
$404$	−12221.9	−1.50511
$405$	0	0
$406$	−575.234	−0.0703162
$407$	−3828.00	−0.466209
$408$	0	0
$409$	6902.51	0.834491	0.417246	−	0.908794i	$-0.362995\pi$
$409$	6902.51	0.834491	0.417246	+	0.908794i	$0.362995\pi$
$410$	1773.33	0.213606
$411$	0	0
$412$	2138.67	0.255740
$413$	5696.60	0.678720
$414$	0	0
$415$	−1435.45	−0.169792
$416$	20173.4	2.37761
$417$	0	0
$418$	11265.4	1.31821
$419$	2241.24	0.261316	0.130658	−	0.991427i	$-0.458291\pi$
$419$	2241.24	0.261316	0.130658	+	0.991427i	$0.458291\pi$
$420$	0	0
$421$	9117.52	1.05549	0.527744	−	0.849403i	$-0.323038\pi$
$421$	9117.52	1.05549	0.527744	+	0.849403i	$0.323038\pi$
$422$	−6030.30	−0.695618
$423$	0	0
$424$	−2220.42	−0.254323
$425$	−705.765	−0.0805521
$426$	0	0
$427$	−8933.21	−1.01243
$428$	15777.4	1.78184
$429$	0	0
$430$	7832.93	0.878460
$431$	1366.55	0.152725	0.0763623	−	0.997080i	$-0.475669\pi$
$431$	1366.55	0.152725	0.0763623	+	0.997080i	$0.475669\pi$
$432$	0	0
$433$	3511.21	0.389695	0.194848	−	0.980834i	$-0.437579\pi$
$433$	3511.21	0.389695	0.194848	+	0.980834i	$0.437579\pi$
$434$	−560.145	−0.0619536
$435$	0	0
$436$	8176.73	0.898152
$437$	1168.18	0.127875
$438$	0	0
$439$	7032.17	0.764526	0.382263	−	0.924054i	$-0.375145\pi$
$439$	7032.17	0.764526	0.382263	+	0.924054i	$0.375145\pi$
$440$	4641.90	0.502941
$441$	0	0
$442$	−25873.8	−2.78437
$443$	−8748.68	−0.938289	−0.469145	−	0.883121i	$-0.655438\pi$
$443$	−8748.68	−0.938289	−0.469145	+	0.883121i	$0.655438\pi$
$444$	0	0
$445$	8380.60	0.892760
$446$	25791.9	2.73830
$447$	0	0
$448$	−7704.68	−0.812526
$449$	805.208	0.0846327	0.0423164	−	0.999104i	$-0.486526\pi$
$449$	805.208	0.0846327	0.0423164	+	0.999104i	$0.486526\pi$
$450$	0	0
$451$	2036.39	0.212616
$452$	−195.907	−0.0203865
$453$	0	0
$454$	18908.4	1.95466
$455$	−9067.75	−0.934291
$456$	0	0
$457$	−10496.9	−1.07445	−0.537226	−	0.843438i	$-0.680528\pi$
$457$	−10496.9	−1.07445	−0.537226	+	0.843438i	$0.680528\pi$
$458$	5502.12	0.561348
$459$	0	0
$460$	2461.91	0.249537
$461$	3325.68	0.335992	0.167996	−	0.985788i	$-0.446270\pi$
$461$	3325.68	0.335992	0.167996	+	0.985788i	$0.446270\pi$
$462$	0	0
$463$	−840.876	−0.0844035	−0.0422018	−	0.999109i	$-0.513437\pi$
$463$	−840.876	−0.0844035	−0.0422018	+	0.999109i	$0.513437\pi$
$464$	569.383	0.0569675
$465$	0	0
$466$	5326.48	0.529495
$467$	−8450.82	−0.837382	−0.418691	−	0.908129i	$-0.637511\pi$
$467$	−8450.82	−0.837382	−0.418691	+	0.908129i	$0.637511\pi$
$468$	0	0
$469$	−4774.23	−0.470050
$470$	−28021.2	−2.75005
$471$	0	0
$472$	4404.38	0.429509
$473$	8994.89	0.874388
$474$	0	0
$475$	−464.109	−0.0448312
$476$	8181.70	0.787831
$477$	0	0
$478$	−14715.9	−1.40814
$479$	5686.86	0.542462	0.271231	−	0.962514i	$-0.412569\pi$
$479$	5686.86	0.542462	0.271231	+	0.962514i	$0.412569\pi$
$480$	0	0
$481$	5781.56	0.548059
$482$	−9069.07	−0.857023
$483$	0	0
$484$	14027.8	1.31741
$485$	17260.7	1.61602
$486$	0	0
$487$	−13768.2	−1.28110	−0.640551	−	0.767916i	$-0.721294\pi$
$487$	−13768.2	−1.28110	−0.640551	+	0.767916i	$0.721294\pi$
$488$	−6906.80	−0.640689
$489$	0	0
$490$	−10465.6	−0.964871
$491$	−8480.14	−0.779436	−0.389718	−	0.920934i	$-0.627428\pi$
$491$	−8480.14	−0.779436	−0.389718	+	0.920934i	$0.627428\pi$
$492$	0	0
$493$	−984.580	−0.0899457
$494$	−17014.6	−1.54964
$495$	0	0
$496$	554.448	0.0501924
$497$	−6696.19	−0.604356
$498$	0	0
$499$	12472.5	1.11893	0.559466	−	0.828853i	$-0.311006\pi$
$499$	12472.5	1.11893	0.559466	+	0.828853i	$0.311006\pi$
$500$	−14358.0	−1.28422
$501$	0	0
$502$	23777.8	2.11405
$503$	4676.36	0.414530	0.207265	−	0.978285i	$-0.433544\pi$
$503$	4676.36	0.414530	0.207265	+	0.978285i	$0.433544\pi$
$504$	0	0
$505$	−13229.3	−1.16574
$506$	5101.48	0.448198
$507$	0	0
$508$	11368.6	0.992916
$509$	−5925.58	−0.516005	−0.258003	−	0.966144i	$-0.583064\pi$
$509$	−5925.58	−0.516005	−0.258003	+	0.966144i	$0.583064\pi$
$510$	0	0
$511$	9856.38	0.853269
$512$	14336.9	1.23752
$513$	0	0
$514$	−4312.25	−0.370049
$515$	2314.95	0.198076
$516$	0	0
$517$	−32178.0	−2.73730
$518$	−3298.99	−0.279825
$519$	0	0
$520$	−7010.82	−0.591240
$521$	−2626.25	−0.220841	−0.110421	−	0.993885i	$-0.535220\pi$
$521$	−2626.25	−0.220841	−0.110421	+	0.993885i	$0.535220\pi$
$522$	0	0
$523$	372.504	0.0311443	0.0155721	−	0.999879i	$-0.495043\pi$
$523$	372.504	0.0311443	0.0155721	+	0.999879i	$0.495043\pi$
$524$	−11482.0	−0.957242
$525$	0	0
$526$	4217.14	0.349574
$527$	−958.755	−0.0792486
$528$	0	0
$529$	529.000	0.0434783
$530$	−12292.7	−1.00748
$531$	0	0
$532$	5380.27	0.438466
$533$	−3075.63	−0.249944
$534$	0	0
$535$	17077.8	1.38007
$536$	−3691.24	−0.297458
$537$	0	0
$538$	15118.3	1.21152
$539$	−12018.1	−0.960399
$540$	0	0
$541$	12899.2	1.02510	0.512550	−	0.858657i	$-0.328701\pi$
$541$	12899.2	1.02510	0.512550	+	0.858657i	$0.328701\pi$
$542$	−26662.9	−2.11305
$543$	0	0
$544$	−19702.5	−1.55283
$545$	8850.70	0.695637
$546$	0	0
$547$	−22144.7	−1.73097	−0.865485	−	0.500936i	$-0.832989\pi$
$547$	−22144.7	−1.73097	−0.865485	+	0.500936i	$0.832989\pi$
$548$	21411.0	1.66904
$549$	0	0
$550$	−2026.79	−0.157132
$551$	−647.457	−0.0500592
$552$	0	0
$553$	−10263.4	−0.789233
$554$	−9588.98	−0.735373
$555$	0	0
$556$	−23788.3	−1.81447
$557$	10236.7	0.778716	0.389358	−	0.921086i	$-0.372697\pi$
$557$	10236.7	0.778716	0.389358	+	0.921086i	$0.372697\pi$
$558$	0	0
$559$	−13585.3	−1.02790
$560$	−5121.47	−0.386467
$561$	0	0
$562$	2143.03	0.160851
$563$	−5750.83	−0.430495	−0.215247	−	0.976560i	$-0.569056\pi$
$563$	−5750.83	−0.430495	−0.215247	+	0.976560i	$0.569056\pi$
$564$	0	0
$565$	−212.055	−0.0157898
$566$	−29729.6	−2.20782
$567$	0	0
$568$	−5177.22	−0.382450
$569$	8285.10	0.610421	0.305210	−	0.952285i	$-0.401273\pi$
$569$	8285.10	0.610421	0.305210	+	0.952285i	$0.401273\pi$
$570$	0	0
$571$	−19799.8	−1.45113	−0.725565	−	0.688154i	$-0.758421\pi$
$571$	−19799.8	−1.45113	−0.725565	+	0.688154i	$0.758421\pi$
$572$	−41177.1	−3.00997
$573$	0	0
$574$	1754.97	0.127615
$575$	−210.169	−0.0152428
$576$	0	0
$577$	−250.868	−0.0181001	−0.00905006	−	0.999959i	$-0.502881\pi$
$577$	−250.868	−0.0181001	−0.00905006	+	0.999959i	$0.502881\pi$
$578$	4458.08	0.320816
$579$	0	0
$580$	−1364.50	−0.0976861
$581$	−1420.59	−0.101439
$582$	0	0
$583$	−14116.3	−1.00281
$584$	7620.56	0.539967
$585$	0	0
$586$	40195.3	2.83354
$587$	9767.11	0.686766	0.343383	−	0.939195i	$-0.388427\pi$
$587$	9767.11	0.686766	0.343383	+	0.939195i	$0.388427\pi$
$588$	0	0
$589$	−630.475	−0.0441057
$590$	24383.7	1.70146
$591$	0	0
$592$	3265.43	0.226703
$593$	18853.8	1.30562	0.652810	−	0.757521i	$-0.273590\pi$
$593$	18853.8	1.30562	0.652810	+	0.757521i	$0.273590\pi$
$594$	0	0
$595$	8856.08	0.610192
$596$	22472.1	1.54445
$597$	0	0
$598$	−7704.93	−0.526886
$599$	11356.0	0.774615	0.387307	−	0.921951i	$-0.373405\pi$
$599$	11356.0	0.774615	0.387307	+	0.921951i	$0.373405\pi$
$600$	0	0
$601$	20359.1	1.38180	0.690902	−	0.722948i	$-0.257213\pi$
$601$	20359.1	1.38180	0.690902	+	0.722948i	$0.257213\pi$
$602$	7751.83	0.524819
$603$	0	0
$604$	−23974.0	−1.61505
$605$	15184.0	1.02036
$606$	0	0
$607$	5975.14	0.399545	0.199772	−	0.979842i	$-0.435980\pi$
$607$	5975.14	0.399545	0.199772	+	0.979842i	$0.435980\pi$
$608$	−12956.3	−0.864226
$609$	0	0
$610$	−38237.6	−2.53802
$611$	48599.5	3.21788
$612$	0	0
$613$	16997.4	1.11993	0.559967	−	0.828515i	$-0.310814\pi$
$613$	16997.4	1.11993	0.559967	+	0.828515i	$0.310814\pi$
$614$	19742.5	1.29763
$615$	0	0
$616$	4593.84	0.300472
$617$	13096.0	0.854497	0.427249	−	0.904134i	$-0.359483\pi$
$617$	13096.0	0.854497	0.427249	+	0.904134i	$0.359483\pi$
$618$	0	0
$619$	702.058	0.0455866	0.0227933	−	0.999740i	$-0.492744\pi$
$619$	702.058	0.0455866	0.0227933	+	0.999740i	$0.492744\pi$
$620$	−1328.71	−0.0860684
$621$	0	0
$622$	−18547.7	−1.19565
$623$	8293.82	0.533362
$624$	0	0
$625$	−14399.3	−0.921554
$626$	−18825.8	−1.20196
$627$	0	0
$628$	−6115.33	−0.388580
$629$	−5646.60	−0.357941
$630$	0	0
$631$	23871.5	1.50604	0.753019	−	0.657999i	$-0.228597\pi$
$631$	23871.5	1.50604	0.753019	+	0.657999i	$0.228597\pi$
$632$	−7935.28	−0.499444
$633$	0	0
$634$	−33387.0	−2.09143
$635$	12305.7	0.769034
$636$	0	0
$637$	18151.3	1.12901
$638$	−2827.48	−0.175456
$639$	0	0
$640$	−11012.4	−0.680161
$641$	−24172.6	−1.48949	−0.744743	−	0.667351i	$-0.767428\pi$
$641$	−24172.6	−1.48949	−0.744743	+	0.667351i	$0.767428\pi$
$642$	0	0
$643$	13648.6	0.837090	0.418545	−	0.908196i	$-0.362540\pi$
$643$	13648.6	0.837090	0.418545	+	0.908196i	$0.362540\pi$
$644$	2436.42	0.149081
$645$	0	0
$646$	16617.4	1.01208
$647$	7009.18	0.425903	0.212952	−	0.977063i	$-0.431692\pi$
$647$	7009.18	0.425903	0.212952	+	0.977063i	$0.431692\pi$
$648$	0	0
$649$	28000.8	1.69357
$650$	3061.12	0.184719
$651$	0	0
$652$	−505.295	−0.0303511
$653$	−4533.57	−0.271688	−0.135844	−	0.990730i	$-0.543375\pi$
$653$	−4533.57	−0.271688	−0.135844	+	0.990730i	$0.543375\pi$
$654$	0	0
$655$	−12428.4	−0.741404
$656$	−1737.12	−0.103389
$657$	0	0
$658$	−27731.1	−1.64296
$659$	7112.62	0.420438	0.210219	−	0.977654i	$-0.432582\pi$
$659$	7112.62	0.420438	0.210219	+	0.977654i	$0.432582\pi$
$660$	0	0
$661$	16088.6	0.946709	0.473354	−	0.880872i	$-0.343043\pi$
$661$	16088.6	0.946709	0.473354	+	0.880872i	$0.343043\pi$
$662$	−538.237	−0.0316000
$663$	0	0
$664$	−1098.34	−0.0641927
$665$	5823.74	0.339601
$666$	0	0
$667$	−293.196	−0.0170204
$668$	19896.9	1.15245
$669$	0	0
$670$	−20435.6	−1.17835
$671$	−43909.9	−2.52626
$672$	0	0
$673$	22629.7	1.29615	0.648076	−	0.761576i	$-0.275574\pi$
$673$	22629.7	1.29615	0.648076	+	0.761576i	$0.275574\pi$
$674$	−7366.82	−0.421008
$675$	0	0
$676$	40343.6	2.29538
$677$	15444.0	0.876749	0.438375	−	0.898792i	$-0.355554\pi$
$677$	15444.0	0.876749	0.438375	+	0.898792i	$0.355554\pi$
$678$	0	0
$679$	17082.0	0.965458
$680$	6847.17	0.386143
$681$	0	0
$682$	−2753.31	−0.154589
$683$	−834.894	−0.0467736	−0.0233868	−	0.999726i	$-0.507445\pi$
$683$	−834.894	−0.0467736	−0.0233868	+	0.999726i	$0.507445\pi$
$684$	0	0
$685$	23175.8	1.29270
$686$	−25835.0	−1.43788
$687$	0	0
$688$	−7672.98	−0.425188
$689$	21320.3	1.17886
$690$	0	0
$691$	4982.99	0.274330	0.137165	−	0.990548i	$-0.456201\pi$
$691$	4982.99	0.274330	0.137165	+	0.990548i	$0.456201\pi$
$692$	37756.2	2.07410
$693$	0	0
$694$	−31987.8	−1.74963
$695$	−25749.0	−1.40535
$696$	0	0
$697$	3003.84	0.163240
$698$	−26421.5	−1.43277
$699$	0	0
$700$	−967.974	−0.0522657
$701$	−21916.9	−1.18087	−0.590436	−	0.807084i	$-0.701044\pi$
$701$	−21916.9	−1.18087	−0.590436	+	0.807084i	$0.701044\pi$
$702$	0	0
$703$	−3713.19	−0.199211
$704$	−37871.2	−2.02745
$705$	0	0
$706$	−7576.52	−0.403890
$707$	−13092.4	−0.696448
$708$	0	0
$709$	−30858.3	−1.63456	−0.817282	−	0.576237i	$-0.804520\pi$
$709$	−30858.3	−1.63456	−0.817282	+	0.576237i	$0.804520\pi$
$710$	−28662.3	−1.51504
$711$	0	0
$712$	6412.45	0.337523
$713$	−285.506	−0.0149962
$714$	0	0
$715$	−44571.2	−2.33128
$716$	30669.7	1.60081
$717$	0	0
$718$	−9466.32	−0.492033
$719$	−19500.3	−1.01146	−0.505728	−	0.862693i	$-0.668776\pi$
$719$	−19500.3	−1.01146	−0.505728	+	0.862693i	$0.668776\pi$
$720$	0	0
$721$	2290.98	0.118337
$722$	−18127.6	−0.934405
$723$	0	0
$724$	47067.4	2.41609
$725$	116.485	0.00596711
$726$	0	0
$727$	26922.8	1.37347	0.686735	−	0.726908i	$-0.259043\pi$
$727$	26922.8	1.37347	0.686735	+	0.726908i	$0.259043\pi$
$728$	−6938.22	−0.353225
$729$	0	0
$730$	42189.1	2.13903
$731$	13268.2	0.671328
$732$	0	0
$733$	−22120.5	−1.11465	−0.557326	−	0.830294i	$-0.688173\pi$
$733$	−22120.5	−1.11465	−0.557326	+	0.830294i	$0.688173\pi$
$734$	44697.6	2.24771
$735$	0	0
$736$	−5867.19	−0.293842
$737$	−23467.0	−1.17289
$738$	0	0
$739$	−27225.8	−1.35523	−0.677617	−	0.735415i	$-0.736987\pi$
$739$	−27225.8	−1.35523	−0.677617	+	0.735415i	$0.736987\pi$
$740$	−7825.48	−0.388744
$741$	0	0
$742$	−12165.4	−0.601897
$743$	191.429	0.00945201	0.00472600	−	0.999989i	$-0.498496\pi$
$743$	191.429	0.00945201	0.00472600	+	0.999989i	$0.498496\pi$
$744$	0	0
$745$	24324.3	1.19621
$746$	−4219.41	−0.207083
$747$	0	0
$748$	40216.0	1.96583
$749$	16901.0	0.824497
$750$	0	0
$751$	−876.709	−0.0425986	−0.0212993	−	0.999773i	$-0.506780\pi$
$751$	−876.709	−0.0425986	−0.0212993	+	0.999773i	$0.506780\pi$
$752$	27449.0	1.33107
$753$	0	0
$754$	4270.43	0.206260
$755$	−25950.1	−1.25089
$756$	0	0
$757$	−26681.0	−1.28103	−0.640513	−	0.767948i	$-0.721278\pi$
$757$	−26681.0	−1.28103	−0.640513	+	0.767948i	$0.721278\pi$
$758$	−47092.2	−2.25655
$759$	0	0
$760$	4502.68	0.214907
$761$	6947.99	0.330965	0.165483	−	0.986213i	$-0.447082\pi$
$761$	6947.99	0.330965	0.165483	+	0.986213i	$0.447082\pi$
$762$	0	0
$763$	8759.06	0.415595
$764$	12859.3	0.608943
$765$	0	0
$766$	−22515.1	−1.06201
$767$	−42290.5	−1.99090
$768$	0	0
$769$	7189.91	0.337159	0.168579	−	0.985688i	$-0.446082\pi$
$769$	7189.91	0.337159	0.168579	+	0.985688i	$0.446082\pi$
$770$	25432.5	1.19029
$771$	0	0
$772$	−11652.2	−0.543230
$773$	27591.1	1.28381	0.641904	−	0.766785i	$-0.278145\pi$
$773$	27591.1	1.28381	0.641904	+	0.766785i	$0.278145\pi$
$774$	0	0
$775$	113.430	0.00525745
$776$	13207.1	0.610963
$777$	0	0
$778$	35744.2	1.64716
$779$	1975.32	0.0908512
$780$	0	0
$781$	−32914.1	−1.50801
$782$	7525.08	0.344113
$783$	0	0
$784$	10251.9	0.467013
$785$	−6619.38	−0.300963
$786$	0	0
$787$	10322.4	0.467541	0.233770	−	0.972292i	$-0.424894\pi$
$787$	10322.4	0.467541	0.233770	+	0.972292i	$0.424894\pi$
$788$	−45337.8	−2.04961
$789$	0	0
$790$	−43931.5	−1.97850
$791$	−209.859	−0.00943330
$792$	0	0
$793$	66318.5	2.96978
$794$	49062.0	2.19288
$795$	0	0
$796$	−4710.81	−0.209761
$797$	−40205.9	−1.78691	−0.893455	−	0.449153i	$-0.851726\pi$
$797$	−40205.9	−1.78691	−0.893455	+	0.449153i	$0.851726\pi$
$798$	0	0
$799$	−47465.0	−2.10162
$800$	2331.00	0.103017
$801$	0	0
$802$	−18078.9	−0.795995
$803$	48447.6	2.12911
$804$	0	0
$805$	2637.24	0.115466
$806$	4158.42	0.181730
$807$	0	0
$808$	−10122.5	−0.440727
$809$	42375.9	1.84160	0.920802	−	0.390031i	$-0.127536\pi$
$809$	42375.9	1.84160	0.920802	+	0.390031i	$0.127536\pi$
$810$	0	0
$811$	7082.68	0.306667	0.153333	−	0.988175i	$-0.450999\pi$
$811$	7082.68	0.306667	0.153333	+	0.988175i	$0.450999\pi$
$812$	−1350.37	−0.0583607
$813$	0	0
$814$	−16215.7	−0.698230
$815$	−546.945	−0.0235075
$816$	0	0
$817$	8725.11	0.373627
$818$	29239.5	1.24980
$819$	0	0
$820$	4162.94	0.177288
$821$	−18154.3	−0.771728	−0.385864	−	0.922556i	$-0.626097\pi$
$821$	−18154.3	−0.771728	−0.385864	+	0.922556i	$0.626097\pi$
$822$	0	0
$823$	10437.1	0.442058	0.221029	−	0.975267i	$-0.429059\pi$
$823$	10437.1	0.442058	0.221029	+	0.975267i	$0.429059\pi$
$824$	1771.30	0.0748860
$825$	0	0
$826$	24131.2	1.01650
$827$	1486.57	0.0625068	0.0312534	−	0.999511i	$-0.490050\pi$
$827$	1486.57	0.0625068	0.0312534	+	0.999511i	$0.490050\pi$
$828$	0	0
$829$	−9987.68	−0.418440	−0.209220	−	0.977869i	$-0.567092\pi$
$829$	−9987.68	−0.418440	−0.209220	+	0.977869i	$0.567092\pi$
$830$	−6080.67	−0.254293
$831$	0	0
$832$	57198.1	2.38340
$833$	−17727.6	−0.737364
$834$	0	0
$835$	21536.9	0.892594
$836$	26445.9	1.09408
$837$	0	0
$838$	9494.03	0.391367
$839$	2835.68	0.116685	0.0583424	−	0.998297i	$-0.481418\pi$
$839$	2835.68	0.116685	0.0583424	+	0.998297i	$0.481418\pi$
$840$	0	0
$841$	−24226.5	−0.993337
$842$	38622.4	1.58078
$843$	0	0
$844$	−14156.3	−0.577345
$845$	43668.9	1.77782
$846$	0	0
$847$	15026.8	0.609596
$848$	12041.7	0.487634
$849$	0	0
$850$	−2989.67	−0.120641
$851$	−1681.49	−0.0677330
$852$	0	0
$853$	17544.2	0.704223	0.352111	−	0.935958i	$-0.385464\pi$
$853$	17544.2	0.704223	0.352111	+	0.935958i	$0.385464\pi$
$854$	−37841.7	−1.51629
$855$	0	0
$856$	13067.2	0.521760
$857$	−12827.7	−0.511301	−0.255651	−	0.966769i	$-0.582290\pi$
$857$	−12827.7	−0.511301	−0.255651	+	0.966769i	$0.582290\pi$
$858$	0	0
$859$	−35061.5	−1.39265	−0.696324	−	0.717728i	$-0.745182\pi$
$859$	−35061.5	−1.39265	−0.696324	+	0.717728i	$0.745182\pi$
$860$	18388.0	0.729100
$861$	0	0
$862$	5788.79	0.228732
$863$	4222.11	0.166538	0.0832691	−	0.996527i	$-0.473464\pi$
$863$	4222.11	0.166538	0.0832691	+	0.996527i	$0.473464\pi$
$864$	0	0
$865$	40868.2	1.60643
$866$	14873.7	0.583637
$867$	0	0
$868$	−1314.95	−0.0514199
$869$	−50448.4	−1.96933
$870$	0	0
$871$	35443.0	1.37881
$872$	6772.15	0.262998
$873$	0	0
$874$	4948.47	0.191515
$875$	−15380.6	−0.594238
$876$	0	0
$877$	17535.0	0.675159	0.337579	−	0.941297i	$-0.390392\pi$
$877$	17535.0	0.675159	0.337579	+	0.941297i	$0.390392\pi$
$878$	29788.7	1.14501
$879$	0	0
$880$	−25173.9	−0.964330
$881$	−9116.63	−0.348634	−0.174317	−	0.984690i	$-0.555772\pi$
$881$	−9116.63	−0.348634	−0.174317	+	0.984690i	$0.555772\pi$
$882$	0	0
$883$	−41050.4	−1.56450	−0.782251	−	0.622963i	$-0.785929\pi$
$883$	−41050.4	−1.56450	−0.782251	+	0.622963i	$0.785929\pi$
$884$	−60739.5	−2.31096
$885$	0	0
$886$	−37060.0	−1.40525
$887$	−22519.5	−0.852461	−0.426230	−	0.904615i	$-0.640159\pi$
$887$	−22519.5	−0.852461	−0.426230	+	0.904615i	$0.640159\pi$
$888$	0	0
$889$	12178.3	0.459444
$890$	35500.8	1.33707
$891$	0	0
$892$	60547.0	2.27272
$893$	−31212.9	−1.16965
$894$	0	0
$895$	33197.7	1.23986
$896$	−10898.4	−0.406349
$897$	0	0
$898$	3410.91	0.126752
$899$	158.241	0.00587055
$900$	0	0
$901$	−20822.6	−0.769924
$902$	8626.30	0.318431
$903$	0	0
$904$	−162.255	−0.00596960
$905$	50947.0	1.87131
$906$	0	0
$907$	−32037.9	−1.17288	−0.586439	−	0.809993i	$-0.699471\pi$
$907$	−32037.9	−1.17288	−0.586439	+	0.809993i	$0.699471\pi$
$908$	44387.9	1.62232
$909$	0	0
$910$	−38411.6	−1.39927
$911$	12881.7	0.468485	0.234242	−	0.972178i	$-0.424739\pi$
$911$	12881.7	0.468485	0.234242	+	0.972178i	$0.424739\pi$
$912$	0	0
$913$	−6982.69	−0.253114
$914$	−44465.6	−1.60918
$915$	0	0
$916$	12916.4	0.465905
$917$	−12299.8	−0.442937
$918$	0	0
$919$	−7311.92	−0.262457	−0.131229	−	0.991352i	$-0.541892\pi$
$919$	−7311.92	−0.262457	−0.131229	+	0.991352i	$0.541892\pi$
$920$	2039.01	0.0730696
$921$	0	0
$922$	14087.8	0.503207
$923$	49711.3	1.77277
$924$	0	0
$925$	668.047	0.0237462
$926$	−3562.01	−0.126409
$927$	0	0
$928$	3251.87	0.115030
$929$	−35059.5	−1.23817	−0.619087	−	0.785322i	$-0.712497\pi$
$929$	−35059.5	−1.23817	−0.619087	+	0.785322i	$0.712497\pi$
$930$	0	0
$931$	−11657.6	−0.410379
$932$	12504.0	0.439467
$933$	0	0
$934$	−35798.2	−1.25413
$935$	43530.8	1.52258
$936$	0	0
$937$	25471.7	0.888072	0.444036	−	0.896009i	$-0.353546\pi$
$937$	25471.7	0.888072	0.444036	+	0.896009i	$0.353546\pi$
$938$	−20224.0	−0.703983
$939$	0	0
$940$	−65780.6	−2.28247
$941$	−22426.5	−0.776923	−0.388461	−	0.921465i	$-0.626993\pi$
$941$	−22426.5	−0.776923	−0.388461	+	0.921465i	$0.626993\pi$
$942$	0	0
$943$	894.508	0.0308899
$944$	−23885.7	−0.823532
$945$	0	0
$946$	38103.0	1.30955
$947$	36090.9	1.23843	0.619216	−	0.785220i	$-0.287450\pi$
$947$	36090.9	1.23843	0.619216	+	0.785220i	$0.287450\pi$
$948$	0	0
$949$	−73171.9	−2.50291
$950$	−1966.00	−0.0671426
$951$	0	0
$952$	6776.27	0.230693
$953$	16379.8	0.556762	0.278381	−	0.960471i	$-0.410202\pi$
$953$	16379.8	0.556762	0.278381	+	0.960471i	$0.410202\pi$
$954$	0	0
$955$	13919.2	0.471639
$956$	−34546.0	−1.16872
$957$	0	0
$958$	24089.9	0.812432
$959$	22935.8	0.772301
$960$	0	0
$961$	−29636.9	−0.994828
$962$	24491.1	0.820815
$963$	0	0
$964$	−21289.9	−0.711308
$965$	−12612.7	−0.420742
$966$	0	0
$967$	−22686.2	−0.754434	−0.377217	−	0.926125i	$-0.623119\pi$
$967$	−22686.2	−0.754434	−0.377217	+	0.926125i	$0.623119\pi$
$968$	11618.1	0.385765
$969$	0	0
$970$	73117.5	2.42027
$971$	−53773.7	−1.77722	−0.888610	−	0.458663i	$-0.848328\pi$
$971$	−53773.7	−1.77722	−0.888610	+	0.458663i	$0.848328\pi$
$972$	0	0
$973$	−25482.4	−0.839597
$974$	−58323.0	−1.91867
$975$	0	0
$976$	37456.8	1.22844
$977$	20398.6	0.667972	0.333986	−	0.942578i	$-0.391606\pi$
$977$	20398.6	0.667972	0.333986	+	0.942578i	$0.391606\pi$
$978$	0	0
$979$	40767.0	1.33087
$980$	−24568.2	−0.800819
$981$	0	0
$982$	−35922.4	−1.16734
$983$	12118.6	0.393208	0.196604	−	0.980483i	$-0.437009\pi$
$983$	12118.6	0.393208	0.196604	+	0.980483i	$0.437009\pi$
$984$	0	0
$985$	−49074.8	−1.58746
$986$	−4170.75	−0.134710
$987$	0	0
$988$	−39942.1	−1.28616
$989$	3951.10	0.127035
$990$	0	0
$991$	−26885.7	−0.861809	−0.430904	−	0.902398i	$-0.641805\pi$
$991$	−26885.7	−0.861809	−0.430904	+	0.902398i	$0.641805\pi$
$992$	3166.57	0.101350
$993$	0	0
$994$	−28365.5	−0.905130
$995$	−5099.10	−0.162465
$996$	0	0
$997$	21022.7	0.667797	0.333899	−	0.942609i	$-0.391636\pi$
$997$	21022.7	0.667797	0.333899	+	0.942609i	$0.391636\pi$
$998$	52834.5	1.67580
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.c.1.2		2
3.2	odd	2		69.4.a.a.1.1	✓	2
12.11	even	2		1104.4.a.h.1.2		2
15.14	odd	2		1725.4.a.n.1.2		2
69.68	even	2		1587.4.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.a.1.1	✓	2	3.2	odd	2
207.4.a.c.1.2		2	1.1	even	1	trivial
1104.4.a.h.1.2		2	12.11	even	2
1587.4.a.b.1.1		2	69.68	even	2
1725.4.a.n.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+4.23607 q^{2} +9.94427 q^{4} +10.7639 q^{5} +10.6525 q^{7} +8.23607 q^{8} +45.5967 q^{10} +52.3607 q^{11} -79.0820 q^{13} +45.1246 q^{14} -44.6656 q^{16} +77.2361 q^{17} +50.7902 q^{19} +107.039 q^{20} +221.803 q^{22} +23.0000 q^{23} -9.13777 q^{25} -334.997 q^{26} +105.931 q^{28} -12.7477 q^{29} -12.4133 q^{31} -255.095 q^{32} +327.177 q^{34} +114.663 q^{35} -73.1084 q^{37} +215.151 q^{38} +88.6525 q^{40} +38.8916 q^{41} +171.787 q^{43} +520.689 q^{44} +97.4296 q^{46} -614.545 q^{47} -229.525 q^{49} -38.7082 q^{50} -786.413 q^{52} -269.597 q^{53} +563.607 q^{55} +87.7345 q^{56} -54.0000 q^{58} +534.768 q^{59} -838.604 q^{61} -52.5836 q^{62} -723.276 q^{64} -851.234 q^{65} -448.180 q^{67} +768.056 q^{68} +485.718 q^{70} -628.604 q^{71} +925.266 q^{73} -309.692 q^{74} +505.072 q^{76} +557.771 q^{77} -963.479 q^{79} -480.778 q^{80} +164.748 q^{82} -133.358 q^{83} +831.364 q^{85} +727.702 q^{86} +431.246 q^{88} +778.581 q^{89} -842.420 q^{91} +228.718 q^{92} -2603.25 q^{94} +546.703 q^{95} +1603.57 q^{97} -972.282 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{4} + 26 q^{5} - 10 q^{7} + 12 q^{8} + 42 q^{10} + 60 q^{11} - 24 q^{13} + 50 q^{14} + 18 q^{16} + 150 q^{17} - 46 q^{19} - 14 q^{20} + 220 q^{22} + 46 q^{23} + 98 q^{25} - 348 q^{26}+ \cdots - 992 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^4 + 26 * q^5 - 10 * q^7 + 12 * q^8 + 42 * q^10 + 60 * q^11 - 24 * q^13 + 50 * q^14 + 18 * q^16 + 150 * q^17 - 46 * q^19 - 14 * q^20 + 220 * q^22 + 46 * q^23 + 98 * q^25 - 348 * q^26 + 270 * q^28 + 216 * q^29 + 324 * q^31 - 300 * q^32 + 310 * q^34 - 200 * q^35 + 140 * q^37 + 238 * q^38 + 146 * q^40 + 364 * q^41 - 126 * q^43 + 460 * q^44 + 92 * q^46 - 120 * q^47 - 146 * q^49 - 64 * q^50 - 1224 * q^52 - 490 * q^53 + 680 * q^55 + 10 * q^56 - 108 * q^58 + 32 * q^59 - 908 * q^61 - 132 * q^62 - 1214 * q^64 - 12 * q^65 - 874 * q^67 + 190 * q^68 + 560 * q^70 - 488 * q^71 + 652 * q^73 - 360 * q^74 + 1274 * q^76 + 400 * q^77 - 1198 * q^79 + 474 * q^80 + 88 * q^82 + 100 * q^83 + 1940 * q^85 + 798 * q^86 + 460 * q^88 - 102 * q^89 - 1980 * q^91 + 46 * q^92 - 2720 * q^94 - 928 * q^95 + 1624 * q^97 - 992 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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