LMFDB - Embedded newform 637.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.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:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.746052.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 8x + 2 $ x^5 - x^4 - 7x^3 + 8x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.00852$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00852 q^{2} +1.75906 q^{3} +2.03417 q^{4} +0.905722 q^{5} -3.53311 q^{6} -0.0686323 q^{8} +0.0942784 q^{9} -1.81916 q^{10} +0.716361 q^{11} +3.57822 q^{12} +1.00000 q^{13} +1.59322 q^{15} -3.93049 q^{16} +2.35227 q^{17} -0.189360 q^{18} +6.63591 q^{19} +1.84239 q^{20} -1.43883 q^{22} +3.75906 q^{23} -0.120728 q^{24} -4.17967 q^{25} -2.00852 q^{26} -5.11133 q^{27} +3.25799 q^{29} -3.20001 q^{30} +1.57050 q^{31} +8.03175 q^{32} +1.26012 q^{33} -4.72459 q^{34} +0.191778 q^{36} +5.20883 q^{37} -13.3284 q^{38} +1.75906 q^{39} -0.0621618 q^{40} +4.92168 q^{41} -9.43766 q^{43} +1.45720 q^{44} +0.0853900 q^{45} -7.55016 q^{46} -8.31986 q^{47} -6.91395 q^{48} +8.39497 q^{50} +4.13778 q^{51} +2.03417 q^{52} +14.0833 q^{53} +10.2662 q^{54} +0.648824 q^{55} +11.6729 q^{57} -6.54376 q^{58} +0.716361 q^{59} +3.24087 q^{60} -11.6527 q^{61} -3.15439 q^{62} -8.27099 q^{64} +0.905722 q^{65} -2.53098 q^{66} +9.39174 q^{67} +4.78492 q^{68} +6.61239 q^{69} +10.9914 q^{71} -0.00647055 q^{72} -3.47300 q^{73} -10.4621 q^{74} -7.35227 q^{75} +13.4986 q^{76} -3.53311 q^{78} +13.0082 q^{79} -3.55993 q^{80} -9.27395 q^{81} -9.88531 q^{82} +3.54083 q^{83} +2.13050 q^{85} +18.9558 q^{86} +5.73099 q^{87} -0.0491655 q^{88} +12.0501 q^{89} -0.171508 q^{90} +7.64656 q^{92} +2.76260 q^{93} +16.7106 q^{94} +6.01029 q^{95} +14.1283 q^{96} +7.43766 q^{97} +0.0675374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * q^99

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 $

$2$	−2.00852	−1.42024	−0.710121	−	0.704080i	$-0.751360\pi$
$2$	−2.00852	−1.42024	−0.710121	+	0.704080i	$0.751360\pi$
$3$	1.75906	1.01559	0.507796	−	0.861477i	$-0.330460\pi$
$3$	1.75906	1.01559	0.507796	+	0.861477i	$0.330460\pi$
$4$	2.03417	1.01709
$5$	0.905722	0.405051	0.202526	−	0.979277i	$-0.435085\pi$
$5$	0.905722	0.405051	0.202526	+	0.979277i	$0.435085\pi$
$6$	−3.53311	−1.44238
$7$	0	0
$7$	0	0
$8$	−0.0686323	−0.0242652
$9$	0.0942784	0.0314261
$10$	−1.81916	−0.575270
$11$	0.716361	0.215991	0.107996	−	0.994151i	$-0.465557\pi$
$11$	0.716361	0.215991	0.107996	+	0.994151i	$0.465557\pi$
$12$	3.57822	1.03294
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.59322	0.411366
$16$	−3.93049	−0.982623
$17$	2.35227	0.570510	0.285255	−	0.958452i	$-0.407922\pi$
$17$	2.35227	0.570510	0.285255	+	0.958452i	$0.407922\pi$
$18$	−0.189360	−0.0446327
$19$	6.63591	1.52238	0.761191	−	0.648528i	$-0.224615\pi$
$19$	6.63591	1.52238	0.761191	+	0.648528i	$0.224615\pi$
$20$	1.84239	0.411971
$21$	0	0
$22$	−1.43883	−0.306759
$23$	3.75906	0.783817	0.391909	−	0.920004i	$-0.371815\pi$
$23$	3.75906	0.783817	0.391909	+	0.920004i	$0.371815\pi$
$24$	−0.120728	−0.0246435
$25$	−4.17967	−0.835934
$26$	−2.00852	−0.393904
$27$	−5.11133	−0.983675
$28$	0	0
$29$	3.25799	0.604994	0.302497	−	0.953150i	$-0.402180\pi$
$29$	3.25799	0.604994	0.302497	+	0.953150i	$0.402180\pi$
$30$	−3.20001	−0.584239
$31$	1.57050	0.282070	0.141035	−	0.990005i	$-0.454957\pi$
$31$	1.57050	0.282070	0.141035	+	0.990005i	$0.454957\pi$
$32$	8.03175	1.41983
$33$	1.26012	0.219359
$34$	−4.72459	−0.810261
$35$	0	0
$36$	0.191778	0.0319631
$37$	5.20883	0.856326	0.428163	−	0.903702i	$-0.359161\pi$
$37$	5.20883	0.856326	0.428163	+	0.903702i	$0.359161\pi$
$38$	−13.3284	−2.16215
$39$	1.75906	0.281674
$40$	−0.0621618	−0.00982864
$41$	4.92168	0.768637	0.384318	−	0.923201i	$-0.374437\pi$
$41$	4.92168	0.768637	0.384318	+	0.923201i	$0.374437\pi$
$42$	0	0
$43$	−9.43766	−1.43923	−0.719615	−	0.694373i	$-0.755682\pi$
$43$	−9.43766	−1.43923	−0.719615	+	0.694373i	$0.755682\pi$
$44$	1.45720	0.219681
$45$	0.0853900	0.0127292
$46$	−7.55016	−1.11321
$47$	−8.31986	−1.21358	−0.606788	−	0.794863i	$-0.707542\pi$
$47$	−8.31986	−1.21358	−0.606788	+	0.794863i	$0.707542\pi$
$48$	−6.91395	−0.997943
$49$	0	0
$50$	8.39497	1.18723
$51$	4.13778	0.579405
$52$	2.03417	0.282089
$53$	14.0833	1.93449	0.967243	−	0.253854i	$-0.0816983\pi$
$53$	14.0833	1.93449	0.967243	+	0.253854i	$0.0816983\pi$
$54$	10.2662	1.39706
$55$	0.648824	0.0874874
$56$	0	0
$57$	11.6729	1.54612
$58$	−6.54376	−0.859238
$59$	0.716361	0.0932623	0.0466311	−	0.998912i	$-0.485151\pi$
$59$	0.716361	0.0932623	0.0466311	+	0.998912i	$0.485151\pi$
$60$	3.24087	0.418395
$61$	−11.6527	−1.49197	−0.745986	−	0.665962i	$-0.768021\pi$
$61$	−11.6527	−1.49197	−0.745986	+	0.665962i	$0.768021\pi$
$62$	−3.15439	−0.400607
$63$	0	0
$64$	−8.27099	−1.03387
$65$	0.905722	0.112341
$66$	−2.53098	−0.311542
$67$	9.39174	1.14738	0.573692	−	0.819071i	$-0.305511\pi$
$67$	9.39174	1.14738	0.573692	+	0.819071i	$0.305511\pi$
$68$	4.78492	0.580257
$69$	6.61239	0.796038
$70$	0	0
$71$	10.9914	1.30444	0.652220	−	0.758030i	$-0.273838\pi$
$71$	10.9914	1.30444	0.652220	+	0.758030i	$0.273838\pi$
$72$	−0.00647055	−0.000762561 0
$73$	−3.47300	−0.406484	−0.203242	−	0.979129i	$-0.565148\pi$
$73$	−3.47300	−0.406484	−0.203242	+	0.979129i	$0.565148\pi$
$74$	−10.4621	−1.21619
$75$	−7.35227	−0.848967
$76$	13.4986	1.54839
$77$	0	0
$78$	−3.53311	−0.400046
$79$	13.0082	1.46353	0.731766	−	0.681556i	$-0.238696\pi$
$79$	13.0082	1.46353	0.731766	+	0.681556i	$0.238696\pi$
$80$	−3.55993	−0.398012
$81$	−9.27395	−1.03044
$82$	−9.88531	−1.09165
$83$	3.54083	0.388656	0.194328	−	0.980937i	$-0.437747\pi$
$83$	3.54083	0.388656	0.194328	+	0.980937i	$0.437747\pi$
$84$	0	0
$85$	2.13050	0.231085
$86$	18.9558	2.04405
$87$	5.73099	0.614427
$88$	−0.0491655	−0.00524106
$89$	12.0501	1.27730	0.638651	−	0.769496i	$-0.279493\pi$
$89$	12.0501	1.27730	0.638651	+	0.769496i	$0.279493\pi$
$90$	−0.171508	−0.0180785
$91$	0	0
$92$	7.64656	0.797209
$93$	2.76260	0.286468
$94$	16.7106	1.72357
$95$	6.01029	0.616642
$96$	14.1283	1.44196
$97$	7.43766	0.755180	0.377590	−	0.925973i	$-0.376753\pi$
$97$	7.43766	0.755180	0.377590	+	0.925973i	$0.376753\pi$
$98$	0	0
$99$	0.0675374	0.00678776
$100$	−8.50216	−0.850216
$101$	−1.19905	−0.119310	−0.0596551	−	0.998219i	$-0.519000\pi$
$101$	−1.19905	−0.119310	−0.0596551	+	0.998219i	$0.519000\pi$
$102$	−8.31083	−0.822894
$103$	−14.4123	−1.42009	−0.710043	−	0.704158i	$-0.751325\pi$
$103$	−14.4123	−1.42009	−0.710043	+	0.704158i	$0.751325\pi$
$104$	−0.0686323	−0.00672995
$105$	0	0
$106$	−28.2866	−2.74744
$107$	13.5932	1.31411	0.657053	−	0.753845i	$-0.271803\pi$
$107$	13.5932	1.31411	0.657053	+	0.753845i	$0.271803\pi$
$108$	−10.3973	−1.00048
$109$	−13.7248	−1.31460	−0.657299	−	0.753630i	$-0.728301\pi$
$109$	−13.7248	−1.31460	−0.657299	+	0.753630i	$0.728301\pi$
$110$	−1.30318	−0.124253
$111$	9.16262	0.869677
$112$	0	0
$113$	−3.25799	−0.306486	−0.153243	−	0.988189i	$-0.548972\pi$
$113$	−3.25799	−0.306486	−0.153243	+	0.988189i	$0.548972\pi$
$114$	−23.4454	−2.19586
$115$	3.40466	0.317486
$116$	6.62731	0.615331
$117$	0.0942784	0.00871604
$118$	−1.43883	−0.132455
$119$	0	0
$120$	−0.109346	−0.00998189
$121$	−10.4868	−0.953348
$122$	23.4047	2.11896
$123$	8.65750	0.780621
$124$	3.19466	0.286889
$125$	−8.31422	−0.743647
$126$	0	0
$127$	−0.950834	−0.0843729	−0.0421865	−	0.999110i	$-0.513432\pi$
$127$	−0.950834	−0.0843729	−0.0421865	+	0.999110i	$0.513432\pi$
$128$	0.548979	0.0485233
$129$	−16.6014	−1.46167
$130$	−1.81916	−0.159551
$131$	−18.8196	−1.64428	−0.822138	−	0.569288i	$-0.807219\pi$
$131$	−18.8196	−1.64428	−0.822138	+	0.569288i	$0.807219\pi$
$132$	2.56330	0.223106
$133$	0	0
$134$	−18.8635	−1.62956
$135$	−4.62944	−0.398439
$136$	−0.161442	−0.0138435
$137$	6.18179	0.528146	0.264073	−	0.964503i	$-0.414934\pi$
$137$	6.18179	0.528146	0.264073	+	0.964503i	$0.414934\pi$
$138$	−13.2811	−1.13057
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−14.6351	−1.23250
$142$	−22.0765	−1.85262
$143$	0.716361	0.0599051
$144$	−0.370560	−0.0308800
$145$	2.95083	0.245053
$146$	6.97560	0.577305
$147$	0	0
$148$	10.5956	0.870956
$149$	−21.0771	−1.72670	−0.863351	−	0.504604i	$-0.831639\pi$
$149$	−21.0771	−1.72670	−0.863351	+	0.504604i	$0.831639\pi$
$150$	14.7672	1.20574
$151$	−15.7234	−1.27955	−0.639777	−	0.768560i	$-0.720973\pi$
$151$	−15.7234	−1.27955	−0.639777	+	0.768560i	$0.720973\pi$
$152$	−0.455438	−0.0369409
$153$	0.221768	0.0179289
$154$	0	0
$155$	1.42244	0.114253
$156$	3.57822	0.286487
$157$	−7.78499	−0.621310	−0.310655	−	0.950523i	$-0.600548\pi$
$157$	−7.78499	−0.621310	−0.310655	+	0.950523i	$0.600548\pi$
$158$	−26.1272	−2.07857
$159$	24.7733	1.96465
$160$	7.27453	0.575102
$161$	0	0
$162$	18.6269	1.46347
$163$	1.68991	0.132364	0.0661820	−	0.997808i	$-0.478918\pi$
$163$	1.68991	0.132364	0.0661820	+	0.997808i	$0.478918\pi$
$164$	10.0115	0.781769
$165$	1.14132	0.0888514
$166$	−7.11184	−0.551986
$167$	−21.8667	−1.69210	−0.846049	−	0.533105i	$-0.821025\pi$
$167$	−21.8667	−1.69210	−0.846049	+	0.533105i	$0.821025\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.27917	−0.328197
$171$	0.625623	0.0478426
$172$	−19.1978	−1.46382
$173$	5.84122	0.444100	0.222050	−	0.975035i	$-0.428725\pi$
$173$	5.84122	0.444100	0.222050	+	0.975035i	$0.428725\pi$
$174$	−11.5108	−0.872634
$175$	0	0
$176$	−2.81565	−0.212238
$177$	1.26012	0.0947164
$178$	−24.2028	−1.81408
$179$	2.53427	0.189421	0.0947103	−	0.995505i	$-0.469808\pi$
$179$	2.53427	0.189421	0.0947103	+	0.995505i	$0.469808\pi$
$180$	0.173698	0.0129467
$181$	−10.7248	−0.797169	−0.398585	−	0.917132i	$-0.630498\pi$
$181$	−10.7248	−0.797169	−0.398585	+	0.917132i	$0.630498\pi$
$182$	0	0
$183$	−20.4977	−1.51523
$184$	−0.257993	−0.0190195
$185$	4.71775	0.346856
$186$	−5.54874	−0.406854
$187$	1.68508	0.123225
$188$	−16.9240	−1.23431
$189$	0	0
$190$	−12.0718	−0.875781
$191$	−1.67861	−0.121460	−0.0607298	−	0.998154i	$-0.519343\pi$
$191$	−1.67861	−0.121460	−0.0607298	+	0.998154i	$0.519343\pi$
$192$	−14.5491	−1.04999
$193$	−6.44816	−0.464148	−0.232074	−	0.972698i	$-0.574551\pi$
$193$	−6.44816	−0.464148	−0.232074	+	0.972698i	$0.574551\pi$
$194$	−14.9387	−1.07254
$195$	1.59322	0.114093
$196$	0	0
$197$	1.87251	0.133411	0.0667054	−	0.997773i	$-0.478751\pi$
$197$	1.87251	0.133411	0.0667054	+	0.997773i	$0.478751\pi$
$198$	−0.135650	−0.00964026
$199$	−11.3967	−0.807888	−0.403944	−	0.914784i	$-0.632361\pi$
$199$	−11.3967	−0.807888	−0.403944	+	0.914784i	$0.632361\pi$
$200$	0.286860	0.0202841
$201$	16.5206	1.16527
$202$	2.40833	0.169449
$203$	0	0
$204$	8.41694	0.589304
$205$	4.45767	0.311337
$206$	28.9475	2.01687
$207$	0.354398	0.0246324
$208$	−3.93049	−0.272531
$209$	4.75371	0.328821
$210$	0	0
$211$	7.53599	0.518799	0.259400	−	0.965770i	$-0.416475\pi$
$211$	7.53599	0.518799	0.259400	+	0.965770i	$0.416475\pi$
$212$	28.6478	1.96754
$213$	19.3345	1.32478
$214$	−27.3023	−1.86635
$215$	−8.54789	−0.582962
$216$	0.350802	0.0238691
$217$	0	0
$218$	27.5666	1.86705
$219$	−6.10920	−0.412822
$220$	1.31982	0.0889821
$221$	2.35227	0.158231
$222$	−18.4033	−1.23515
$223$	17.6349	1.18092	0.590459	−	0.807067i	$-0.298947\pi$
$223$	17.6349	1.18092	0.590459	+	0.807067i	$0.298947\pi$
$224$	0	0
$225$	−0.394052	−0.0262702
$226$	6.54376	0.435284
$227$	−5.32904	−0.353701	−0.176851	−	0.984238i	$-0.556591\pi$
$227$	−5.32904	−0.353701	−0.176851	+	0.984238i	$0.556591\pi$
$228$	23.7447	1.57253
$229$	−8.51900	−0.562951	−0.281476	−	0.959568i	$-0.590824\pi$
$229$	−8.51900	−0.562951	−0.281476	+	0.959568i	$0.590824\pi$
$230$	−6.83834	−0.450907
$231$	0	0
$232$	−0.223604	−0.0146803
$233$	4.75371	0.311426	0.155713	−	0.987802i	$-0.450233\pi$
$233$	4.75371	0.311426	0.155713	+	0.987802i	$0.450233\pi$
$234$	−0.189360	−0.0123789
$235$	−7.53548	−0.491561
$236$	1.45720	0.0948557
$237$	22.8821	1.48635
$238$	0	0
$239$	14.8314	0.959365	0.479682	−	0.877442i	$-0.340752\pi$
$239$	14.8314	0.959365	0.479682	+	0.877442i	$0.340752\pi$
$240$	−6.26212	−0.404218
$241$	−6.12131	−0.394308	−0.197154	−	0.980373i	$-0.563170\pi$
$241$	−6.12131	−0.394308	−0.197154	+	0.980373i	$0.563170\pi$
$242$	21.0630	1.35398
$243$	−0.979411	−0.0628292
$244$	−23.7035	−1.51746
$245$	0	0
$246$	−17.3888	−1.10867
$247$	6.63591	0.422233
$248$	−0.107787	−0.00684448
$249$	6.22852	0.394716
$250$	16.6993	1.05616
$251$	−13.9708	−0.881832	−0.440916	−	0.897548i	$-0.645346\pi$
$251$	−13.9708	−0.881832	−0.440916	+	0.897548i	$0.645346\pi$
$252$	0	0
$253$	2.69284	0.169298
$254$	1.90977	0.119830
$255$	3.74767	0.234688
$256$	15.4393	0.964959
$257$	17.2651	1.07696	0.538482	−	0.842637i	$-0.318998\pi$
$257$	17.2651	1.07696	0.538482	+	0.842637i	$0.318998\pi$
$258$	33.3443	2.07592
$259$	0	0
$260$	1.84239	0.114260
$261$	0.307158	0.0190126
$262$	37.7996	2.33527
$263$	−2.60672	−0.160737	−0.0803687	−	0.996765i	$-0.525610\pi$
$263$	−2.60672	−0.160737	−0.0803687	+	0.996765i	$0.525610\pi$
$264$	−0.0864849	−0.00532278
$265$	12.7555	0.783565
$266$	0	0
$267$	21.1967	1.29722
$268$	19.1044	1.16699
$269$	−14.4895	−0.883443	−0.441721	−	0.897152i	$-0.645632\pi$
$269$	−14.4895	−0.883443	−0.441721	+	0.897152i	$0.645632\pi$
$270$	9.29834	0.565879
$271$	−8.63591	−0.524594	−0.262297	−	0.964987i	$-0.584480\pi$
$271$	−8.63591	−0.524594	−0.262297	+	0.964987i	$0.584480\pi$
$272$	−9.24558	−0.560596
$273$	0	0
$274$	−12.4163	−0.750095
$275$	−2.99415	−0.180554
$276$	13.4507	0.809639
$277$	12.2270	0.734647	0.367324	−	0.930093i	$-0.380274\pi$
$277$	12.2270	0.734647	0.367324	+	0.930093i	$0.380274\pi$
$278$	8.03410	0.481853
$279$	0.148064	0.00886437
$280$	0	0
$281$	−24.1822	−1.44259	−0.721293	−	0.692630i	$-0.756452\pi$
$281$	−24.1822	−1.44259	−0.721293	+	0.692630i	$0.756452\pi$
$282$	29.3950	1.75044
$283$	−30.7683	−1.82899	−0.914493	−	0.404601i	$-0.867410\pi$
$283$	−30.7683	−1.82899	−0.914493	+	0.404601i	$0.867410\pi$
$284$	22.3584	1.32673
$285$	10.5724	0.626257
$286$	−1.43883	−0.0850797
$287$	0	0
$288$	0.757221	0.0446197
$289$	−11.4668	−0.674519
$290$	−5.92682	−0.348035
$291$	13.0833	0.766954
$292$	−7.06467	−0.413429
$293$	−31.8295	−1.85950	−0.929749	−	0.368193i	$-0.879976\pi$
$293$	−31.8295	−1.85950	−0.929749	+	0.368193i	$0.879976\pi$
$294$	0	0
$295$	0.648824	0.0377760
$296$	−0.357494	−0.0207789
$297$	−3.66156	−0.212465
$298$	42.3338	2.45233
$299$	3.75906	0.217392
$300$	−14.9558	−0.863472
$301$	0	0
$302$	31.5809	1.81728
$303$	−2.10920	−0.121170
$304$	−26.0824	−1.49593
$305$	−10.5541	−0.604324
$306$	−0.445427	−0.0254634
$307$	28.7884	1.64304	0.821520	−	0.570179i	$-0.193126\pi$
$307$	28.7884	1.64304	0.821520	+	0.570179i	$0.193126\pi$
$308$	0	0
$309$	−25.3521	−1.44223
$310$	−2.85700	−0.162266
$311$	5.51862	0.312932	0.156466	−	0.987683i	$-0.449990\pi$
$311$	5.51862	0.312932	0.156466	+	0.987683i	$0.449990\pi$
$312$	−0.120728	−0.00683488
$313$	−4.84799	−0.274024	−0.137012	−	0.990569i	$-0.543750\pi$
$313$	−4.84799	−0.274024	−0.137012	+	0.990569i	$0.543750\pi$
$314$	15.6363	0.882410
$315$	0	0
$316$	26.4608	1.48854
$317$	−7.65511	−0.429954	−0.214977	−	0.976619i	$-0.568968\pi$
$317$	−7.65511	−0.429954	−0.214977	+	0.976619i	$0.568968\pi$
$318$	−49.7577	−2.79027
$319$	2.33390	0.130673
$320$	−7.49121	−0.418772
$321$	23.9112	1.33459
$322$	0	0
$323$	15.6095	0.868534
$324$	−18.8648	−1.04804
$325$	−4.17967	−0.231846
$326$	−3.39423	−0.187989
$327$	−24.1427	−1.33510
$328$	−0.337786	−0.0186511
$329$	0	0
$330$	−2.29236	−0.126190
$331$	11.3432	0.623477	0.311739	−	0.950168i	$-0.399089\pi$
$331$	11.3432	0.623477	0.311739	+	0.950168i	$0.399089\pi$
$332$	7.20265	0.395297
$333$	0.491080	0.0269110
$334$	43.9199	2.40319
$335$	8.50631	0.464749
$336$	0	0
$337$	1.74149	0.0948649	0.0474324	−	0.998874i	$-0.484896\pi$
$337$	1.74149	0.0948649	0.0474324	+	0.998874i	$0.484896\pi$
$338$	−2.00852	−0.109249
$339$	−5.73099	−0.311265
$340$	4.33381	0.235034
$341$	1.12504	0.0609246
$342$	−1.25658	−0.0679480
$343$	0	0
$344$	0.647729	0.0349232
$345$	5.98898	0.322436
$346$	−11.7322	−0.630729
$347$	21.0503	1.13004	0.565019	−	0.825078i	$-0.308869\pi$
$347$	21.0503	1.13004	0.565019	+	0.825078i	$0.308869\pi$
$348$	11.6578	0.624925
$349$	−8.35601	−0.447287	−0.223643	−	0.974671i	$-0.571795\pi$
$349$	−8.35601	−0.447287	−0.223643	+	0.974671i	$0.571795\pi$
$350$	0	0
$351$	−5.11133	−0.272822
$352$	5.75364	0.306670
$353$	−8.53355	−0.454195	−0.227097	−	0.973872i	$-0.572924\pi$
$353$	−8.53355	−0.454195	−0.227097	+	0.973872i	$0.572924\pi$
$354$	−2.53098	−0.134520
$355$	9.95515	0.528365
$356$	24.5119	1.29913
$357$	0	0
$358$	−5.09015	−0.269023
$359$	−16.1713	−0.853488	−0.426744	−	0.904372i	$-0.640339\pi$
$359$	−16.1713	−0.853488	−0.426744	+	0.904372i	$0.640339\pi$
$360$	−0.00586051	−0.000308876 0
$361$	25.0353	1.31765
$362$	21.5410	1.13217
$363$	−18.4469	−0.968212
$364$	0	0
$365$	−3.14557	−0.164647
$366$	41.1701	2.15200
$367$	28.1540	1.46963	0.734813	−	0.678269i	$-0.237270\pi$
$367$	28.1540	1.46963	0.734813	+	0.678269i	$0.237270\pi$
$368$	−14.7749	−0.770197
$369$	0.464008	0.0241553
$370$	−9.47571	−0.492619
$371$	0	0
$372$	5.61959	0.291362
$373$	−28.5037	−1.47586	−0.737932	−	0.674875i	$-0.764197\pi$
$373$	−28.5037	−1.47586	−0.737932	+	0.674875i	$0.764197\pi$
$374$	−3.38452	−0.175009
$375$	−14.6252	−0.755241
$376$	0.571012	0.0294477
$377$	3.25799	0.167795
$378$	0	0
$379$	−7.26263	−0.373056	−0.186528	−	0.982450i	$-0.559724\pi$
$379$	−7.26263	−0.373056	−0.186528	+	0.982450i	$0.559724\pi$
$380$	12.2259	0.627178
$381$	−1.67257	−0.0856884
$382$	3.37152	0.172502
$383$	12.9325	0.660822	0.330411	−	0.943837i	$-0.392813\pi$
$383$	12.9325	0.660822	0.330411	+	0.943837i	$0.392813\pi$
$384$	0.965684	0.0492799
$385$	0	0
$386$	12.9513	0.659203
$387$	−0.889768	−0.0452294
$388$	15.1295	0.768083
$389$	−21.1357	−1.07162	−0.535811	−	0.844338i	$-0.679994\pi$
$389$	−21.1357	−1.07162	−0.535811	+	0.844338i	$0.679994\pi$
$390$	−3.20001	−0.162039
$391$	8.84232	0.447175
$392$	0	0
$393$	−33.1047	−1.66991
$394$	−3.76098	−0.189476
$395$	11.7818	0.592805
$396$	0.137383	0.00690373
$397$	19.2073	0.963988	0.481994	−	0.876175i	$-0.339913\pi$
$397$	19.2073	0.963988	0.481994	+	0.876175i	$0.339913\pi$
$398$	22.8905	1.14740
$399$	0	0
$400$	16.4282	0.821408
$401$	16.6692	0.832420	0.416210	−	0.909268i	$-0.363358\pi$
$401$	16.6692	0.832420	0.416210	+	0.909268i	$0.363358\pi$
$402$	−33.1820	−1.65497
$403$	1.57050	0.0782321
$404$	−2.43908	−0.121349
$405$	−8.39961	−0.417380
$406$	0	0
$407$	3.73140	0.184959
$408$	−0.283985	−0.0140594
$409$	12.3483	0.610585	0.305293	−	0.952259i	$-0.401246\pi$
$409$	12.3483	0.610585	0.305293	+	0.952259i	$0.401246\pi$
$410$	−8.95333	−0.442174
$411$	10.8741	0.536381
$412$	−29.3171	−1.44435
$413$	0	0
$414$	−0.711817	−0.0349839
$415$	3.20700	0.157426
$416$	8.03175	0.393789
$417$	−7.03622	−0.344565
$418$	−9.54794	−0.467005
$419$	4.35934	0.212968	0.106484	−	0.994314i	$-0.466041\pi$
$419$	4.35934	0.212968	0.106484	+	0.994314i	$0.466041\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−15.1362	−0.736820
$423$	−0.784383	−0.0381380
$424$	−0.966567	−0.0469407
$425$	−9.83171	−0.476908
$426$	−38.8338	−1.88150
$427$	0	0
$428$	27.6509	1.33656
$429$	1.26012	0.0608391
$430$	17.1687	0.827946
$431$	−23.3626	−1.12533	−0.562667	−	0.826683i	$-0.690225\pi$
$431$	−23.3626	−1.12533	−0.562667	+	0.826683i	$0.690225\pi$
$432$	20.0900	0.966582
$433$	−2.71285	−0.130371	−0.0651856	−	0.997873i	$-0.520764\pi$
$433$	−2.71285	−0.130371	−0.0651856	+	0.997873i	$0.520764\pi$
$434$	0	0
$435$	5.19068	0.248874
$436$	−27.9186	−1.33706
$437$	24.9448	1.19327
$438$	12.2705	0.586306
$439$	−8.83519	−0.421681	−0.210840	−	0.977521i	$-0.567620\pi$
$439$	−8.83519	−0.421681	−0.210840	+	0.977521i	$0.567620\pi$
$440$	−0.0445303	−0.00212290
$441$	0	0
$442$	−4.72459	−0.224726
$443$	−2.90558	−0.138048	−0.0690240	−	0.997615i	$-0.521989\pi$
$443$	−2.90558	−0.138048	−0.0690240	+	0.997615i	$0.521989\pi$
$444$	18.6383	0.884536
$445$	10.9140	0.517373
$446$	−35.4201	−1.67719
$447$	−37.0758	−1.75362
$448$	0	0
$449$	−15.2777	−0.720998	−0.360499	−	0.932760i	$-0.617394\pi$
$449$	−15.2777	−0.720998	−0.360499	+	0.932760i	$0.617394\pi$
$450$	0.791464	0.0373100
$451$	3.52570	0.166019
$452$	−6.62731	−0.311723
$453$	−27.6584	−1.29950
$454$	10.7035	0.502341
$455$	0	0
$456$	−0.801141	−0.0375169
$457$	23.6600	1.10677	0.553384	−	0.832926i	$-0.313336\pi$
$457$	23.6600	1.10677	0.553384	+	0.832926i	$0.313336\pi$
$458$	17.1106	0.799527
$459$	−12.0232	−0.561196
$460$	6.92566	0.322910
$461$	−26.6170	−1.23968	−0.619839	−	0.784729i	$-0.712802\pi$
$461$	−26.6170	−1.23968	−0.619839	+	0.784729i	$0.712802\pi$
$462$	0	0
$463$	−1.44250	−0.0670385	−0.0335193	−	0.999438i	$-0.510672\pi$
$463$	−1.44250	−0.0670385	−0.0335193	+	0.999438i	$0.510672\pi$
$464$	−12.8055	−0.594481
$465$	2.50214	0.116034
$466$	−9.54794	−0.442300
$467$	−8.38959	−0.388224	−0.194112	−	0.980979i	$-0.562183\pi$
$467$	−8.38959	−0.388224	−0.194112	+	0.980979i	$0.562183\pi$
$468$	0.191778	0.00886496
$469$	0	0
$470$	15.1352	0.698135
$471$	−13.6942	−0.630997
$472$	−0.0491655	−0.00226303
$473$	−6.76077	−0.310861
$474$	−45.9592	−2.11098
$475$	−27.7359	−1.27261
$476$	0	0
$477$	1.32775	0.0607934
$478$	−29.7893	−1.36253
$479$	12.6122	0.576265	0.288132	−	0.957591i	$-0.406966\pi$
$479$	12.6122	0.576265	0.288132	+	0.957591i	$0.406966\pi$
$480$	12.7963	0.584069
$481$	5.20883	0.237502
$482$	12.2948	0.560013
$483$	0	0
$484$	−21.3320	−0.969636
$485$	6.73645	0.305886
$486$	1.96717	0.0892327
$487$	21.5680	0.977341	0.488671	−	0.872468i	$-0.337482\pi$
$487$	21.5680	0.977341	0.488671	+	0.872468i	$0.337482\pi$
$488$	0.799750	0.0362030
$489$	2.97265	0.134428
$490$	0	0
$491$	39.2347	1.77064	0.885318	−	0.464987i	$-0.153941\pi$
$491$	39.2347	1.77064	0.885318	+	0.464987i	$0.153941\pi$
$492$	17.6108	0.793958
$493$	7.66368	0.345155
$494$	−13.3284	−0.599673
$495$	0.0611701	0.00274939
$496$	−6.17283	−0.277168
$497$	0	0
$498$	−12.5101	−0.560592
$499$	9.16814	0.410422	0.205211	−	0.978718i	$-0.434212\pi$
$499$	9.16814	0.410422	0.205211	+	0.978718i	$0.434212\pi$
$500$	−16.9125	−0.756352
$501$	−38.4648	−1.71848
$502$	28.0608	1.25241
$503$	24.9370	1.11188	0.555942	−	0.831221i	$-0.312358\pi$
$503$	24.9370	1.11188	0.555942	+	0.831221i	$0.312358\pi$
$504$	0	0
$505$	−1.08601	−0.0483267
$506$	−5.40864	−0.240443
$507$	1.75906	0.0781224
$508$	−1.93416	−0.0858144
$509$	5.89807	0.261428	0.130714	−	0.991420i	$-0.458273\pi$
$509$	5.89807	0.261428	0.130714	+	0.991420i	$0.458273\pi$
$510$	−7.52730	−0.333314
$511$	0	0
$512$	−32.1083	−1.41900
$513$	−33.9183	−1.49753
$514$	−34.6773	−1.52955
$515$	−13.0535	−0.575208
$516$	−33.7700	−1.48664
$517$	−5.96003	−0.262122
$518$	0	0
$519$	10.2750	0.451024
$520$	−0.0621618	−0.00272597
$521$	−37.1895	−1.62930	−0.814652	−	0.579950i	$-0.803072\pi$
$521$	−37.1895	−1.62930	−0.814652	+	0.579950i	$0.803072\pi$
$522$	−0.616935	−0.0270025
$523$	−5.09080	−0.222605	−0.111303	−	0.993787i	$-0.535502\pi$
$523$	−5.09080	−0.222605	−0.111303	+	0.993787i	$0.535502\pi$
$524$	−38.2823	−1.67237
$525$	0	0
$526$	5.23567	0.228286
$527$	3.69424	0.160924
$528$	−4.95289	−0.215547
$529$	−8.86950	−0.385630
$530$	−25.6198	−1.11285
$531$	0.0675374	0.00293087
$532$	0	0
$533$	4.92168	0.213181
$534$	−42.5741	−1.84236
$535$	12.3117	0.532280
$536$	−0.644577	−0.0278415
$537$	4.45793	0.192374
$538$	29.1026	1.25470
$539$	0	0
$540$	−9.41707	−0.405246
$541$	−0.766850	−0.0329694	−0.0164847	−	0.999864i	$-0.505247\pi$
$541$	−0.766850	−0.0329694	−0.0164847	+	0.999864i	$0.505247\pi$
$542$	17.3454	0.745050
$543$	−18.8655	−0.809598
$544$	18.8929	0.810025
$545$	−12.4309	−0.532480
$546$	0	0
$547$	14.1428	0.604702	0.302351	−	0.953197i	$-0.402229\pi$
$547$	14.1428	0.604702	0.302351	+	0.953197i	$0.402229\pi$
$548$	12.5748	0.537170
$549$	−1.09859	−0.0468869
$550$	6.01383	0.256430
$551$	21.6197	0.921032
$552$	−0.453824	−0.0193160
$553$	0	0
$554$	−24.5582	−1.04338
$555$	8.29878	0.352264
$556$	−8.13668	−0.345072
$557$	−24.8627	−1.05347	−0.526733	−	0.850031i	$-0.676583\pi$
$557$	−24.8627	−1.05347	−0.526733	+	0.850031i	$0.676583\pi$
$558$	−0.297391	−0.0125895
$559$	−9.43766	−0.399171
$560$	0	0
$561$	2.96414	0.125146
$562$	48.5705	2.04882
$563$	44.0094	1.85478	0.927388	−	0.374101i	$-0.122049\pi$
$563$	44.0094	1.85478	0.927388	+	0.374101i	$0.122049\pi$
$564$	−29.7703	−1.25356
$565$	−2.95083	−0.124143
$566$	61.7989	2.59760
$567$	0	0
$568$	−0.754366	−0.0316525
$569$	−33.2616	−1.39440	−0.697199	−	0.716877i	$-0.745571\pi$
$569$	−33.2616	−1.39440	−0.697199	+	0.716877i	$0.745571\pi$
$570$	−21.2350	−0.889436
$571$	−12.3540	−0.516998	−0.258499	−	0.966011i	$-0.583228\pi$
$571$	−12.3540	−0.516998	−0.258499	+	0.966011i	$0.583228\pi$
$572$	1.45720	0.0609286
$573$	−2.95276	−0.123353
$574$	0	0
$575$	−15.7116	−0.655219
$576$	−0.779776	−0.0324907
$577$	−25.9659	−1.08097	−0.540486	−	0.841353i	$-0.681760\pi$
$577$	−25.9659	−1.08097	−0.540486	+	0.841353i	$0.681760\pi$
$578$	23.0314	0.957979
$579$	−11.3427	−0.471385
$580$	6.00250	0.249240
$581$	0	0
$582$	−26.2781	−1.08926
$583$	10.0887	0.417831
$584$	0.238360	0.00986341
$585$	0.0853900	0.00353044
$586$	63.9303	2.64094
$587$	−23.9747	−0.989543	−0.494771	−	0.869023i	$-0.664748\pi$
$587$	−23.9747	−0.989543	−0.494771	+	0.869023i	$0.664748\pi$
$588$	0	0
$589$	10.4217	0.429418
$590$	−1.30318	−0.0536510
$591$	3.29385	0.135491
$592$	−20.4733	−0.841445
$593$	47.0480	1.93203	0.966015	−	0.258484i	$-0.0832230\pi$
$593$	47.0480	1.93203	0.966015	+	0.258484i	$0.0832230\pi$
$594$	7.35433	0.301752
$595$	0	0
$596$	−42.8744	−1.75620
$597$	−20.0474	−0.820484
$598$	−7.55016	−0.308749
$599$	20.1736	0.824271	0.412135	−	0.911123i	$-0.364783\pi$
$599$	20.1736	0.824271	0.412135	+	0.911123i	$0.364783\pi$
$600$	0.504604	0.0206004
$601$	29.5773	1.20648	0.603242	−	0.797558i	$-0.293875\pi$
$601$	29.5773	1.20648	0.603242	+	0.797558i	$0.293875\pi$
$602$	0	0
$603$	0.885439	0.0360579
$604$	−31.9841	−1.30142
$605$	−9.49815	−0.386155
$606$	4.23638	0.172091
$607$	−15.4420	−0.626771	−0.313385	−	0.949626i	$-0.601463\pi$
$607$	−15.4420	−0.626771	−0.313385	+	0.949626i	$0.601463\pi$
$608$	53.2980	2.16152
$609$	0	0
$610$	21.1981	0.858287
$611$	−8.31986	−0.336586
$612$	0.451115	0.0182352
$613$	1.99485	0.0805711	0.0402855	−	0.999188i	$-0.487173\pi$
$613$	1.99485	0.0805711	0.0402855	+	0.999188i	$0.487173\pi$
$614$	−57.8222	−2.33351
$615$	7.84129	0.316191
$616$	0	0
$617$	−2.85584	−0.114972	−0.0574858	−	0.998346i	$-0.518308\pi$
$617$	−2.85584	−0.114972	−0.0574858	+	0.998346i	$0.518308\pi$
$618$	50.9202	2.04831
$619$	31.9823	1.28548	0.642738	−	0.766086i	$-0.277798\pi$
$619$	31.9823	1.28548	0.642738	+	0.766086i	$0.277798\pi$
$620$	2.89348	0.116205
$621$	−19.2138	−0.771022
$622$	−11.0843	−0.444440
$623$	0	0
$624$	−6.91395	−0.276780
$625$	13.3680	0.534719
$626$	9.73730	0.389181
$627$	8.36204	0.333948
$628$	−15.8360	−0.631925
$629$	12.2526	0.488542
$630$	0	0
$631$	32.1115	1.27834	0.639169	−	0.769066i	$-0.279278\pi$
$631$	32.1115	1.27834	0.639169	+	0.769066i	$0.279278\pi$
$632$	−0.892781	−0.0355129
$633$	13.2562	0.526888
$634$	15.3755	0.610638
$635$	−0.861191	−0.0341753
$636$	50.3930	1.99821
$637$	0	0
$638$	−4.68769	−0.185588
$639$	1.03625	0.0409935
$640$	0.497222	0.0196544
$641$	33.0248	1.30440	0.652200	−	0.758047i	$-0.273846\pi$
$641$	33.0248	1.30440	0.652200	+	0.758047i	$0.273846\pi$
$642$	−48.0263	−1.89545
$643$	15.7942	0.622863	0.311432	−	0.950269i	$-0.399192\pi$
$643$	15.7942	0.622863	0.311432	+	0.950269i	$0.399192\pi$
$644$	0	0
$645$	−15.0362	−0.592051
$646$	−31.3520	−1.23353
$647$	−4.64072	−0.182445	−0.0912227	−	0.995831i	$-0.529078\pi$
$647$	−4.64072	−0.182445	−0.0912227	+	0.995831i	$0.529078\pi$
$648$	0.636493	0.0250038
$649$	0.513173	0.0201438
$650$	8.39497	0.329278
$651$	0	0
$652$	3.43757	0.134626
$653$	26.8285	1.04988	0.524941	−	0.851139i	$-0.324087\pi$
$653$	26.8285	1.04988	0.524941	+	0.851139i	$0.324087\pi$
$654$	48.4912	1.89616
$655$	−17.0453	−0.666016
$656$	−19.3446	−0.755280
$657$	−0.327429	−0.0127742
$658$	0	0
$659$	−42.9889	−1.67461	−0.837306	−	0.546735i	$-0.815871\pi$
$659$	−42.9889	−1.67461	−0.837306	+	0.546735i	$0.815871\pi$
$660$	2.32163	0.0903695
$661$	−29.4698	−1.14624	−0.573122	−	0.819470i	$-0.694268\pi$
$661$	−29.4698	−1.14624	−0.573122	+	0.819470i	$0.694268\pi$
$662$	−22.7830	−0.885488
$663$	4.13778	0.160698
$664$	−0.243015	−0.00943082
$665$	0	0
$666$	−0.986346	−0.0382201
$667$	12.2470	0.474205
$668$	−44.4806	−1.72101
$669$	31.0207	1.19933
$670$	−17.0851	−0.660056
$671$	−8.34752	−0.322252
$672$	0	0
$673$	−20.1702	−0.777504	−0.388752	−	0.921342i	$-0.627094\pi$
$673$	−20.1702	−0.777504	−0.388752	+	0.921342i	$0.627094\pi$
$674$	−3.49782	−0.134731
$675$	21.3637	0.822287
$676$	2.03417	0.0782373
$677$	−6.20481	−0.238470	−0.119235	−	0.992866i	$-0.538044\pi$
$677$	−6.20481	−0.238470	−0.119235	+	0.992866i	$0.538044\pi$
$678$	11.5108	0.442071
$679$	0	0
$680$	−0.146221	−0.00560733
$681$	−9.37409	−0.359216
$682$	−2.25968	−0.0865276
$683$	−1.76952	−0.0677087	−0.0338543	−	0.999427i	$-0.510778\pi$
$683$	−1.76952	−0.0677087	−0.0338543	+	0.999427i	$0.510778\pi$
$684$	1.27262	0.0486600
$685$	5.59899	0.213926
$686$	0	0
$687$	−14.9854	−0.571729
$688$	37.0946	1.41422
$689$	14.0833	0.536530
$690$	−12.0290	−0.457937
$691$	44.9317	1.70928	0.854641	−	0.519219i	$-0.173777\pi$
$691$	44.9317	1.70928	0.854641	+	0.519219i	$0.173777\pi$
$692$	11.8820	0.451688
$693$	0	0
$694$	−42.2800	−1.60493
$695$	−3.62289	−0.137424
$696$	−0.393331	−0.0149092
$697$	11.5771	0.438515
$698$	16.7832	0.635255
$699$	8.36204	0.316281
$700$	0	0
$701$	38.5707	1.45679	0.728397	−	0.685156i	$-0.240266\pi$
$701$	38.5707	1.45679	0.728397	+	0.685156i	$0.240266\pi$
$702$	10.2662	0.387474
$703$	34.5653	1.30366
$704$	−5.92501	−0.223307
$705$	−13.2553	−0.499225
$706$	17.1398	0.645066
$707$	0	0
$708$	2.56330	0.0963346
$709$	8.77731	0.329639	0.164819	−	0.986324i	$-0.447296\pi$
$709$	8.77731	0.329639	0.164819	+	0.986324i	$0.447296\pi$
$710$	−19.9952	−0.750405
$711$	1.22639	0.0459932
$712$	−0.827023	−0.0309940
$713$	5.90360	0.221091
$714$	0	0
$715$	0.648824	0.0242646
$716$	5.15515	0.192657
$717$	26.0893	0.974323
$718$	32.4804	1.21216
$719$	4.20437	0.156796	0.0783982	−	0.996922i	$-0.475019\pi$
$719$	4.20437	0.156796	0.0783982	+	0.996922i	$0.475019\pi$
$720$	−0.335625	−0.0125080
$721$	0	0
$722$	−50.2840	−1.87138
$723$	−10.7677	−0.400456
$724$	−21.8161	−0.810789
$725$	−13.6173	−0.505735
$726$	37.0511	1.37509
$727$	−28.9856	−1.07502	−0.537509	−	0.843258i	$-0.680634\pi$
$727$	−28.9856	−1.07502	−0.537509	+	0.843258i	$0.680634\pi$
$728$	0	0
$729$	26.0990	0.966630
$730$	6.31796	0.233838
$731$	−22.1999	−0.821094
$732$	−41.6958	−1.54112
$733$	−24.0345	−0.887733	−0.443867	−	0.896093i	$-0.646394\pi$
$733$	−24.0345	−0.887733	−0.443867	+	0.896093i	$0.646394\pi$
$734$	−56.5480	−2.08722
$735$	0	0
$736$	30.1918	1.11288
$737$	6.72788	0.247825
$738$	−0.931971	−0.0343063
$739$	11.8055	0.434273	0.217136	−	0.976141i	$-0.430328\pi$
$739$	11.8055	0.434273	0.217136	+	0.976141i	$0.430328\pi$
$740$	9.59670	0.352782
$741$	11.6729	0.428816
$742$	0	0
$743$	47.2786	1.73448	0.867241	−	0.497888i	$-0.165891\pi$
$743$	47.2786	1.73448	0.867241	+	0.497888i	$0.165891\pi$
$744$	−0.189603	−0.00695120
$745$	−19.0900	−0.699402
$746$	57.2503	2.09608
$747$	0.333824	0.0122140
$748$	3.42773	0.125330
$749$	0	0
$750$	29.3750	1.07262
$751$	5.47700	0.199859	0.0999294	−	0.994995i	$-0.468138\pi$
$751$	5.47700	0.199859	0.0999294	+	0.994995i	$0.468138\pi$
$752$	32.7012	1.19249
$753$	−24.5755	−0.895581
$754$	−6.54376	−0.238310
$755$	−14.2410	−0.518285
$756$	0	0
$757$	10.7453	0.390546	0.195273	−	0.980749i	$-0.437441\pi$
$757$	10.7453	0.390546	0.195273	+	0.980749i	$0.437441\pi$
$758$	14.5872	0.529830
$759$	4.73686	0.171937
$760$	−0.412500	−0.0149630
$761$	−33.0399	−1.19770	−0.598848	−	0.800863i	$-0.704375\pi$
$761$	−33.0399	−1.19770	−0.598848	+	0.800863i	$0.704375\pi$
$762$	3.35940	0.121698
$763$	0	0
$764$	−3.41457	−0.123535
$765$	0.200860	0.00726212
$766$	−25.9753	−0.938527
$767$	0.716361	0.0258663
$768$	27.1587	0.980004
$769$	2.98332	0.107581	0.0537907	−	0.998552i	$-0.482870\pi$
$769$	2.98332	0.107581	0.0537907	+	0.998552i	$0.482870\pi$
$770$	0	0
$771$	30.3702	1.09376
$772$	−13.1167	−0.472079
$773$	−21.9085	−0.787995	−0.393998	−	0.919111i	$-0.628908\pi$
$773$	−21.9085	−0.787995	−0.393998	+	0.919111i	$0.628908\pi$
$774$	1.78712	0.0642367
$775$	−6.56417	−0.235792
$776$	−0.510464	−0.0183246
$777$	0	0
$778$	42.4516	1.52196
$779$	32.6598	1.17016
$780$	3.24087	0.116042
$781$	7.87381	0.281747
$782$	−17.7600	−0.635097
$783$	−16.6527	−0.595118
$784$	0	0
$785$	−7.05104	−0.251662
$786$	66.4917	2.37168
$787$	−13.3632	−0.476347	−0.238174	−	0.971223i	$-0.576549\pi$
$787$	−13.3632	−0.476347	−0.238174	+	0.971223i	$0.576549\pi$
$788$	3.80900	0.135690
$789$	−4.58537	−0.163244
$790$	−23.6640	−0.841927
$791$	0	0
$792$	−0.00463525	−0.000164706 0
$793$	−11.6527	−0.413798
$794$	−38.5784	−1.36910
$795$	22.4377	0.795782
$796$	−23.1827	−0.821691
$797$	32.5388	1.15258	0.576292	−	0.817244i	$-0.304499\pi$
$797$	32.5388	1.15258	0.576292	+	0.817244i	$0.304499\pi$
$798$	0	0
$799$	−19.5706	−0.692357
$800$	−33.5701	−1.18688
$801$	1.13606	0.0401407
$802$	−33.4805	−1.18224
$803$	−2.48792	−0.0877969
$804$	33.6057	1.18518
$805$	0	0
$806$	−3.15439	−0.111109
$807$	−25.4879	−0.897217
$808$	0.0822938	0.00289509
$809$	7.68827	0.270305	0.135153	−	0.990825i	$-0.456848\pi$
$809$	7.68827	0.270305	0.135153	+	0.990825i	$0.456848\pi$
$810$	16.8708	0.592781
$811$	−48.3178	−1.69667	−0.848334	−	0.529461i	$-0.822394\pi$
$811$	−48.3178	−1.69667	−0.848334	+	0.529461i	$0.822394\pi$
$812$	0	0
$813$	−15.1911	−0.532773
$814$	−7.49461	−0.262686
$815$	1.53059	0.0536142
$816$	−16.2635	−0.569336
$817$	−62.6275	−2.19106
$818$	−24.8019	−0.867178
$819$	0	0
$820$	9.06766	0.316656
$821$	−3.73442	−0.130332	−0.0651661	−	0.997874i	$-0.520758\pi$
$821$	−3.73442	−0.130332	−0.0651661	+	0.997874i	$0.520758\pi$
$822$	−21.8409	−0.761790
$823$	14.2318	0.496089	0.248045	−	0.968749i	$-0.420212\pi$
$823$	14.2318	0.496089	0.248045	+	0.968749i	$0.420212\pi$
$824$	0.989150	0.0344587
$825$	−5.26688	−0.183369
$826$	0	0
$827$	48.3016	1.67961	0.839805	−	0.542888i	$-0.182669\pi$
$827$	48.3016	1.67961	0.839805	+	0.542888i	$0.182669\pi$
$828$	0.720906	0.0250532
$829$	11.5101	0.399763	0.199882	−	0.979820i	$-0.435944\pi$
$829$	11.5101	0.399763	0.199882	+	0.979820i	$0.435944\pi$
$830$	−6.44135	−0.223582
$831$	21.5079	0.746102
$832$	−8.27099	−0.286745
$833$	0	0
$834$	14.1324	0.489366
$835$	−19.8052	−0.685386
$836$	9.66985	0.334439
$837$	−8.02734	−0.277465
$838$	−8.75583	−0.302465
$839$	13.1103	0.452616	0.226308	−	0.974056i	$-0.427334\pi$
$839$	13.1103	0.452616	0.226308	+	0.974056i	$0.427334\pi$
$840$	0	0
$841$	−18.3855	−0.633982
$842$	20.0852	0.692183
$843$	−42.5378	−1.46508
$844$	15.3295	0.527663
$845$	0.905722	0.0311578
$846$	1.57545	0.0541652
$847$	0	0
$848$	−55.3541	−1.90087
$849$	−54.1232	−1.85750
$850$	19.7472	0.677325
$851$	19.5803	0.671203
$852$	39.3297	1.34741
$853$	8.80346	0.301425	0.150712	−	0.988578i	$-0.451843\pi$
$853$	8.80346	0.301425	0.150712	+	0.988578i	$0.451843\pi$
$854$	0	0
$855$	0.566640	0.0193787
$856$	−0.932934	−0.0318870
$857$	16.9651	0.579516	0.289758	−	0.957100i	$-0.406425\pi$
$857$	16.9651	0.579516	0.289758	+	0.957100i	$0.406425\pi$
$858$	−2.53098	−0.0864063
$859$	14.5410	0.496132	0.248066	−	0.968743i	$-0.420205\pi$
$859$	14.5410	0.496132	0.248066	+	0.968743i	$0.420205\pi$
$860$	−17.3879	−0.592922
$861$	0	0
$862$	46.9243	1.59825
$863$	39.0444	1.32909	0.664544	−	0.747249i	$-0.268626\pi$
$863$	39.0444	1.32909	0.664544	+	0.747249i	$0.268626\pi$
$864$	−41.0529	−1.39665
$865$	5.29052	0.179883
$866$	5.44882	0.185159
$867$	−20.1708	−0.685036
$868$	0	0
$869$	9.31854	0.316110
$870$	−10.4256	−0.353461
$871$	9.39174	0.318227
$872$	0.941966	0.0318990
$873$	0.701211	0.0237324
$874$	−50.1022	−1.69473
$875$	0	0
$876$	−12.4272	−0.419875
$877$	32.5941	1.10062	0.550312	−	0.834959i	$-0.314509\pi$
$877$	32.5941	1.10062	0.550312	+	0.834959i	$0.314509\pi$
$878$	17.7457	0.598888
$879$	−55.9899	−1.88849
$880$	−2.55020	−0.0859671
$881$	−43.4141	−1.46266	−0.731330	−	0.682024i	$-0.761100\pi$
$881$	−43.4141	−1.46266	−0.731330	+	0.682024i	$0.761100\pi$
$882$	0	0
$883$	28.2902	0.952040	0.476020	−	0.879434i	$-0.342079\pi$
$883$	28.2902	0.952040	0.476020	+	0.879434i	$0.342079\pi$
$884$	4.78492	0.160934
$885$	1.14132	0.0383650
$886$	5.83592	0.196062
$887$	−50.2650	−1.68773	−0.843866	−	0.536554i	$-0.819726\pi$
$887$	−50.2650	−1.68773	−0.843866	+	0.536554i	$0.819726\pi$
$888$	−0.628852	−0.0211029
$889$	0	0
$890$	−21.9210	−0.734794
$891$	−6.64349	−0.222565
$892$	35.8724	1.20110
$893$	−55.2099	−1.84753
$894$	74.4676	2.49057
$895$	2.29535	0.0767250
$896$	0	0
$897$	6.61239	0.220781
$898$	30.6856	1.02399
$899$	5.11668	0.170651
$900$	−0.801570	−0.0267190
$901$	33.1277	1.10364
$902$	−7.08145	−0.235786
$903$	0	0
$904$	0.223604	0.00743695
$905$	−9.71369	−0.322894
$906$	55.5526	1.84561
$907$	26.8277	0.890798	0.445399	−	0.895332i	$-0.353062\pi$
$907$	26.8277	0.890798	0.445399	+	0.895332i	$0.353062\pi$
$908$	−10.8402	−0.359744
$909$	−0.113045	−0.00374946
$910$	0	0
$911$	−22.3560	−0.740687	−0.370344	−	0.928895i	$-0.620760\pi$
$911$	−22.3560	−0.740687	−0.370344	+	0.928895i	$0.620760\pi$
$912$	−45.8804	−1.51925
$913$	2.53651	0.0839463
$914$	−47.5217	−1.57188
$915$	−18.5652	−0.613747
$916$	−17.3291	−0.572569
$917$	0	0
$918$	24.1489	0.797034
$919$	−8.62244	−0.284428	−0.142214	−	0.989836i	$-0.545422\pi$
$919$	−8.62244	−0.284428	−0.142214	+	0.989836i	$0.545422\pi$
$920$	−0.233670	−0.00770386
$921$	50.6404	1.66866
$922$	53.4609	1.76064
$923$	10.9914	0.361786
$924$	0	0
$925$	−21.7712	−0.715832
$926$	2.89729	0.0952109
$927$	−1.35877	−0.0446278
$928$	26.1674	0.858987
$929$	41.3861	1.35783	0.678916	−	0.734216i	$-0.262450\pi$
$929$	41.3861	1.35783	0.678916	+	0.734216i	$0.262450\pi$
$930$	−5.02562	−0.164796
$931$	0	0
$932$	9.66985	0.316747
$933$	9.70757	0.317812
$934$	16.8507	0.551372
$935$	1.52621	0.0499124
$936$	−0.00647055	−0.000211496 0
$937$	21.3818	0.698514	0.349257	−	0.937027i	$-0.386434\pi$
$937$	21.3818	0.698514	0.349257	+	0.937027i	$0.386434\pi$
$938$	0	0
$939$	−8.52788	−0.278297
$940$	−15.3285	−0.499959
$941$	−53.1480	−1.73258	−0.866288	−	0.499545i	$-0.833501\pi$
$941$	−53.1480	−1.73258	−0.866288	+	0.499545i	$0.833501\pi$
$942$	27.5052	0.896168
$943$	18.5009	0.602471
$944$	−2.81565	−0.0916416
$945$	0	0
$946$	13.5792	0.441497
$947$	−8.86936	−0.288216	−0.144108	−	0.989562i	$-0.546031\pi$
$947$	−8.86936	−0.288216	−0.144108	+	0.989562i	$0.546031\pi$
$948$	46.5461	1.51175
$949$	−3.47300	−0.112738
$950$	55.7082	1.80741
$951$	−13.4658	−0.436658
$952$	0	0
$953$	−39.8167	−1.28979	−0.644894	−	0.764272i	$-0.723099\pi$
$953$	−39.8167	−1.28979	−0.644894	+	0.764272i	$0.723099\pi$
$954$	−2.66681	−0.0863413
$955$	−1.52035	−0.0491973
$956$	30.1696	0.975756
$957$	4.10546	0.132711
$958$	−25.3319	−0.818435
$959$	0	0
$960$	−13.1775	−0.425301
$961$	−28.5335	−0.920437
$962$	−10.4621	−0.337310
$963$	1.28155	0.0412973
$964$	−12.4518	−0.401045
$965$	−5.84024	−0.188004
$966$	0	0
$967$	−22.1611	−0.712652	−0.356326	−	0.934362i	$-0.615971\pi$
$967$	−22.1611	−0.712652	−0.356326	+	0.934362i	$0.615971\pi$
$968$	0.719735	0.0231332
$969$	27.4579	0.882075
$970$	−13.5303	−0.434433
$971$	36.0423	1.15665	0.578327	−	0.815805i	$-0.303706\pi$
$971$	36.0423	1.15665	0.578327	+	0.815805i	$0.303706\pi$
$972$	−1.99229	−0.0639027
$973$	0	0
$974$	−43.3199	−1.38806
$975$	−7.35227	−0.235461
$976$	45.8007	1.46604
$977$	32.9416	1.05389	0.526947	−	0.849898i	$-0.323336\pi$
$977$	32.9416	1.05389	0.526947	+	0.849898i	$0.323336\pi$
$978$	−5.97064	−0.190920
$979$	8.63219	0.275886
$980$	0	0
$981$	−1.29395	−0.0413128
$982$	−78.8038	−2.51473
$983$	−4.19945	−0.133942	−0.0669709	−	0.997755i	$-0.521333\pi$
$983$	−4.19945	−0.133942	−0.0669709	+	0.997755i	$0.521333\pi$
$984$	−0.594185	−0.0189419
$985$	1.69597	0.0540382
$986$	−15.3927	−0.490203
$987$	0	0
$988$	13.4986	0.429447
$989$	−35.4767	−1.12809
$990$	−0.122862	−0.00390480
$991$	−13.4139	−0.426105	−0.213053	−	0.977041i	$-0.568341\pi$
$991$	−13.4139	−0.426105	−0.213053	+	0.977041i	$0.568341\pi$
$992$	12.6139	0.400491
$993$	19.9533	0.633198
$994$	0	0
$995$	−10.3222	−0.327236
$996$	12.6699	0.401460
$997$	−47.8868	−1.51659	−0.758295	−	0.651911i	$-0.773967\pi$
$997$	−47.8868	−1.51659	−0.758295	+	0.651911i	$0.773967\pi$
$998$	−18.4144	−0.582899
$999$	−26.6240	−0.842347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.l.1.1		5
3.2	odd	2		5733.2.a.bl.1.5		5
7.2	even	3		91.2.e.c.53.5	✓	10
7.3	odd	6		637.2.e.m.79.5		10
7.4	even	3		91.2.e.c.79.5	yes	10
7.5	odd	6		637.2.e.m.508.5		10
7.6	odd	2		637.2.a.k.1.1		5
13.12	even	2		8281.2.a.bw.1.5		5
21.2	odd	6		819.2.j.h.235.1		10
21.11	odd	6		819.2.j.h.352.1		10
21.20	even	2		5733.2.a.bm.1.5		5
28.11	odd	6		1456.2.r.p.625.4		10
28.23	odd	6		1456.2.r.p.417.4		10
91.25	even	6		1183.2.e.f.170.1		10
91.51	even	6		1183.2.e.f.508.1		10
91.90	odd	2		8281.2.a.bx.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.c.53.5	✓	10	7.2	even	3
91.2.e.c.79.5	yes	10	7.4	even	3
637.2.a.k.1.1		5	7.6	odd	2
637.2.a.l.1.1		5	1.1	even	1	trivial
637.2.e.m.79.5		10	7.3	odd	6
637.2.e.m.508.5		10	7.5	odd	6
819.2.j.h.235.1		10	21.2	odd	6
819.2.j.h.352.1		10	21.11	odd	6
1183.2.e.f.170.1		10	91.25	even	6
1183.2.e.f.508.1		10	91.51	even	6
1456.2.r.p.417.4		10	28.23	odd	6
1456.2.r.p.625.4		10	28.11	odd	6
5733.2.a.bl.1.5		5	3.2	odd	2
5733.2.a.bm.1.5		5	21.20	even	2
8281.2.a.bw.1.5		5	13.12	even	2
8281.2.a.bx.1.5		5	91.90	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00852 q^{2} +1.75906 q^{3} +2.03417 q^{4} +0.905722 q^{5} -3.53311 q^{6} -0.0686323 q^{8} +0.0942784 q^{9} -1.81916 q^{10} +0.716361 q^{11} +3.57822 q^{12} +1.00000 q^{13} +1.59322 q^{15} -3.93049 q^{16} +2.35227 q^{17} -0.189360 q^{18} +6.63591 q^{19} +1.84239 q^{20} -1.43883 q^{22} +3.75906 q^{23} -0.120728 q^{24} -4.17967 q^{25} -2.00852 q^{26} -5.11133 q^{27} +3.25799 q^{29} -3.20001 q^{30} +1.57050 q^{31} +8.03175 q^{32} +1.26012 q^{33} -4.72459 q^{34} +0.191778 q^{36} +5.20883 q^{37} -13.3284 q^{38} +1.75906 q^{39} -0.0621618 q^{40} +4.92168 q^{41} -9.43766 q^{43} +1.45720 q^{44} +0.0853900 q^{45} -7.55016 q^{46} -8.31986 q^{47} -6.91395 q^{48} +8.39497 q^{50} +4.13778 q^{51} +2.03417 q^{52} +14.0833 q^{53} +10.2662 q^{54} +0.648824 q^{55} +11.6729 q^{57} -6.54376 q^{58} +0.716361 q^{59} +3.24087 q^{60} -11.6527 q^{61} -3.15439 q^{62} -8.27099 q^{64} +0.905722 q^{65} -2.53098 q^{66} +9.39174 q^{67} +4.78492 q^{68} +6.61239 q^{69} +10.9914 q^{71} -0.00647055 q^{72} -3.47300 q^{73} -10.4621 q^{74} -7.35227 q^{75} +13.4986 q^{76} -3.53311 q^{78} +13.0082 q^{79} -3.55993 q^{80} -9.27395 q^{81} -9.88531 q^{82} +3.54083 q^{83} +2.13050 q^{85} +18.9558 q^{86} +5.73099 q^{87} -0.0491655 q^{88} +12.0501 q^{89} -0.171508 q^{90} +7.64656 q^{92} +2.76260 q^{93} +16.7106 q^{94} +6.01029 q^{95} +14.1283 q^{96} +7.43766 q^{97} +0.0675374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26}+ \cdots + 11 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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