LMFDB - Embedded newform 605.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,4,Mod(1,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, 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("605.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	605.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:	$35.6961555535$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.43845 q^{2} +0.561553 q^{3} -5.93087 q^{4} +5.00000 q^{5} +0.807764 q^{6} +31.0540 q^{7} -20.0388 q^{8} -26.6847 q^{9} +7.19224 q^{10} -3.33050 q^{12} +45.6155 q^{13} +44.6695 q^{14} +2.80776 q^{15} +18.6222 q^{16} +40.4536 q^{17} -38.3845 q^{18} -91.2699 q^{19} -29.6543 q^{20} +17.4384 q^{21} +32.2462 q^{23} -11.2529 q^{24} +25.0000 q^{25} +65.6155 q^{26} -30.1468 q^{27} -184.177 q^{28} -35.8702 q^{29} +4.03882 q^{30} +311.702 q^{31} +187.098 q^{32} +58.1904 q^{34} +155.270 q^{35} +158.263 q^{36} -368.147 q^{37} -131.287 q^{38} +25.6155 q^{39} -100.194 q^{40} +393.602 q^{41} +25.0843 q^{42} +351.602 q^{43} -133.423 q^{45} +46.3845 q^{46} -230.155 q^{47} +10.4573 q^{48} +621.349 q^{49} +35.9612 q^{50} +22.7168 q^{51} -270.540 q^{52} +406.902 q^{53} -43.3645 q^{54} -622.285 q^{56} -51.2529 q^{57} -51.5975 q^{58} -368.725 q^{59} -16.6525 q^{60} +322.825 q^{61} +448.366 q^{62} -828.665 q^{63} +120.153 q^{64} +228.078 q^{65} +442.233 q^{67} -239.925 q^{68} +18.1080 q^{69} +223.348 q^{70} +667.745 q^{71} +534.729 q^{72} +84.5701 q^{73} -529.560 q^{74} +14.0388 q^{75} +541.310 q^{76} +36.8466 q^{78} +411.983 q^{79} +93.1109 q^{80} +703.557 q^{81} +566.176 q^{82} +835.619 q^{83} -103.425 q^{84} +202.268 q^{85} +505.761 q^{86} -20.1430 q^{87} -799.853 q^{89} -191.922 q^{90} +1416.54 q^{91} -191.248 q^{92} +175.037 q^{93} -331.066 q^{94} -456.349 q^{95} +105.065 q^{96} +768.945 q^{97} +893.778 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{2} - 3 q^{3} + 17 q^{4} + 10 q^{5} - 19 q^{6} + 25 q^{7} + 63 q^{8} - 41 q^{9} + 35 q^{10} - 85 q^{12} + 50 q^{13} + 11 q^{14} - 15 q^{15} + 297 q^{16} + 151 q^{17} - 118 q^{18} + 3 q^{19} + 85 q^{20}+ \cdots - 810 q^{98}+O(q^{100}) $ 2 * q + 7 * q^2 - 3 * q^3 + 17 * q^4 + 10 * q^5 - 19 * q^6 + 25 * q^7 + 63 * q^8 - 41 * q^9 + 35 * q^10 - 85 * q^12 + 50 * q^13 + 11 * q^14 - 15 * q^15 + 297 * q^16 + 151 * q^17 - 118 * q^18 + 3 * q^19 + 85 * q^20 + 39 * q^21 + 48 * q^23 - 307 * q^24 + 50 * q^25 + 90 * q^26 + 117 * q^27 - 323 * q^28 + 221 * q^29 - 95 * q^30 + 141 * q^31 + 1071 * q^32 + 673 * q^34 + 125 * q^35 - 170 * q^36 - 559 * q^37 + 393 * q^38 + 10 * q^39 + 315 * q^40 + 144 * q^41 + 145 * q^42 + 60 * q^43 - 205 * q^45 + 134 * q^46 - 48 * q^47 - 981 * q^48 + 315 * q^49 + 175 * q^50 - 371 * q^51 - 170 * q^52 + 117 * q^53 + 775 * q^54 - 1125 * q^56 - 387 * q^57 + 1377 * q^58 - 86 * q^59 - 425 * q^60 + 155 * q^61 - 501 * q^62 - 742 * q^63 + 2809 * q^64 + 250 * q^65 + 266 * q^67 + 2295 * q^68 - 38 * q^69 + 55 * q^70 + 1587 * q^71 - 654 * q^72 - 70 * q^73 - 1591 * q^74 - 75 * q^75 + 2703 * q^76 - 50 * q^78 + 1294 * q^79 + 1485 * q^80 + 566 * q^81 - 822 * q^82 + 558 * q^83 + 391 * q^84 + 755 * q^85 - 1116 * q^86 - 935 * q^87 - 1777 * q^89 - 590 * q^90 + 1390 * q^91 + 170 * q^92 + 783 * q^93 + 682 * q^94 + 15 * q^95 - 3043 * q^96 - 334 * q^97 - 810 * 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 $

$2$	1.43845	0.508568	0.254284	−	0.967130i	$-0.418160\pi$
$2$	1.43845	0.508568	0.254284	+	0.967130i	$0.418160\pi$
$3$	0.561553	0.108071	0.0540354	−	0.998539i	$-0.482792\pi$
$3$	0.561553	0.108071	0.0540354	+	0.998539i	$0.482792\pi$
$4$	−5.93087	−0.741359
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0.807764	0.0549614
$7$	31.0540	1.67676	0.838379	−	0.545088i	$-0.183504\pi$
$7$	31.0540	1.67676	0.838379	+	0.545088i	$0.183504\pi$
$8$	−20.0388	−0.885599
$9$	−26.6847	−0.988321
$10$	7.19224	0.227438
$11$	0	0
$11$	0	0
$12$	−3.33050	−0.0801193
$13$	45.6155	0.973190	0.486595	−	0.873628i	$-0.338239\pi$
$13$	45.6155	0.973190	0.486595	+	0.873628i	$0.338239\pi$
$14$	44.6695	0.852745
$15$	2.80776	0.0483308
$16$	18.6222	0.290971
$17$	40.4536	0.577144	0.288572	−	0.957458i	$-0.406820\pi$
$17$	40.4536	0.577144	0.288572	+	0.957458i	$0.406820\pi$
$18$	−38.3845	−0.502628
$19$	−91.2699	−1.10204	−0.551020	−	0.834492i	$-0.685761\pi$
$19$	−91.2699	−1.10204	−0.551020	+	0.834492i	$0.685761\pi$
$20$	−29.6543	−0.331546
$21$	17.4384	0.181209
$22$	0	0
$23$	32.2462	0.292339	0.146170	−	0.989260i	$-0.453305\pi$
$23$	32.2462	0.292339	0.146170	+	0.989260i	$0.453305\pi$
$24$	−11.2529	−0.0957075
$25$	25.0000	0.200000
$26$	65.6155	0.494933
$27$	−30.1468	−0.214880
$28$	−184.177	−1.24308
$29$	−35.8702	−0.229688	−0.114844	−	0.993384i	$-0.536637\pi$
$29$	−35.8702	−0.229688	−0.114844	+	0.993384i	$0.536637\pi$
$30$	4.03882	0.0245795
$31$	311.702	1.80591	0.902956	−	0.429733i	$-0.141392\pi$
$31$	311.702	1.80591	0.902956	+	0.429733i	$0.141392\pi$
$32$	187.098	1.03358
$33$	0	0
$34$	58.1904	0.293517
$35$	155.270	0.749869
$36$	158.263	0.732700
$37$	−368.147	−1.63576	−0.817878	−	0.575392i	$-0.804850\pi$
$37$	−368.147	−1.63576	−0.817878	+	0.575392i	$0.804850\pi$
$38$	−131.287	−0.560462
$39$	25.6155	0.105174
$40$	−100.194	−0.396052
$41$	393.602	1.49928	0.749638	−	0.661848i	$-0.230227\pi$
$41$	393.602	1.49928	0.749638	+	0.661848i	$0.230227\pi$
$42$	25.0843	0.0921569
$43$	351.602	1.24695	0.623475	−	0.781843i	$-0.285720\pi$
$43$	351.602	1.24695	0.623475	+	0.781843i	$0.285720\pi$
$44$	0	0
$45$	−133.423	−0.441990
$46$	46.3845	0.148674
$47$	−230.155	−0.714289	−0.357145	−	0.934049i	$-0.616250\pi$
$47$	−230.155	−0.714289	−0.357145	+	0.934049i	$0.616250\pi$
$48$	10.4573	0.0314455
$49$	621.349	1.81151
$50$	35.9612	0.101714
$51$	22.7168	0.0623724
$52$	−270.540	−0.721483
$53$	406.902	1.05457	0.527286	−	0.849688i	$-0.323210\pi$
$53$	406.902	1.05457	0.527286	+	0.849688i	$0.323210\pi$
$54$	−43.3645	−0.109281
$55$	0	0
$56$	−622.285	−1.48493
$57$	−51.2529	−0.119098
$58$	−51.5975	−0.116812
$59$	−368.725	−0.813626	−0.406813	−	0.913511i	$-0.633360\pi$
$59$	−368.725	−0.813626	−0.406813	+	0.913511i	$0.633360\pi$
$60$	−16.6525	−0.0358304
$61$	322.825	0.677598	0.338799	−	0.940859i	$-0.389979\pi$
$61$	322.825	0.677598	0.338799	+	0.940859i	$0.389979\pi$
$62$	448.366	0.918429
$63$	−828.665	−1.65717
$64$	120.153	0.234673
$65$	228.078	0.435224
$66$	0	0
$67$	442.233	0.806378	0.403189	−	0.915117i	$-0.367902\pi$
$67$	442.233	0.806378	0.403189	+	0.915117i	$0.367902\pi$
$68$	−239.925	−0.427870
$69$	18.1080	0.0315933
$70$	223.348	0.381359
$71$	667.745	1.11615	0.558076	−	0.829790i	$-0.311540\pi$
$71$	667.745	1.11615	0.558076	+	0.829790i	$0.311540\pi$
$72$	534.729	0.875256
$73$	84.5701	0.135591	0.0677957	−	0.997699i	$-0.478403\pi$
$73$	84.5701	0.135591	0.0677957	+	0.997699i	$0.478403\pi$
$74$	−529.560	−0.831893
$75$	14.0388	0.0216142
$76$	541.310	0.817006
$77$	0	0
$78$	36.8466	0.0534879
$79$	411.983	0.586730	0.293365	−	0.956000i	$-0.405225\pi$
$79$	411.983	0.586730	0.293365	+	0.956000i	$0.405225\pi$
$80$	93.1109	0.130126
$81$	703.557	0.965098
$82$	566.176	0.762484
$83$	835.619	1.10507	0.552537	−	0.833488i	$-0.313660\pi$
$83$	835.619	1.10507	0.552537	+	0.833488i	$0.313660\pi$
$84$	−103.425	−0.134341
$85$	202.268	0.258106
$86$	505.761	0.634159
$87$	−20.1430	−0.0248225
$88$	0	0
$89$	−799.853	−0.952632	−0.476316	−	0.879274i	$-0.658028\pi$
$89$	−799.853	−0.952632	−0.476316	+	0.879274i	$0.658028\pi$
$90$	−191.922	−0.224782
$91$	1416.54	1.63180
$92$	−191.248	−0.216728
$93$	175.037	0.195167
$94$	−331.066	−0.363265
$95$	−456.349	−0.492847
$96$	105.065	0.111700
$97$	768.945	0.804892	0.402446	−	0.915444i	$-0.368160\pi$
$97$	768.945	0.804892	0.402446	+	0.915444i	$0.368160\pi$
$98$	893.778	0.921278
$99$	0	0
$100$	−148.272	−0.148272
$101$	1412.31	1.39139	0.695696	−	0.718337i	$-0.255096\pi$
$101$	1412.31	1.39139	0.695696	+	0.718337i	$0.255096\pi$
$102$	32.6770	0.0317206
$103$	24.4185	0.0233595	0.0116797	−	0.999932i	$-0.496282\pi$
$103$	24.4185	0.0233595	0.0116797	+	0.999932i	$0.496282\pi$
$104$	−914.081	−0.861856
$105$	87.1922	0.0810390
$106$	585.308	0.536322
$107$	−532.155	−0.480798	−0.240399	−	0.970674i	$-0.577278\pi$
$107$	−532.155	−0.480798	−0.240399	+	0.970674i	$0.577278\pi$
$108$	178.797	0.159303
$109$	1234.62	1.08491	0.542455	−	0.840085i	$-0.317495\pi$
$109$	1234.62	1.08491	0.542455	+	0.840085i	$0.317495\pi$
$110$	0	0
$111$	−206.734	−0.176778
$112$	578.292	0.487888
$113$	−1304.45	−1.08595	−0.542976	−	0.839748i	$-0.682703\pi$
$113$	−1304.45	−1.08595	−0.542976	+	0.839748i	$0.682703\pi$
$114$	−73.7245	−0.0605696
$115$	161.231	0.130738
$116$	212.742	0.170281
$117$	−1217.23	−0.961824
$118$	−530.392	−0.413784
$119$	1256.25	0.967729
$120$	−56.2643	−0.0428017
$121$	0	0
$122$	464.366	0.344605
$123$	221.028	0.162028
$124$	−1848.66	−1.33883
$125$	125.000	0.0894427
$126$	−1191.99	−0.842785
$127$	−1167.88	−0.816004	−0.408002	−	0.912981i	$-0.633774\pi$
$127$	−1167.88	−0.816004	−0.408002	+	0.912981i	$0.633774\pi$
$128$	−1323.95	−0.914231
$129$	197.443	0.134759
$130$	328.078	0.221341
$131$	−1549.95	−1.03374	−0.516869	−	0.856065i	$-0.672902\pi$
$131$	−1549.95	−1.03374	−0.516869	+	0.856065i	$0.672902\pi$
$132$	0	0
$133$	−2834.29	−1.84785
$134$	636.129	0.410098
$135$	−150.734	−0.0960971
$136$	−810.642	−0.511118
$137$	−2440.68	−1.52205	−0.761026	−	0.648722i	$-0.775304\pi$
$137$	−2440.68	−1.52205	−0.761026	+	0.648722i	$0.775304\pi$
$138$	26.0473	0.0160674
$139$	1861.43	1.13586	0.567930	−	0.823077i	$-0.307744\pi$
$139$	1861.43	1.13586	0.567930	+	0.823077i	$0.307744\pi$
$140$	−920.885	−0.555922
$141$	−129.244	−0.0771939
$142$	960.516	0.567639
$143$	0	0
$144$	−496.926	−0.287573
$145$	−179.351	−0.102719
$146$	121.650	0.0689575
$147$	348.920	0.195772
$148$	2183.43	1.21268
$149$	−814.376	−0.447760	−0.223880	−	0.974617i	$-0.571872\pi$
$149$	−814.376	−0.447760	−0.223880	+	0.974617i	$0.571872\pi$
$150$	20.1941	0.0109923
$151$	−3666.43	−1.97596	−0.987979	−	0.154591i	$-0.950594\pi$
$151$	−3666.43	−1.97596	−0.987979	+	0.154591i	$0.950594\pi$
$152$	1828.94	0.975965
$153$	−1079.49	−0.570403
$154$	0	0
$155$	1558.51	0.807628
$156$	−151.922	−0.0779713
$157$	2671.96	1.35825	0.679127	−	0.734021i	$-0.262359\pi$
$157$	2671.96	1.35825	0.679127	+	0.734021i	$0.262359\pi$
$158$	592.616	0.298392
$159$	228.497	0.113969
$160$	935.488	0.462230
$161$	1001.37	0.490182
$162$	1012.03	0.490818
$163$	−1728.53	−0.830605	−0.415302	−	0.909683i	$-0.636324\pi$
$163$	−1728.53	−0.830605	−0.415302	+	0.909683i	$0.636324\pi$
$164$	−2334.40	−1.11150
$165$	0	0
$166$	1201.99	0.562005
$167$	0.0653990	3.03037e−5 0	1.51519e−5	−	1.00000i	$-0.499995\pi$
$167$	0.0653990	3.03037e−5 0	1.51519e−5		1.00000i	$0.499995\pi$
$168$	−349.446	−0.160478
$169$	−116.224	−0.0529010
$170$	290.952	0.131265
$171$	2435.51	1.08917
$172$	−2085.31	−0.924437
$173$	1816.75	0.798411	0.399205	−	0.916861i	$-0.369286\pi$
$173$	1816.75	0.798411	0.399205	+	0.916861i	$0.369286\pi$
$174$	−28.9747	−0.0126239
$175$	776.349	0.335351
$176$	0	0
$177$	−207.059	−0.0879293
$178$	−1150.55	−0.484478
$179$	2381.23	0.994312	0.497156	−	0.867661i	$-0.334378\pi$
$179$	2381.23	0.994312	0.497156	+	0.867661i	$0.334378\pi$
$180$	791.316	0.327673
$181$	1275.61	0.523840	0.261920	−	0.965090i	$-0.415644\pi$
$181$	1275.61	0.523840	0.261920	+	0.965090i	$0.415644\pi$
$182$	2037.62	0.829883
$183$	181.283	0.0732286
$184$	−646.176	−0.258895
$185$	−1840.73	−0.731532
$186$	251.781	0.0992554
$187$	0	0
$188$	1365.02	0.529545
$189$	−936.177	−0.360301
$190$	−656.434	−0.250646
$191$	−3699.04	−1.40133	−0.700663	−	0.713492i	$-0.747112\pi$
$191$	−3699.04	−1.40133	−0.700663	+	0.713492i	$0.747112\pi$
$192$	67.4720	0.0253613
$193$	−1231.22	−0.459196	−0.229598	−	0.973285i	$-0.573741\pi$
$193$	−1231.22	−0.459196	−0.229598	+	0.973285i	$0.573741\pi$
$194$	1106.09	0.409342
$195$	128.078	0.0470350
$196$	−3685.14	−1.34298
$197$	−728.087	−0.263320	−0.131660	−	0.991295i	$-0.542031\pi$
$197$	−728.087	−0.263320	−0.131660	+	0.991295i	$0.542031\pi$
$198$	0	0
$199$	−1650.81	−0.588055	−0.294027	−	0.955797i	$-0.594996\pi$
$199$	−1650.81	−0.588055	−0.294027	+	0.955797i	$0.594996\pi$
$200$	−500.971	−0.177120
$201$	248.337	0.0871460
$202$	2031.54	0.707617
$203$	−1113.91	−0.385130
$204$	−134.731	−0.0462403
$205$	1968.01	0.670497
$206$	35.1247	0.0118799
$207$	−860.479	−0.288925
$208$	849.460	0.283171
$209$	0	0
$210$	125.421	0.0412138
$211$	−1587.01	−0.517792	−0.258896	−	0.965905i	$-0.583359\pi$
$211$	−1587.01	−0.517792	−0.258896	+	0.965905i	$0.583359\pi$
$212$	−2413.29	−0.781817
$213$	374.974	0.120624
$214$	−765.477	−0.244518
$215$	1758.01	0.557653
$216$	604.106	0.190297
$217$	9679.58	3.02808
$218$	1775.94	0.551751
$219$	47.4906	0.0146535
$220$	0	0
$221$	1845.31	0.561670
$222$	−297.376	−0.0899034
$223$	−1998.59	−0.600160	−0.300080	−	0.953914i	$-0.597013\pi$
$223$	−1998.59	−0.600160	−0.300080	+	0.953914i	$0.597013\pi$
$224$	5810.12	1.73306
$225$	−667.116	−0.197664
$226$	−1876.39	−0.552280
$227$	−4212.00	−1.23154	−0.615771	−	0.787925i	$-0.711155\pi$
$227$	−4212.00	−1.23154	−0.615771	+	0.787925i	$0.711155\pi$
$228$	303.974	0.0882946
$229$	1371.82	0.395863	0.197931	−	0.980216i	$-0.436578\pi$
$229$	1371.82	0.395863	0.197931	+	0.980216i	$0.436578\pi$
$230$	231.922	0.0664892
$231$	0	0
$232$	718.797	0.203411
$233$	1718.37	0.483152	0.241576	−	0.970382i	$-0.422336\pi$
$233$	1718.37	0.483152	0.241576	+	0.970382i	$0.422336\pi$
$234$	−1750.93	−0.489153
$235$	−1150.78	−0.319440
$236$	2186.86	0.603189
$237$	231.350	0.0634085
$238$	1807.04	0.492156
$239$	1794.77	0.485749	0.242875	−	0.970058i	$-0.421910\pi$
$239$	1794.77	0.485749	0.242875	+	0.970058i	$0.421910\pi$
$240$	52.2867	0.0140629
$241$	−5185.02	−1.38588	−0.692939	−	0.720996i	$-0.743685\pi$
$241$	−5185.02	−1.38588	−0.692939	+	0.720996i	$0.743685\pi$
$242$	0	0
$243$	1209.05	0.319179
$244$	−1914.63	−0.502343
$245$	3106.75	0.810134
$246$	317.938	0.0824023
$247$	−4163.32	−1.07249
$248$	−6246.13	−1.59931
$249$	469.244	0.119426
$250$	179.806	0.0454877
$251$	−3827.69	−0.962556	−0.481278	−	0.876568i	$-0.659827\pi$
$251$	−3827.69	−0.962556	−0.481278	+	0.876568i	$0.659827\pi$
$252$	4914.70	1.22856
$253$	0	0
$254$	−1679.93	−0.414993
$255$	113.584	0.0278938
$256$	−2865.65	−0.699621
$257$	−2671.37	−0.648387	−0.324193	−	0.945991i	$-0.605093\pi$
$257$	−2671.37	−0.648387	−0.324193	+	0.945991i	$0.605093\pi$
$258$	284.012	0.0685341
$259$	−11432.4	−2.74276
$260$	−1352.70	−0.322657
$261$	957.185	0.227005
$262$	−2229.52	−0.525726
$263$	−6659.10	−1.56128	−0.780642	−	0.624979i	$-0.785108\pi$
$263$	−6659.10	−1.56128	−0.780642	+	0.624979i	$0.785108\pi$
$264$	0	0
$265$	2034.51	0.471619
$266$	−4076.98	−0.939758
$267$	−449.160	−0.102952
$268$	−2622.83	−0.597816
$269$	8473.05	1.92049	0.960244	−	0.279162i	$-0.0900567\pi$
$269$	8473.05	1.92049	0.960244	+	0.279162i	$0.0900567\pi$
$270$	−216.823	−0.0488719
$271$	2643.21	0.592486	0.296243	−	0.955113i	$-0.404266\pi$
$271$	2643.21	0.592486	0.296243	+	0.955113i	$0.404266\pi$
$272$	753.334	0.167932
$273$	795.464	0.176350
$274$	−3510.78	−0.774066
$275$	0	0
$276$	−107.396	−0.0234220
$277$	386.962	0.0839361	0.0419680	−	0.999119i	$-0.486637\pi$
$277$	386.962	0.0839361	0.0419680	+	0.999119i	$0.486637\pi$
$278$	2677.57	0.577662
$279$	−8317.65	−1.78482
$280$	−3111.43	−0.664083
$281$	1106.42	0.234887	0.117444	−	0.993080i	$-0.462530\pi$
$281$	1106.42	0.234887	0.117444	+	0.993080i	$0.462530\pi$
$282$	−185.911	−0.0392583
$283$	−1165.71	−0.244855	−0.122428	−	0.992477i	$-0.539068\pi$
$283$	−1165.71	−0.244855	−0.122428	+	0.992477i	$0.539068\pi$
$284$	−3960.31	−0.827469
$285$	−256.264	−0.0532624
$286$	0	0
$287$	12222.9	2.51392
$288$	−4992.63	−1.02151
$289$	−3276.51	−0.666905
$290$	−257.987	−0.0522398
$291$	431.803	0.0869854
$292$	−501.574	−0.100522
$293$	3646.82	0.727131	0.363566	−	0.931569i	$-0.381559\pi$
$293$	3646.82	0.727131	0.363566	+	0.931569i	$0.381559\pi$
$294$	501.904	0.0995633
$295$	−1843.63	−0.363865
$296$	7377.23	1.44862
$297$	0	0
$298$	−1171.44	−0.227716
$299$	1470.93	0.284502
$300$	−83.2624	−0.0160239
$301$	10918.6	2.09083
$302$	−5273.96	−1.00491
$303$	793.089	0.150369
$304$	−1699.64	−0.320662
$305$	1614.12	0.303031
$306$	−1552.79	−0.290089
$307$	816.066	0.151711	0.0758556	−	0.997119i	$-0.475831\pi$
$307$	816.066	0.151711	0.0758556	+	0.997119i	$0.475831\pi$
$308$	0	0
$309$	13.7123	0.00252448
$310$	2241.83	0.410734
$311$	3146.99	0.573793	0.286896	−	0.957962i	$-0.407376\pi$
$311$	3146.99	0.573793	0.286896	+	0.957962i	$0.407376\pi$
$312$	−513.305	−0.0931416
$313$	−5085.49	−0.918367	−0.459184	−	0.888341i	$-0.651858\pi$
$313$	−5085.49	−0.918367	−0.459184	+	0.888341i	$0.651858\pi$
$314$	3843.48	0.690764
$315$	−4143.32	−0.741111
$316$	−2443.42	−0.434978
$317$	−4801.89	−0.850791	−0.425396	−	0.905007i	$-0.639865\pi$
$317$	−4801.89	−0.850791	−0.425396	+	0.905007i	$0.639865\pi$
$318$	328.681	0.0579608
$319$	0	0
$320$	600.763	0.104949
$321$	−298.833	−0.0519603
$322$	1440.42	0.249291
$323$	−3692.20	−0.636035
$324$	−4172.70	−0.715484
$325$	1140.39	0.194638
$326$	−2486.39	−0.422419
$327$	693.305	0.117247
$328$	−7887.32	−1.32776
$329$	−7147.24	−1.19769
$330$	0	0
$331$	−2597.31	−0.431302	−0.215651	−	0.976470i	$-0.569187\pi$
$331$	−2597.31	−0.431302	−0.215651	+	0.976470i	$0.569187\pi$
$332$	−4955.95	−0.819256
$333$	9823.87	1.61665
$334$	0.0940730	1.54115e−5 0
$335$	2211.16	0.360623
$336$	324.742	0.0527265
$337$	−2695.45	−0.435698	−0.217849	−	0.975982i	$-0.569904\pi$
$337$	−2695.45	−0.435698	−0.217849	+	0.975982i	$0.569904\pi$
$338$	−167.182	−0.0269038
$339$	−732.519	−0.117360
$340$	−1199.63	−0.191349
$341$	0	0
$342$	3503.35	0.553916
$343$	8643.85	1.36071
$344$	−7045.69	−1.10430
$345$	90.5398	0.0141290
$346$	2613.30	0.406046
$347$	9239.79	1.42945	0.714723	−	0.699407i	$-0.246553\pi$
$347$	9239.79	1.42945	0.714723	+	0.699407i	$0.246553\pi$
$348$	119.466	0.0184024
$349$	−3857.67	−0.591680	−0.295840	−	0.955237i	$-0.595600\pi$
$349$	−3857.67	−0.591680	−0.295840	+	0.955237i	$0.595600\pi$
$350$	1116.74	0.170549
$351$	−1375.16	−0.209119
$352$	0	0
$353$	−3378.21	−0.509360	−0.254680	−	0.967025i	$-0.581970\pi$
$353$	−3378.21	−0.509360	−0.254680	+	0.967025i	$0.581970\pi$
$354$	−297.843	−0.0447180
$355$	3338.73	0.499158
$356$	4743.83	0.706242
$357$	705.448	0.104583
$358$	3425.28	0.505675
$359$	9892.96	1.45440	0.727201	−	0.686425i	$-0.240821\pi$
$359$	9892.96	1.45440	0.727201	+	0.686425i	$0.240821\pi$
$360$	2673.65	0.391426
$361$	1471.19	0.214491
$362$	1834.89	0.266408
$363$	0	0
$364$	−8401.33	−1.20975
$365$	422.850	0.0606383
$366$	260.766	0.0372417
$367$	5295.58	0.753208	0.376604	−	0.926374i	$-0.377092\pi$
$367$	5295.58	0.753208	0.376604	+	0.926374i	$0.377092\pi$
$368$	600.495	0.0850623
$369$	−10503.1	−1.48177
$370$	−2647.80	−0.372034
$371$	12635.9	1.76826
$372$	−1038.12	−0.144688
$373$	−7786.10	−1.08083	−0.540414	−	0.841399i	$-0.681732\pi$
$373$	−7786.10	−1.08083	−0.540414	+	0.841399i	$0.681732\pi$
$374$	0	0
$375$	70.1941	0.00966615
$376$	4612.04	0.632574
$377$	−1636.24	−0.223530
$378$	−1346.64	−0.183237
$379$	−1940.17	−0.262955	−0.131477	−	0.991319i	$-0.541972\pi$
$379$	−1940.17	−0.262955	−0.131477	+	0.991319i	$0.541972\pi$
$380$	2706.55	0.365376
$381$	−655.826	−0.0881863
$382$	−5320.88	−0.712669
$383$	10805.6	1.44161	0.720807	−	0.693136i	$-0.243772\pi$
$383$	10805.6	1.44161	0.720807	+	0.693136i	$0.243772\pi$
$384$	−743.466	−0.0988017
$385$	0	0
$386$	−1771.04	−0.233533
$387$	−9382.39	−1.23239
$388$	−4560.51	−0.596714
$389$	−9225.11	−1.20239	−0.601197	−	0.799101i	$-0.705309\pi$
$389$	−9225.11	−1.20239	−0.601197	+	0.799101i	$0.705309\pi$
$390$	184.233	0.0239205
$391$	1304.48	0.168722
$392$	−12451.1	−1.60428
$393$	−870.378	−0.111717
$394$	−1047.31	−0.133916
$395$	2059.91	0.262394
$396$	0	0
$397$	13364.3	1.68951	0.844753	−	0.535156i	$-0.179747\pi$
$397$	13364.3	1.68951	0.844753	+	0.535156i	$0.179747\pi$
$398$	−2374.61	−0.299066
$399$	−1591.60	−0.199699
$400$	465.554	0.0581943
$401$	−6030.88	−0.751042	−0.375521	−	0.926814i	$-0.622536\pi$
$401$	−6030.88	−0.751042	−0.375521	+	0.926814i	$0.622536\pi$
$402$	357.220	0.0443197
$403$	14218.4	1.75750
$404$	−8376.25	−1.03152
$405$	3517.78	0.431605
$406$	−1602.31	−0.195865
$407$	0	0
$408$	−455.219	−0.0552370
$409$	−931.381	−0.112601	−0.0563005	−	0.998414i	$-0.517931\pi$
$409$	−931.381	−0.112601	−0.0563005	+	0.998414i	$0.517931\pi$
$410$	2830.88	0.340993
$411$	−1370.57	−0.164489
$412$	−144.823	−0.0173178
$413$	−11450.4	−1.36425
$414$	−1237.75	−0.146938
$415$	4178.10	0.494204
$416$	8534.55	1.00587
$417$	1045.29	0.122753
$418$	0	0
$419$	−13161.8	−1.53460	−0.767300	−	0.641289i	$-0.778400\pi$
$419$	−13161.8	−1.53460	−0.767300	+	0.641289i	$0.778400\pi$
$420$	−517.126	−0.0600789
$421$	−1127.05	−0.130473	−0.0652365	−	0.997870i	$-0.520780\pi$
$421$	−1127.05	−0.130473	−0.0652365	+	0.997870i	$0.520780\pi$
$422$	−2282.83	−0.263332
$423$	6141.62	0.705947
$424$	−8153.84	−0.933929
$425$	1011.34	0.115429
$426$	539.381	0.0613453
$427$	10025.0	1.13617
$428$	3156.14	0.356444
$429$	0	0
$430$	2528.81	0.283604
$431$	4386.13	0.490191	0.245096	−	0.969499i	$-0.421181\pi$
$431$	4386.13	0.490191	0.245096	+	0.969499i	$0.421181\pi$
$432$	−561.398	−0.0625238
$433$	10905.5	1.21035	0.605177	−	0.796091i	$-0.293102\pi$
$433$	10905.5	1.21035	0.605177	+	0.796091i	$0.293102\pi$
$434$	13923.6	1.53998
$435$	−100.715	−0.0111010
$436$	−7322.38	−0.804308
$437$	−2943.11	−0.322169
$438$	68.3127	0.00745229
$439$	3900.94	0.424104	0.212052	−	0.977258i	$-0.431985\pi$
$439$	3900.94	0.424104	0.212052	+	0.977258i	$0.431985\pi$
$440$	0	0
$441$	−16580.5	−1.79036
$442$	2654.38	0.285647
$443$	−15537.5	−1.66638	−0.833192	−	0.552985i	$-0.813489\pi$
$443$	−15537.5	−1.66638	−0.833192	+	0.552985i	$0.813489\pi$
$444$	1226.11	0.131056
$445$	−3999.27	−0.426030
$446$	−2874.87	−0.305222
$447$	−457.315	−0.0483898
$448$	3731.22	0.393490
$449$	−3464.47	−0.364139	−0.182070	−	0.983286i	$-0.558280\pi$
$449$	−3464.47	−0.364139	−0.182070	+	0.983286i	$0.558280\pi$
$450$	−959.612	−0.100526
$451$	0	0
$452$	7736.54	0.805080
$453$	−2058.89	−0.213543
$454$	−6058.74	−0.626323
$455$	7082.72	0.729765
$456$	1027.05	0.105473
$457$	14410.6	1.47506	0.737528	−	0.675317i	$-0.235993\pi$
$457$	14410.6	1.47506	0.737528	+	0.675317i	$0.235993\pi$
$458$	1973.29	0.201323
$459$	−1219.55	−0.124016
$460$	−956.240	−0.0969238
$461$	3219.12	0.325226	0.162613	−	0.986690i	$-0.448008\pi$
$461$	3219.12	0.325226	0.162613	+	0.986690i	$0.448008\pi$
$462$	0	0
$463$	−1338.77	−0.134380	−0.0671899	−	0.997740i	$-0.521403\pi$
$463$	−1338.77	−0.134380	−0.0671899	+	0.997740i	$0.521403\pi$
$464$	−667.982	−0.0668325
$465$	875.185	0.0872811
$466$	2471.79	0.245716
$467$	−12221.6	−1.21102	−0.605512	−	0.795836i	$-0.707032\pi$
$467$	−12221.6	−1.21102	−0.605512	+	0.795836i	$0.707032\pi$
$468$	7219.26	0.713057
$469$	13733.1	1.35210
$470$	−1655.33	−0.162457
$471$	1500.45	0.146788
$472$	7388.82	0.720547
$473$	0	0
$474$	332.785	0.0322475
$475$	−2281.75	−0.220408
$476$	−7450.63	−0.717435
$477$	−10858.1	−1.04226
$478$	2581.68	0.247036
$479$	20290.0	1.93544	0.967718	−	0.252036i	$-0.0811001\pi$
$479$	20290.0	1.93544	0.967718	+	0.252036i	$0.0811001\pi$
$480$	525.326	0.0499536
$481$	−16793.2	−1.59190
$482$	−7458.38	−0.704813
$483$	562.324	0.0529744
$484$	0	0
$485$	3844.72	0.359959
$486$	1739.15	0.162324
$487$	17360.7	1.61538	0.807688	−	0.589610i	$-0.200718\pi$
$487$	17360.7	1.61538	0.807688	+	0.589610i	$0.200718\pi$
$488$	−6469.03	−0.600080
$489$	−970.658	−0.0897642
$490$	4468.89	0.412008
$491$	−5129.01	−0.471424	−0.235712	−	0.971823i	$-0.575742\pi$
$491$	−5129.01	−0.471424	−0.235712	+	0.971823i	$0.575742\pi$
$492$	−1310.89	−0.120121
$493$	−1451.08	−0.132563
$494$	−5988.72	−0.545436
$495$	0	0
$496$	5804.56	0.525469
$497$	20736.1	1.87152
$498$	674.983	0.0607364
$499$	13009.6	1.16712	0.583558	−	0.812072i	$-0.301660\pi$
$499$	13009.6	1.16712	0.583558	+	0.812072i	$0.301660\pi$
$500$	−741.359	−0.0663091
$501$	0.0367250	3.27495e−6 0
$502$	−5505.93	−0.489525
$503$	−3972.02	−0.352095	−0.176047	−	0.984382i	$-0.556331\pi$
$503$	−3972.02	−0.352095	−0.176047	+	0.984382i	$0.556331\pi$
$504$	16605.5	1.46759
$505$	7061.57	0.622249
$506$	0	0
$507$	−65.2657	−0.00571706
$508$	6926.54	0.604952
$509$	11174.1	0.973048	0.486524	−	0.873667i	$-0.338265\pi$
$509$	11174.1	0.973048	0.486524	+	0.873667i	$0.338265\pi$
$510$	163.385	0.0141859
$511$	2626.24	0.227354
$512$	6469.49	0.558426
$513$	2751.49	0.236806
$514$	−3842.62	−0.329749
$515$	122.093	0.0104467
$516$	−1171.01	−0.0999047
$517$	0	0
$518$	−16444.9	−1.39488
$519$	1020.20	0.0862850
$520$	−4570.41	−0.385434
$521$	−18614.6	−1.56530	−0.782648	−	0.622464i	$-0.786132\pi$
$521$	−18614.6	−1.56530	−0.782648	+	0.622464i	$0.786132\pi$
$522$	1376.86	0.115447
$523$	−3792.30	−0.317066	−0.158533	−	0.987354i	$-0.550676\pi$
$523$	−3792.30	−0.317066	−0.158533	+	0.987354i	$0.550676\pi$
$524$	9192.54	0.766370
$525$	435.961	0.0362417
$526$	−9578.76	−0.794019
$527$	12609.5	1.04227
$528$	0	0
$529$	−11127.2	−0.914538
$530$	2926.54	0.239850
$531$	9839.31	0.804124
$532$	16809.8	1.36992
$533$	17954.4	1.45908
$534$	−646.093	−0.0523580
$535$	−2660.78	−0.215019
$536$	−8861.83	−0.714128
$537$	1337.19	0.107456
$538$	12188.0	0.976698
$539$	0	0
$540$	893.983	0.0712424
$541$	−5187.61	−0.412260	−0.206130	−	0.978525i	$-0.566087\pi$
$541$	−5187.61	−0.412260	−0.206130	+	0.978525i	$0.566087\pi$
$542$	3802.12	0.301319
$543$	716.320	0.0566119
$544$	7568.77	0.596523
$545$	6173.11	0.485187
$546$	1144.23	0.0896862
$547$	15642.1	1.22268	0.611342	−	0.791366i	$-0.290630\pi$
$547$	15642.1	1.22268	0.611342	+	0.791366i	$0.290630\pi$
$548$	14475.3	1.12839
$549$	−8614.47	−0.669684
$550$	0	0
$551$	3273.87	0.253125
$552$	−362.862	−0.0279790
$553$	12793.7	0.983804
$554$	556.624	0.0426872
$555$	−1033.67	−0.0790573
$556$	−11039.9	−0.842080
$557$	−15798.8	−1.20182	−0.600912	−	0.799315i	$-0.705196\pi$
$557$	−15798.8	−1.20182	−0.600912	+	0.799315i	$0.705196\pi$
$558$	−11964.5	−0.907702
$559$	16038.5	1.21352
$560$	2891.46	0.218190
$561$	0	0
$562$	1591.52	0.119456
$563$	−5028.68	−0.376436	−0.188218	−	0.982127i	$-0.560271\pi$
$563$	−5028.68	−0.376436	−0.188218	+	0.982127i	$0.560271\pi$
$564$	766.531	0.0572284
$565$	−6522.26	−0.485652
$566$	−1676.81	−0.124526
$567$	21848.2	1.61824
$568$	−13380.8	−0.988463
$569$	−24194.1	−1.78255	−0.891273	−	0.453467i	$-0.850187\pi$
$569$	−24194.1	−1.78255	−0.891273	+	0.453467i	$0.850187\pi$
$570$	−368.623	−0.0270875
$571$	8184.61	0.599852	0.299926	−	0.953963i	$-0.403038\pi$
$571$	8184.61	0.599852	0.299926	+	0.953963i	$0.403038\pi$
$572$	0	0
$573$	−2077.21	−0.151443
$574$	17582.0	1.27850
$575$	806.155	0.0584678
$576$	−3206.23	−0.231932
$577$	550.908	0.0397480	0.0198740	−	0.999802i	$-0.493673\pi$
$577$	550.908	0.0397480	0.0198740	+	0.999802i	$0.493673\pi$
$578$	−4713.08	−0.339167
$579$	−691.393	−0.0496258
$580$	1063.71	0.0761519
$581$	25949.3	1.85294
$582$	621.126	0.0442380
$583$	0	0
$584$	−1694.68	−0.120080
$585$	−6086.17	−0.430141
$586$	5245.76	0.369796
$587$	3080.03	0.216570	0.108285	−	0.994120i	$-0.465464\pi$
$587$	3080.03	0.216570	0.108285	+	0.994120i	$0.465464\pi$
$588$	−2069.40	−0.145137
$589$	−28449.0	−1.99019
$590$	−2651.96	−0.185050
$591$	−408.859	−0.0284572
$592$	−6855.69	−0.475958
$593$	−5257.89	−0.364108	−0.182054	−	0.983289i	$-0.558274\pi$
$593$	−5257.89	−0.364108	−0.182054	+	0.983289i	$0.558274\pi$
$594$	0	0
$595$	6281.23	0.432782
$596$	4829.96	0.331951
$597$	−927.018	−0.0635516
$598$	2115.85	0.144688
$599$	−1181.80	−0.0806125	−0.0403062	−	0.999187i	$-0.512833\pi$
$599$	−1181.80	−0.0806125	−0.0403062	+	0.999187i	$0.512833\pi$
$600$	−281.321	−0.0191415
$601$	27879.8	1.89225	0.946123	−	0.323807i	$-0.104963\pi$
$601$	27879.8	1.89225	0.946123	+	0.323807i	$0.104963\pi$
$602$	15705.9	1.06333
$603$	−11800.8	−0.796960
$604$	21745.1	1.46489
$605$	0	0
$606$	1140.82	0.0764728
$607$	4500.25	0.300922	0.150461	−	0.988616i	$-0.451924\pi$
$607$	4500.25	0.300922	0.150461	+	0.988616i	$0.451924\pi$
$608$	−17076.4	−1.13904
$609$	−625.521	−0.0416214
$610$	2321.83	0.154112
$611$	−10498.7	−0.695139
$612$	6402.32	0.422873
$613$	−8963.16	−0.590569	−0.295284	−	0.955409i	$-0.595414\pi$
$613$	−8963.16	−0.590569	−0.295284	+	0.955409i	$0.595414\pi$
$614$	1173.87	0.0771555
$615$	1105.14	0.0724612
$616$	0	0
$617$	−9394.42	−0.612974	−0.306487	−	0.951875i	$-0.599154\pi$
$617$	−9394.42	−0.612974	−0.306487	+	0.951875i	$0.599154\pi$
$618$	19.7244	0.00128387
$619$	4816.38	0.312741	0.156371	−	0.987698i	$-0.450021\pi$
$619$	4816.38	0.312741	0.156371	+	0.987698i	$0.450021\pi$
$620$	−9243.31	−0.598742
$621$	−972.119	−0.0628177
$622$	4526.78	0.291813
$623$	−24838.6	−1.59733
$624$	477.017	0.0306025
$625$	625.000	0.0400000
$626$	−7315.21	−0.467052
$627$	0	0
$628$	−15847.1	−1.00695
$629$	−14892.9	−0.944066
$630$	−5959.95	−0.376905
$631$	−24847.7	−1.56763	−0.783814	−	0.620996i	$-0.786728\pi$
$631$	−24847.7	−1.56763	−0.783814	+	0.620996i	$0.786728\pi$
$632$	−8255.65	−0.519608
$633$	−891.189	−0.0559583
$634$	−6907.26	−0.432685
$635$	−5839.39	−0.364928
$636$	−1355.19	−0.0844916
$637$	28343.2	1.76295
$638$	0	0
$639$	−17818.6	−1.10312
$640$	−6619.74	−0.408856
$641$	−30497.0	−1.87919	−0.939594	−	0.342290i	$-0.888798\pi$
$641$	−30497.0	−1.87919	−0.939594	+	0.342290i	$0.888798\pi$
$642$	−429.856	−0.0264253
$643$	5407.75	0.331665	0.165833	−	0.986154i	$-0.446969\pi$
$643$	5407.75	0.331665	0.165833	+	0.986154i	$0.446969\pi$
$644$	−5939.01	−0.363400
$645$	987.216	0.0602660
$646$	−5311.03	−0.323467
$647$	22118.8	1.34402	0.672011	−	0.740542i	$-0.265431\pi$
$647$	22118.8	1.34402	0.672011	+	0.740542i	$0.265431\pi$
$648$	−14098.4	−0.854690
$649$	0	0
$650$	1640.39	0.0989866
$651$	5435.59	0.327247
$652$	10251.7	0.615776
$653$	−12862.4	−0.770817	−0.385409	−	0.922746i	$-0.625940\pi$
$653$	−12862.4	−0.770817	−0.385409	+	0.922746i	$0.625940\pi$
$654$	997.283	0.0596282
$655$	−7749.74	−0.462301
$656$	7329.73	0.436247
$657$	−2256.72	−0.134008
$658$	−10280.9	−0.609106
$659$	−13216.0	−0.781218	−0.390609	−	0.920557i	$-0.627736\pi$
$659$	−13216.0	−0.781218	−0.390609	+	0.920557i	$0.627736\pi$
$660$	0	0
$661$	32098.3	1.88878	0.944388	−	0.328833i	$-0.106655\pi$
$661$	32098.3	1.88878	0.944388	+	0.328833i	$0.106655\pi$
$662$	−3736.09	−0.219347
$663$	1036.24	0.0607002
$664$	−16744.8	−0.978652
$665$	−14171.5	−0.826385
$666$	14131.1	0.822177
$667$	−1156.68	−0.0671466
$668$	−0.387873	−2.24659e−5 0
$669$	−1122.32	−0.0648599
$670$	3180.64	0.183401
$671$	0	0
$672$	3262.69	0.187293
$673$	19956.4	1.14304	0.571518	−	0.820590i	$-0.306355\pi$
$673$	19956.4	1.14304	0.571518	+	0.820590i	$0.306355\pi$
$674$	−3877.26	−0.221582
$675$	−753.669	−0.0429759
$676$	689.307	0.0392186
$677$	1483.58	0.0842223	0.0421111	−	0.999113i	$-0.486592\pi$
$677$	1483.58	0.0842223	0.0421111	+	0.999113i	$0.486592\pi$
$678$	−1053.69	−0.0596854
$679$	23878.8	1.34961
$680$	−4053.21	−0.228579
$681$	−2365.26	−0.133094
$682$	0	0
$683$	23228.2	1.30132	0.650659	−	0.759370i	$-0.274493\pi$
$683$	23228.2	1.30132	0.650659	+	0.759370i	$0.274493\pi$
$684$	−14444.7	−0.807464
$685$	−12203.4	−0.680682
$686$	12433.7	0.692015
$687$	770.350	0.0427812
$688$	6547.60	0.362827
$689$	18561.1	1.02630
$690$	130.237	0.00718554
$691$	−7333.20	−0.403717	−0.201858	−	0.979415i	$-0.564698\pi$
$691$	−7333.20	−0.403717	−0.201858	+	0.979415i	$0.564698\pi$
$692$	−10774.9	−0.591909
$693$	0	0
$694$	13290.9	0.726971
$695$	9307.16	0.507972
$696$	403.643	0.0219828
$697$	15922.6	0.865298
$698$	−5549.06	−0.300909
$699$	964.958	0.0522147
$700$	−4604.43	−0.248616
$701$	19481.6	1.04966	0.524829	−	0.851208i	$-0.324129\pi$
$701$	19481.6	1.04966	0.524829	+	0.851208i	$0.324129\pi$
$702$	−1978.10	−0.106351
$703$	33600.7	1.80267
$704$	0	0
$705$	−646.222	−0.0345222
$706$	−4859.38	−0.259044
$707$	43858.0	2.33303
$708$	1228.04	0.0651872
$709$	14807.5	0.784353	0.392177	−	0.919890i	$-0.371722\pi$
$709$	14807.5	0.784353	0.392177	+	0.919890i	$0.371722\pi$
$710$	4802.58	0.253856
$711$	−10993.6	−0.579878
$712$	16028.1	0.843650
$713$	10051.2	0.527939
$714$	1014.75	0.0531877
$715$	0	0
$716$	−14122.8	−0.737142
$717$	1007.86	0.0524953
$718$	14230.5	0.739662
$719$	22891.1	1.18734	0.593669	−	0.804709i	$-0.297679\pi$
$719$	22891.1	1.18734	0.593669	+	0.804709i	$0.297679\pi$
$720$	−2484.63	−0.128607
$721$	758.292	0.0391682
$722$	2116.23	0.109083
$723$	−2911.66	−0.149773
$724$	−7565.45	−0.388353
$725$	−896.756	−0.0459375
$726$	0	0
$727$	24318.7	1.24062	0.620309	−	0.784357i	$-0.287007\pi$
$727$	24318.7	1.24062	0.620309	+	0.784357i	$0.287007\pi$
$728$	−28385.9	−1.44512
$729$	−18317.1	−0.930605
$730$	608.248	0.0308387
$731$	14223.6	0.719669
$732$	−1075.17	−0.0542887
$733$	34939.0	1.76057	0.880287	−	0.474441i	$-0.157350\pi$
$733$	34939.0	1.76057	0.880287	+	0.474441i	$0.157350\pi$
$734$	7617.42	0.383057
$735$	1744.60	0.0875519
$736$	6033.19	0.302155
$737$	0	0
$738$	−15108.2	−0.753579
$739$	12663.1	0.630337	0.315168	−	0.949036i	$-0.397939\pi$
$739$	12663.1	0.630337	0.315168	+	0.949036i	$0.397939\pi$
$740$	10917.2	0.542328
$741$	−2337.93	−0.115905
$742$	18176.1	0.899281
$743$	1711.32	0.0844985	0.0422492	−	0.999107i	$-0.486548\pi$
$743$	1711.32	0.0844985	0.0422492	+	0.999107i	$0.486548\pi$
$744$	−3507.53	−0.172839
$745$	−4071.88	−0.200244
$746$	−11199.9	−0.549675
$747$	−22298.2	−1.09217
$748$	0	0
$749$	−16525.5	−0.806182
$750$	100.971	0.00491590
$751$	−5165.32	−0.250979	−0.125489	−	0.992095i	$-0.540050\pi$
$751$	−5165.32	−0.250979	−0.125489	+	0.992095i	$0.540050\pi$
$752$	−4285.99	−0.207838
$753$	−2149.45	−0.104024
$754$	−2353.65	−0.113680
$755$	−18332.1	−0.883675
$756$	5552.34	0.267112
$757$	−40684.0	−1.95335	−0.976675	−	0.214722i	$-0.931115\pi$
$757$	−40684.0	−1.95335	−0.976675	+	0.214722i	$0.931115\pi$
$758$	−2790.83	−0.133730
$759$	0	0
$760$	9144.70	0.436465
$761$	−26564.1	−1.26537	−0.632687	−	0.774408i	$-0.718048\pi$
$761$	−26564.1	−1.26537	−0.632687	+	0.774408i	$0.718048\pi$
$762$	−943.370	−0.0448487
$763$	38339.9	1.81913
$764$	21938.5	1.03889
$765$	−5397.45	−0.255092
$766$	15543.2	0.733158
$767$	−16819.6	−0.791813
$768$	−1609.21	−0.0756087
$769$	−24381.9	−1.14335	−0.571674	−	0.820481i	$-0.693706\pi$
$769$	−24381.9	−1.14335	−0.571674	+	0.820481i	$0.693706\pi$
$770$	0	0
$771$	−1500.11	−0.0700717
$772$	7302.19	0.340429
$773$	−18945.0	−0.881508	−0.440754	−	0.897628i	$-0.645289\pi$
$773$	−18945.0	−0.881508	−0.440754	+	0.897628i	$0.645289\pi$
$774$	−13496.1	−0.626752
$775$	7792.54	0.361182
$776$	−15408.8	−0.712812
$777$	−6419.91	−0.296413
$778$	−13269.8	−0.611499
$779$	−35924.0	−1.65226
$780$	−759.612	−0.0348698
$781$	0	0
$782$	1876.42	0.0858064
$783$	1081.37	0.0493552
$784$	11570.9	0.527099
$785$	13359.8	0.607430
$786$	−1251.99	−0.0568156
$787$	7610.70	0.344717	0.172359	−	0.985034i	$-0.444861\pi$
$787$	7610.70	0.344717	0.172359	+	0.985034i	$0.444861\pi$
$788$	4318.19	0.195215
$789$	−3739.44	−0.168729
$790$	2963.08	0.133445
$791$	−40508.4	−1.82088
$792$	0	0
$793$	14725.8	0.659432
$794$	19223.8	0.859229
$795$	1142.49	0.0509683
$796$	9790.75	0.435960
$797$	3732.50	0.165887	0.0829435	−	0.996554i	$-0.473568\pi$
$797$	3732.50	0.165887	0.0829435	+	0.996554i	$0.473568\pi$
$798$	−2289.44	−0.101561
$799$	−9310.61	−0.412247
$800$	4677.44	0.206716
$801$	21343.8	0.941506
$802$	−8675.11	−0.381956
$803$	0	0
$804$	−1472.86	−0.0646065
$805$	5006.87	0.219216
$806$	20452.5	0.893806
$807$	4758.07	0.207549
$808$	−28301.1	−1.23221
$809$	27621.8	1.20041	0.600204	−	0.799847i	$-0.295086\pi$
$809$	27621.8	1.20041	0.600204	+	0.799847i	$0.295086\pi$
$810$	5060.15	0.219501
$811$	29996.0	1.29877	0.649384	−	0.760461i	$-0.275027\pi$
$811$	29996.0	1.29877	0.649384	+	0.760461i	$0.275027\pi$
$812$	6606.48	0.285520
$813$	1484.30	0.0640305
$814$	0	0
$815$	−8642.63	−0.371458
$816$	423.037	0.0181486
$817$	−32090.7	−1.37419
$818$	−1339.74	−0.0572653
$819$	−37800.0	−1.61275
$820$	−11672.0	−0.497079
$821$	−18075.1	−0.768362	−0.384181	−	0.923258i	$-0.625516\pi$
$821$	−18075.1	−0.768362	−0.384181	+	0.923258i	$0.625516\pi$
$822$	−1971.49	−0.0836540
$823$	−26723.7	−1.13187	−0.565935	−	0.824450i	$-0.691485\pi$
$823$	−26723.7	−1.13187	−0.565935	+	0.824450i	$0.691485\pi$
$824$	−489.318	−0.0206871
$825$	0	0
$826$	−16470.8	−0.693816
$827$	−36788.1	−1.54685	−0.773427	−	0.633885i	$-0.781459\pi$
$827$	−36788.1	−1.54685	−0.773427	+	0.633885i	$0.781459\pi$
$828$	5103.39	0.214197
$829$	−1260.15	−0.0527948	−0.0263974	−	0.999652i	$-0.508404\pi$
$829$	−1260.15	−0.0527948	−0.0263974	+	0.999652i	$0.508404\pi$
$830$	6009.97	0.251336
$831$	217.300	0.00907105
$832$	5480.82	0.228381
$833$	25135.8	1.04550
$834$	1503.60	0.0624284
$835$	0.326995	1.35522e−5 0
$836$	0	0
$837$	−9396.80	−0.388054
$838$	−18932.6	−0.780448
$839$	−4023.16	−0.165548	−0.0827742	−	0.996568i	$-0.526378\pi$
$839$	−4023.16	−0.165548	−0.0827742	+	0.996568i	$0.526378\pi$
$840$	−1747.23	−0.0717680
$841$	−23102.3	−0.947244
$842$	−1621.20	−0.0663543
$843$	621.312	0.0253845
$844$	9412.34	0.383870
$845$	−581.118	−0.0236581
$846$	8834.39	0.359022
$847$	0	0
$848$	7577.41	0.306851
$849$	−654.605	−0.0264617
$850$	1454.76	0.0587033
$851$	−11871.3	−0.478195
$852$	−2223.92	−0.0894253
$853$	−40707.7	−1.63400	−0.817001	−	0.576636i	$-0.804365\pi$
$853$	−40707.7	−1.63400	−0.817001	+	0.576636i	$0.804365\pi$
$854$	14420.4	0.577818
$855$	12177.5	0.487091
$856$	10663.8	0.425794
$857$	−24321.4	−0.969431	−0.484715	−	0.874672i	$-0.661077\pi$
$857$	−24321.4	−0.969431	−0.484715	+	0.874672i	$0.661077\pi$
$858$	0	0
$859$	16912.7	0.671776	0.335888	−	0.941902i	$-0.390964\pi$
$859$	16912.7	0.671776	0.335888	+	0.941902i	$0.390964\pi$
$860$	−10426.5	−0.413421
$861$	6863.81	0.271682
$862$	6309.22	0.249295
$863$	−25793.8	−1.01742	−0.508708	−	0.860939i	$-0.669877\pi$
$863$	−25793.8	−1.01742	−0.508708	+	0.860939i	$0.669877\pi$
$864$	−5640.39	−0.222095
$865$	9083.76	0.357060
$866$	15686.9	0.615547
$867$	−1839.93	−0.0720731
$868$	−57408.3	−2.24489
$869$	0	0
$870$	−144.873	−0.00564560
$871$	20172.7	0.784759
$872$	−24740.4	−0.960796
$873$	−20519.0	−0.795492
$874$	−4233.51	−0.163845
$875$	3881.75	0.149974
$876$	−281.660	−0.0108635
$877$	−20143.8	−0.775607	−0.387803	−	0.921742i	$-0.626766\pi$
$877$	−20143.8	−0.775607	−0.387803	+	0.921742i	$0.626766\pi$
$878$	5611.30	0.215686
$879$	2047.88	0.0785817
$880$	0	0
$881$	−18231.3	−0.697196	−0.348598	−	0.937272i	$-0.613342\pi$
$881$	−18231.3	−0.697196	−0.348598	+	0.937272i	$0.613342\pi$
$882$	−23850.2	−0.910518
$883$	−34487.5	−1.31438	−0.657189	−	0.753725i	$-0.728255\pi$
$883$	−34487.5	−1.31438	−0.657189	+	0.753725i	$0.728255\pi$
$884$	−10944.3	−0.416399
$885$	−1035.29	−0.0393232
$886$	−22349.8	−0.847469
$887$	16039.9	0.607180	0.303590	−	0.952803i	$-0.401815\pi$
$887$	16039.9	0.607180	0.303590	+	0.952803i	$0.401815\pi$
$888$	4142.70	0.156554
$889$	−36267.3	−1.36824
$890$	−5752.73	−0.216665
$891$	0	0
$892$	11853.4	0.444934
$893$	21006.2	0.787175
$894$	−657.824	−0.0246095
$895$	11906.2	0.444670
$896$	−41113.8	−1.53294
$897$	826.004	0.0307463
$898$	−4983.46	−0.185189
$899$	−11180.8	−0.414795
$900$	3956.58	0.146540
$901$	16460.7	0.608640
$902$	0	0
$903$	6131.40	0.225958
$904$	26139.7	0.961718
$905$	6378.03	0.234268
$906$	−2961.61	−0.108601
$907$	−23029.2	−0.843077	−0.421539	−	0.906810i	$-0.638510\pi$
$907$	−23029.2	−0.843077	−0.421539	+	0.906810i	$0.638510\pi$
$908$	24980.8	0.913015
$909$	−37687.1	−1.37514
$910$	10188.1	0.371135
$911$	−9206.00	−0.334806	−0.167403	−	0.985889i	$-0.553538\pi$
$911$	−9206.00	−0.334806	−0.167403	+	0.985889i	$0.553538\pi$
$912$	−954.440	−0.0346542
$913$	0	0
$914$	20728.9	0.750166
$915$	906.416	0.0327488
$916$	−8136.10	−0.293476
$917$	−48132.0	−1.73333
$918$	−1754.25	−0.0630707
$919$	−13329.5	−0.478456	−0.239228	−	0.970963i	$-0.576894\pi$
$919$	−13329.5	−0.478456	−0.239228	+	0.970963i	$0.576894\pi$
$920$	−3230.88	−0.115781
$921$	458.264	0.0163956
$922$	4630.53	0.165400
$923$	30459.6	1.08623
$924$	0	0
$925$	−9203.67	−0.327151
$926$	−1925.75	−0.0683412
$927$	−651.600	−0.0230867
$928$	−6711.24	−0.237400
$929$	1819.10	0.0642442	0.0321221	−	0.999484i	$-0.489773\pi$
$929$	1819.10	0.0642442	0.0321221	+	0.999484i	$0.489773\pi$
$930$	1258.91	0.0443884
$931$	−56710.5	−1.99636
$932$	−10191.5	−0.358189
$933$	1767.20	0.0620103
$934$	−17580.1	−0.615888
$935$	0	0
$936$	24391.9	0.851790
$937$	−4012.08	−0.139882	−0.0699408	−	0.997551i	$-0.522281\pi$
$937$	−4012.08	−0.139882	−0.0699408	+	0.997551i	$0.522281\pi$
$938$	19754.3	0.687635
$939$	−2855.77	−0.0992488
$940$	6825.10	0.236820
$941$	−8444.53	−0.292544	−0.146272	−	0.989244i	$-0.546727\pi$
$941$	−8444.53	−0.292544	−0.146272	+	0.989244i	$0.546727\pi$
$942$	2158.32	0.0746515
$943$	12692.2	0.438297
$944$	−6866.47	−0.236742
$945$	−4680.89	−0.161131
$946$	0	0
$947$	45316.0	1.55499	0.777493	−	0.628892i	$-0.216491\pi$
$947$	45316.0	1.55499	0.777493	+	0.628892i	$0.216491\pi$
$948$	−1372.11	−0.0470084
$949$	3857.71	0.131956
$950$	−3282.17	−0.112092
$951$	−2696.51	−0.0919458
$952$	−25173.7	−0.857020
$953$	−96.7143	−0.00328739	−0.00164370	−	0.999999i	$-0.500523\pi$
$953$	−96.7143	−0.00328739	−0.00164370	+	0.999999i	$0.500523\pi$
$954$	−15618.7	−0.530058
$955$	−18495.2	−0.626692
$956$	−10644.6	−0.360114
$957$	0	0
$958$	29186.1	0.984300
$959$	−75792.7	−2.55211
$960$	337.360	0.0113419
$961$	67366.9	2.26132
$962$	−24156.1	−0.809590
$963$	14200.4	0.475183
$964$	30751.7	1.02743
$965$	−6156.08	−0.205359
$966$	808.873	0.0269411
$967$	−7666.63	−0.254956	−0.127478	−	0.991841i	$-0.540688\pi$
$967$	−7666.63	−0.254956	−0.127478	+	0.991841i	$0.540688\pi$
$968$	0	0
$969$	−2073.36	−0.0687368
$970$	5530.43	0.183063
$971$	−5221.18	−0.172560	−0.0862799	−	0.996271i	$-0.527498\pi$
$971$	−5221.18	−0.172560	−0.0862799	+	0.996271i	$0.527498\pi$
$972$	−7170.70	−0.236626
$973$	57804.9	1.90456
$974$	24972.4	0.821529
$975$	640.388	0.0210347
$976$	6011.70	0.197162
$977$	−45769.9	−1.49878	−0.749390	−	0.662129i	$-0.769653\pi$
$977$	−45769.9	−1.49878	−0.749390	+	0.662129i	$0.769653\pi$
$978$	−1396.24	−0.0456512
$979$	0	0
$980$	−18425.7	−0.600600
$981$	−32945.4	−1.07224
$982$	−7377.81	−0.239751
$983$	−14777.8	−0.479488	−0.239744	−	0.970836i	$-0.577064\pi$
$983$	−14777.8	−0.479488	−0.239744	+	0.970836i	$0.577064\pi$
$984$	−4429.15	−0.143492
$985$	−3640.44	−0.117760
$986$	−2087.30	−0.0674171
$987$	−4013.55	−0.129435
$988$	24692.1	0.795103
$989$	11337.8	0.364532
$990$	0	0
$991$	−16783.9	−0.538001	−0.269001	−	0.963140i	$-0.586693\pi$
$991$	−16783.9	−0.538001	−0.269001	+	0.963140i	$0.586693\pi$
$992$	58318.6	1.86655
$993$	−1458.53	−0.0466112
$994$	29827.9	0.951793
$995$	−8254.06	−0.262986
$996$	−2783.03	−0.0885377
$997$	8720.36	0.277008	0.138504	−	0.990362i	$-0.455771\pi$
$997$	8720.36	0.277008	0.138504	+	0.990362i	$0.455771\pi$
$998$	18713.7	0.593557
$999$	11098.4	0.351490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.4.a.g.1.1		2
11.10	odd	2		55.4.a.b.1.2	✓	2
33.32	even	2		495.4.a.e.1.1		2
44.43	even	2		880.4.a.r.1.1		2
55.32	even	4		275.4.b.b.199.2		4
55.43	even	4		275.4.b.b.199.3		4
55.54	odd	2		275.4.a.c.1.1		2
165.164	even	2		2475.4.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.b.1.2	✓	2	11.10	odd	2
275.4.a.c.1.1		2	55.54	odd	2
275.4.b.b.199.2		4	55.32	even	4
275.4.b.b.199.3		4	55.43	even	4
495.4.a.e.1.1		2	33.32	even	2
605.4.a.g.1.1		2	1.1	even	1	trivial
880.4.a.r.1.1		2	44.43	even	2
2475.4.a.l.1.2		2	165.164	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:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43845 q^{2} +0.561553 q^{3} -5.93087 q^{4} +5.00000 q^{5} +0.807764 q^{6} +31.0540 q^{7} -20.0388 q^{8} -26.6847 q^{9} +7.19224 q^{10} -3.33050 q^{12} +45.6155 q^{13} +44.6695 q^{14} +2.80776 q^{15} +18.6222 q^{16} +40.4536 q^{17} -38.3845 q^{18} -91.2699 q^{19} -29.6543 q^{20} +17.4384 q^{21} +32.2462 q^{23} -11.2529 q^{24} +25.0000 q^{25} +65.6155 q^{26} -30.1468 q^{27} -184.177 q^{28} -35.8702 q^{29} +4.03882 q^{30} +311.702 q^{31} +187.098 q^{32} +58.1904 q^{34} +155.270 q^{35} +158.263 q^{36} -368.147 q^{37} -131.287 q^{38} +25.6155 q^{39} -100.194 q^{40} +393.602 q^{41} +25.0843 q^{42} +351.602 q^{43} -133.423 q^{45} +46.3845 q^{46} -230.155 q^{47} +10.4573 q^{48} +621.349 q^{49} +35.9612 q^{50} +22.7168 q^{51} -270.540 q^{52} +406.902 q^{53} -43.3645 q^{54} -622.285 q^{56} -51.2529 q^{57} -51.5975 q^{58} -368.725 q^{59} -16.6525 q^{60} +322.825 q^{61} +448.366 q^{62} -828.665 q^{63} +120.153 q^{64} +228.078 q^{65} +442.233 q^{67} -239.925 q^{68} +18.1080 q^{69} +223.348 q^{70} +667.745 q^{71} +534.729 q^{72} +84.5701 q^{73} -529.560 q^{74} +14.0388 q^{75} +541.310 q^{76} +36.8466 q^{78} +411.983 q^{79} +93.1109 q^{80} +703.557 q^{81} +566.176 q^{82} +835.619 q^{83} -103.425 q^{84} +202.268 q^{85} +505.761 q^{86} -20.1430 q^{87} -799.853 q^{89} -191.922 q^{90} +1416.54 q^{91} -191.248 q^{92} +175.037 q^{93} -331.066 q^{94} -456.349 q^{95} +105.065 q^{96} +768.945 q^{97} +893.778 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{2} - 3 q^{3} + 17 q^{4} + 10 q^{5} - 19 q^{6} + 25 q^{7} + 63 q^{8} - 41 q^{9} + 35 q^{10} - 85 q^{12} + 50 q^{13} + 11 q^{14} - 15 q^{15} + 297 q^{16} + 151 q^{17} - 118 q^{18} + 3 q^{19} + 85 q^{20}+ \cdots - 810 q^{98}+O(q^{100}) \) 2 * q + 7 * q^2 - 3 * q^3 + 17 * q^4 + 10 * q^5 - 19 * q^6 + 25 * q^7 + 63 * q^8 - 41 * q^9 + 35 * q^10 - 85 * q^12 + 50 * q^13 + 11 * q^14 - 15 * q^15 + 297 * q^16 + 151 * q^17 - 118 * q^18 + 3 * q^19 + 85 * q^20 + 39 * q^21 + 48 * q^23 - 307 * q^24 + 50 * q^25 + 90 * q^26 + 117 * q^27 - 323 * q^28 + 221 * q^29 - 95 * q^30 + 141 * q^31 + 1071 * q^32 + 673 * q^34 + 125 * q^35 - 170 * q^36 - 559 * q^37 + 393 * q^38 + 10 * q^39 + 315 * q^40 + 144 * q^41 + 145 * q^42 + 60 * q^43 - 205 * q^45 + 134 * q^46 - 48 * q^47 - 981 * q^48 + 315 * q^49 + 175 * q^50 - 371 * q^51 - 170 * q^52 + 117 * q^53 + 775 * q^54 - 1125 * q^56 - 387 * q^57 + 1377 * q^58 - 86 * q^59 - 425 * q^60 + 155 * q^61 - 501 * q^62 - 742 * q^63 + 2809 * q^64 + 250 * q^65 + 266 * q^67 + 2295 * q^68 - 38 * q^69 + 55 * q^70 + 1587 * q^71 - 654 * q^72 - 70 * q^73 - 1591 * q^74 - 75 * q^75 + 2703 * q^76 - 50 * q^78 + 1294 * q^79 + 1485 * q^80 + 566 * q^81 - 822 * q^82 + 558 * q^83 + 391 * q^84 + 755 * q^85 - 1116 * q^86 - 935 * q^87 - 1777 * q^89 - 590 * q^90 + 1390 * q^91 + 170 * q^92 + 783 * q^93 + 682 * q^94 + 15 * q^95 - 3043 * q^96 - 334 * q^97 - 810 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more