LMFDB - Embedded newform 2254.4.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, 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("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 165 x^{9} + 798 x^{8} + 8769 x^{7} - 38472 x^{6} - 184213 x^{5} + 644009 x^{4} + \cdots + 2848203 $ x^11 - 5x^10 - 165x^9 + 798x^8 + 8769x^7 - 38472x^6 - 184213x^5 + 644009x^4 + 1428336x^3 - 2764269x^2 - 1864863x + 2848203
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.96837$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -7.96837 q^{3} +4.00000 q^{4} +0.313034 q^{5} -15.9367 q^{6} +8.00000 q^{8} +36.4949 q^{9} +0.626067 q^{10} -34.3808 q^{11} -31.8735 q^{12} +78.0974 q^{13} -2.49437 q^{15} +16.0000 q^{16} +19.1637 q^{17} +72.9898 q^{18} -8.01538 q^{19} +1.25213 q^{20} -68.7616 q^{22} -23.0000 q^{23} -63.7470 q^{24} -124.902 q^{25} +156.195 q^{26} -75.6590 q^{27} -18.7342 q^{29} -4.98873 q^{30} +337.914 q^{31} +32.0000 q^{32} +273.959 q^{33} +38.3274 q^{34} +145.980 q^{36} -65.1957 q^{37} -16.0308 q^{38} -622.309 q^{39} +2.50427 q^{40} -253.181 q^{41} +120.965 q^{43} -137.523 q^{44} +11.4241 q^{45} -46.0000 q^{46} +152.324 q^{47} -127.494 q^{48} -249.804 q^{50} -152.704 q^{51} +312.389 q^{52} -319.234 q^{53} -151.318 q^{54} -10.7623 q^{55} +63.8695 q^{57} -37.4684 q^{58} -557.902 q^{59} -9.97747 q^{60} +498.101 q^{61} +675.827 q^{62} +64.0000 q^{64} +24.4471 q^{65} +547.918 q^{66} -760.191 q^{67} +76.6548 q^{68} +183.272 q^{69} +924.726 q^{71} +291.959 q^{72} -739.009 q^{73} -130.391 q^{74} +995.265 q^{75} -32.0615 q^{76} -1244.62 q^{78} -2.73346 q^{79} +5.00854 q^{80} -382.484 q^{81} -506.362 q^{82} +1335.30 q^{83} +5.99888 q^{85} +241.930 q^{86} +149.281 q^{87} -275.046 q^{88} -1012.85 q^{89} +22.8483 q^{90} -92.0000 q^{92} -2692.62 q^{93} +304.648 q^{94} -2.50908 q^{95} -254.988 q^{96} -1239.00 q^{97} -1254.72 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 6 * q^3 + 44 * q^4 + 27 * q^5 + 12 * q^6 + 88 * q^8 + 59 * q^9 + 54 * q^10 + 56 * q^11 + 24 * q^12 + 103 * q^13 + 62 * q^15 + 176 * q^16 + 157 * q^17 + 118 * q^18 + 266 * q^19 + 108 * q^20 + 112 * q^22 - 253 * q^23 + 48 * q^24 + 142 * q^25 + 206 * q^26 + 531 * q^27 - 195 * q^29 + 124 * q^30 + 316 * q^31 + 352 * q^32 + 682 * q^33 + 314 * q^34 + 236 * q^36 + 347 * q^37 + 532 * q^38 - 99 * q^39 + 216 * q^40 + 540 * q^41 + 193 * q^43 + 224 * q^44 + 609 * q^45 - 506 * q^46 + 731 * q^47 + 96 * q^48 + 284 * q^50 - 303 * q^51 + 412 * q^52 + 413 * q^53 + 1062 * q^54 + 1809 * q^55 + 343 * q^57 - 390 * q^58 + 1061 * q^59 + 248 * q^60 + 1157 * q^61 + 632 * q^62 + 704 * q^64 - 37 * q^65 + 1364 * q^66 - 841 * q^67 + 628 * q^68 - 138 * q^69 + 869 * q^71 + 472 * q^72 + 409 * q^73 + 694 * q^74 + 2024 * q^75 + 1064 * q^76 - 198 * q^78 + 247 * q^79 + 432 * q^80 + 1019 * q^81 + 1080 * q^82 + 2863 * q^83 + 785 * q^85 + 386 * q^86 + 670 * q^87 + 448 * q^88 + 1610 * q^89 + 1218 * q^90 - 1012 * q^92 - 1290 * q^93 + 1462 * q^94 - 209 * q^95 + 192 * q^96 + 1783 * q^97 + 101 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−7.96837	−1.53351	−0.766757	−	0.641938i	$-0.778131\pi$
$3$	−7.96837	−1.53351	−0.766757	+	0.641938i	$0.778131\pi$
$4$	4.00000	0.500000
$5$	0.313034	0.0279986	0.0139993	−	0.999902i	$-0.495544\pi$
$5$	0.313034	0.0279986	0.0139993	+	0.999902i	$0.495544\pi$
$6$	−15.9367	−1.08436
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	36.4949	1.35166
$10$	0.626067	0.0197980
$11$	−34.3808	−0.942382	−0.471191	−	0.882031i	$-0.656176\pi$
$11$	−34.3808	−0.942382	−0.471191	+	0.882031i	$0.656176\pi$
$12$	−31.8735	−0.766757
$13$	78.0974	1.66618	0.833089	−	0.553139i	$-0.186570\pi$
$13$	78.0974	1.66618	0.833089	+	0.553139i	$0.186570\pi$
$14$	0	0
$15$	−2.49437	−0.0429362
$16$	16.0000	0.250000
$17$	19.1637	0.273405	0.136702	−	0.990612i	$-0.456350\pi$
$17$	19.1637	0.273405	0.136702	+	0.990612i	$0.456350\pi$
$18$	72.9898	0.955770
$19$	−8.01538	−0.0967819	−0.0483909	−	0.998828i	$-0.515409\pi$
$19$	−8.01538	−0.0967819	−0.0483909	+	0.998828i	$0.515409\pi$
$20$	1.25213	0.0139993
$21$	0	0
$22$	−68.7616	−0.666364
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−63.7470	−0.542179
$25$	−124.902	−0.999216
$26$	156.195	1.17817
$27$	−75.6590	−0.539280
$28$	0	0
$29$	−18.7342	−0.119960	−0.0599802	−	0.998200i	$-0.519104\pi$
$29$	−18.7342	−0.119960	−0.0599802	+	0.998200i	$0.519104\pi$
$30$	−4.98873	−0.0303605
$31$	337.914	1.95778	0.978888	−	0.204397i	$-0.0655233\pi$
$31$	337.914	1.95778	0.978888	+	0.204397i	$0.0655233\pi$
$32$	32.0000	0.176777
$33$	273.959	1.44515
$34$	38.3274	0.193326
$35$	0	0
$36$	145.980	0.675832
$37$	−65.1957	−0.289678	−0.144839	−	0.989455i	$-0.546266\pi$
$37$	−65.1957	−0.289678	−0.144839	+	0.989455i	$0.546266\pi$
$38$	−16.0308	−0.0684351
$39$	−622.309	−2.55511
$40$	2.50427	0.00989899
$41$	−253.181	−0.964396	−0.482198	−	0.876062i	$-0.660161\pi$
$41$	−253.181	−0.964396	−0.482198	+	0.876062i	$0.660161\pi$
$42$	0	0
$43$	120.965	0.429000	0.214500	−	0.976724i	$-0.431188\pi$
$43$	120.965	0.429000	0.214500	+	0.976724i	$0.431188\pi$
$44$	−137.523	−0.471191
$45$	11.4241	0.0378446
$46$	−46.0000	−0.147442
$47$	152.324	0.472740	0.236370	−	0.971663i	$-0.424042\pi$
$47$	152.324	0.472740	0.236370	+	0.971663i	$0.424042\pi$
$48$	−127.494	−0.383378
$49$	0	0
$50$	−249.804	−0.706552
$51$	−152.704	−0.419270
$52$	312.389	0.833089
$53$	−319.234	−0.827361	−0.413681	−	0.910422i	$-0.635757\pi$
$53$	−319.234	−0.827361	−0.413681	+	0.910422i	$0.635757\pi$
$54$	−151.318	−0.381329
$55$	−10.7623	−0.0263853
$56$	0	0
$57$	63.8695	0.148416
$58$	−37.4684	−0.0848249
$59$	−557.902	−1.23106	−0.615531	−	0.788112i	$-0.711059\pi$
$59$	−557.902	−1.23106	−0.615531	+	0.788112i	$0.711059\pi$
$60$	−9.97747	−0.0214681
$61$	498.101	1.04550	0.522748	−	0.852487i	$-0.324907\pi$
$61$	498.101	1.04550	0.522748	+	0.852487i	$0.324907\pi$
$62$	675.827	1.38436
$63$	0	0
$64$	64.0000	0.125000
$65$	24.4471	0.0466506
$66$	547.918	1.02188
$67$	−760.191	−1.38615	−0.693076	−	0.720865i	$-0.743745\pi$
$67$	−760.191	−1.38615	−0.693076	+	0.720865i	$0.743745\pi$
$68$	76.6548	0.136702
$69$	183.272	0.319760
$70$	0	0
$71$	924.726	1.54570	0.772850	−	0.634588i	$-0.218830\pi$
$71$	924.726	1.54570	0.772850	+	0.634588i	$0.218830\pi$
$72$	291.959	0.477885
$73$	−739.009	−1.18486	−0.592428	−	0.805623i	$-0.701831\pi$
$73$	−739.009	−1.18486	−0.592428	+	0.805623i	$0.701831\pi$
$74$	−130.391	−0.204834
$75$	995.265	1.53231
$76$	−32.0615	−0.0483909
$77$	0	0
$78$	−1244.62	−1.80673
$79$	−2.73346	−0.00389289	−0.00194644	−	0.999998i	$-0.500620\pi$
$79$	−2.73346	−0.00389289	−0.00194644	+	0.999998i	$0.500620\pi$
$80$	5.00854	0.00699964
$81$	−382.484	−0.524670
$82$	−506.362	−0.681931
$83$	1335.30	1.76588	0.882941	−	0.469484i	$-0.155560\pi$
$83$	1335.30	1.76588	0.882941	+	0.469484i	$0.155560\pi$
$84$	0	0
$85$	5.99888	0.00765494
$86$	241.930	0.303349
$87$	149.281	0.183961
$88$	−275.046	−0.333182
$89$	−1012.85	−1.20631	−0.603157	−	0.797622i	$-0.706091\pi$
$89$	−1012.85	−1.20631	−0.603157	+	0.797622i	$0.706091\pi$
$90$	22.8483	0.0267602
$91$	0	0
$92$	−92.0000	−0.104257
$93$	−2692.62	−3.00228
$94$	304.648	0.334277
$95$	−2.50908	−0.00270975
$96$	−254.988	−0.271089
$97$	−1239.00	−1.29692	−0.648462	−	0.761247i	$-0.724587\pi$
$97$	−1239.00	−1.29692	−0.648462	+	0.761247i	$0.724587\pi$
$98$	0	0
$99$	−1254.72	−1.27378
$100$	−499.608	−0.499608
$101$	558.524	0.550250	0.275125	−	0.961408i	$-0.411281\pi$
$101$	558.524	0.550250	0.275125	+	0.961408i	$0.411281\pi$
$102$	−305.407	−0.296469
$103$	610.024	0.583567	0.291784	−	0.956484i	$-0.405751\pi$
$103$	610.024	0.583567	0.291784	+	0.956484i	$0.405751\pi$
$104$	624.779	0.589083
$105$	0	0
$106$	−638.468	−0.585033
$107$	1481.38	1.33841	0.669206	−	0.743077i	$-0.266634\pi$
$107$	1481.38	1.33841	0.669206	+	0.743077i	$0.266634\pi$
$108$	−302.636	−0.269640
$109$	−1940.72	−1.70538	−0.852692	−	0.522414i	$-0.825032\pi$
$109$	−1940.72	−1.70538	−0.852692	+	0.522414i	$0.825032\pi$
$110$	−21.5247	−0.0186573
$111$	519.503	0.444226
$112$	0	0
$113$	799.964	0.665967	0.332984	−	0.942933i	$-0.391945\pi$
$113$	799.964	0.665967	0.332984	+	0.942933i	$0.391945\pi$
$114$	127.739	0.104946
$115$	−7.19977	−0.00583811
$116$	−74.9368	−0.0599802
$117$	2850.16	2.25211
$118$	−1115.80	−0.870493
$119$	0	0
$120$	−19.9549	−0.0151802
$121$	−148.961	−0.111917
$122$	996.202	0.739278
$123$	2017.44	1.47891
$124$	1351.65	0.978888
$125$	−78.2277	−0.0559752
$126$	0	0
$127$	−573.172	−0.400478	−0.200239	−	0.979747i	$-0.564172\pi$
$127$	−573.172	−0.400478	−0.200239	+	0.979747i	$0.564172\pi$
$128$	128.000	0.0883883
$129$	−963.895	−0.657877
$130$	48.8942	0.0329870
$131$	1325.17	0.883821	0.441910	−	0.897059i	$-0.354301\pi$
$131$	1325.17	0.883821	0.441910	+	0.897059i	$0.354301\pi$
$132$	1095.84	0.722577
$133$	0	0
$134$	−1520.38	−0.980157
$135$	−23.6838	−0.0150991
$136$	153.310	0.0966632
$137$	2251.31	1.40396	0.701980	−	0.712196i	$-0.252299\pi$
$137$	2251.31	1.40396	0.701980	+	0.712196i	$0.252299\pi$
$138$	366.545	0.226104
$139$	−511.025	−0.311831	−0.155916	−	0.987770i	$-0.549833\pi$
$139$	−511.025	−0.311831	−0.155916	+	0.987770i	$0.549833\pi$
$140$	0	0
$141$	−1213.78	−0.724953
$142$	1849.45	1.09298
$143$	−2685.05	−1.57018
$144$	583.919	0.337916
$145$	−5.86443	−0.00335872
$146$	−1478.02	−0.837820
$147$	0	0
$148$	−260.783	−0.144839
$149$	−30.4343	−0.0167334	−0.00836668	−	0.999965i	$-0.502663\pi$
$149$	−30.4343	−0.0167334	−0.00836668	+	0.999965i	$0.502663\pi$
$150$	1990.53	1.08351
$151$	1669.60	0.899802	0.449901	−	0.893078i	$-0.351459\pi$
$151$	1669.60	0.899802	0.449901	+	0.893078i	$0.351459\pi$
$152$	−64.1231	−0.0342176
$153$	699.378	0.369551
$154$	0	0
$155$	105.778	0.0548149
$156$	−2489.23	−1.27755
$157$	2362.20	1.20079	0.600396	−	0.799703i	$-0.295010\pi$
$157$	2362.20	1.20079	0.600396	+	0.799703i	$0.295010\pi$
$158$	−5.46692	−0.00275269
$159$	2543.77	1.26877
$160$	10.0171	0.00494950
$161$	0	0
$162$	−764.968	−0.370997
$163$	2766.83	1.32954	0.664770	−	0.747048i	$-0.268530\pi$
$163$	2766.83	1.32954	0.664770	+	0.747048i	$0.268530\pi$
$164$	−1012.72	−0.482198
$165$	85.7583	0.0404623
$166$	2670.60	1.24867
$167$	−66.2807	−0.0307123	−0.0153561	−	0.999882i	$-0.504888\pi$
$167$	−66.2807	−0.0307123	−0.0153561	+	0.999882i	$0.504888\pi$
$168$	0	0
$169$	3902.20	1.77615
$170$	11.9978	0.00541286
$171$	−292.521	−0.130816
$172$	483.860	0.214500
$173$	243.513	0.107017	0.0535086	−	0.998567i	$-0.482960\pi$
$173$	243.513	0.107017	0.0535086	+	0.998567i	$0.482960\pi$
$174$	298.562	0.130080
$175$	0	0
$176$	−550.093	−0.235595
$177$	4445.57	1.88785
$178$	−2025.70	−0.852993
$179$	−548.942	−0.229217	−0.114608	−	0.993411i	$-0.536561\pi$
$179$	−548.942	−0.229217	−0.114608	+	0.993411i	$0.536561\pi$
$180$	45.6965	0.0189223
$181$	−1486.29	−0.610359	−0.305180	−	0.952295i	$-0.598716\pi$
$181$	−1486.29	−0.610359	−0.305180	+	0.952295i	$0.598716\pi$
$182$	0	0
$183$	−3969.05	−1.60328
$184$	−184.000	−0.0737210
$185$	−20.4084	−0.00811058
$186$	−5385.24	−2.12293
$187$	−658.863	−0.257652
$188$	609.297	0.236370
$189$	0	0
$190$	−5.01817	−0.00191609
$191$	4483.82	1.69863	0.849314	−	0.527888i	$-0.177016\pi$
$191$	4483.82	1.69863	0.849314	+	0.527888i	$0.177016\pi$
$192$	−509.976	−0.191689
$193$	−3136.30	−1.16972	−0.584859	−	0.811135i	$-0.698850\pi$
$193$	−3136.30	−1.16972	−0.584859	+	0.811135i	$0.698850\pi$
$194$	−2478.00	−0.917063
$195$	−194.804	−0.0715393
$196$	0	0
$197$	−1243.82	−0.449839	−0.224919	−	0.974377i	$-0.572212\pi$
$197$	−1243.82	−0.449839	−0.224919	+	0.974377i	$0.572212\pi$
$198$	−2509.45	−0.900700
$199$	3416.94	1.21719	0.608594	−	0.793482i	$-0.291734\pi$
$199$	3416.94	1.21719	0.608594	+	0.793482i	$0.291734\pi$
$200$	−999.216	−0.353276
$201$	6057.48	2.12568
$202$	1117.05	0.389086
$203$	0	0
$204$	−610.814	−0.209635
$205$	−79.2542	−0.0270017
$206$	1220.05	0.412644
$207$	−839.383	−0.281841
$208$	1249.56	0.416544
$209$	275.575	0.0912054
$210$	0	0
$211$	−1423.47	−0.464435	−0.232217	−	0.972664i	$-0.574598\pi$
$211$	−1423.47	−0.464435	−0.232217	+	0.972664i	$0.574598\pi$
$212$	−1276.94	−0.413681
$213$	−7368.56	−2.37035
$214$	2962.75	0.946400
$215$	37.8661	0.0120114
$216$	−605.272	−0.190664
$217$	0	0
$218$	−3881.43	−1.20589
$219$	5888.70	1.81699
$220$	−43.0494	−0.0131927
$221$	1496.64	0.455541
$222$	1039.01	0.314115
$223$	3447.48	1.03525	0.517625	−	0.855608i	$-0.326816\pi$
$223$	3447.48	1.03525	0.517625	+	0.855608i	$0.326816\pi$
$224$	0	0
$225$	−4558.29	−1.35060
$226$	1599.93	0.470910
$227$	3054.66	0.893148	0.446574	−	0.894747i	$-0.352644\pi$
$227$	3054.66	0.893148	0.446574	+	0.894747i	$0.352644\pi$
$228$	255.478	0.0742081
$229$	−2613.27	−0.754103	−0.377052	−	0.926192i	$-0.623062\pi$
$229$	−2613.27	−0.754103	−0.377052	+	0.926192i	$0.623062\pi$
$230$	−14.3995	−0.00412816
$231$	0	0
$232$	−149.874	−0.0424124
$233$	2510.83	0.705964	0.352982	−	0.935630i	$-0.385168\pi$
$233$	2510.83	0.705964	0.352982	+	0.935630i	$0.385168\pi$
$234$	5700.31	1.59248
$235$	47.6826	0.0132360
$236$	−2231.61	−0.615531
$237$	21.7812	0.00596979
$238$	0	0
$239$	2312.80	0.625953	0.312977	−	0.949761i	$-0.398674\pi$
$239$	2312.80	0.625953	0.312977	+	0.949761i	$0.398674\pi$
$240$	−39.9099	−0.0107340
$241$	847.100	0.226417	0.113209	−	0.993571i	$-0.463887\pi$
$241$	847.100	0.226417	0.113209	+	0.993571i	$0.463887\pi$
$242$	−297.923	−0.0791372
$243$	5090.57	1.34387
$244$	1992.40	0.522748
$245$	0	0
$246$	4034.88	1.04575
$247$	−625.980	−0.161256
$248$	2703.31	0.692178
$249$	−10640.2	−2.70800
$250$	−156.455	−0.0395804
$251$	6614.70	1.66341	0.831705	−	0.555217i	$-0.187365\pi$
$251$	6614.70	1.66341	0.831705	+	0.555217i	$0.187365\pi$
$252$	0	0
$253$	790.758	0.196500
$254$	−1146.34	−0.283181
$255$	−47.8013	−0.0117390
$256$	256.000	0.0625000
$257$	2128.48	0.516619	0.258310	−	0.966062i	$-0.416835\pi$
$257$	2128.48	0.516619	0.258310	+	0.966062i	$0.416835\pi$
$258$	−1927.79	−0.465190
$259$	0	0
$260$	97.7884	0.0233253
$261$	−683.703	−0.162146
$262$	2650.34	0.624956
$263$	2933.91	0.687881	0.343941	−	0.938991i	$-0.388238\pi$
$263$	2933.91	0.687881	0.343941	+	0.938991i	$0.388238\pi$
$264$	2191.67	0.510939
$265$	−99.9309	−0.0231649
$266$	0	0
$267$	8070.77	1.84990
$268$	−3040.77	−0.693076
$269$	4576.75	1.03736	0.518679	−	0.854969i	$-0.326424\pi$
$269$	4576.75	1.03736	0.518679	+	0.854969i	$0.326424\pi$
$270$	−47.3676	−0.0106767
$271$	−4699.17	−1.05334	−0.526668	−	0.850071i	$-0.676559\pi$
$271$	−4699.17	−1.05334	−0.526668	+	0.850071i	$0.676559\pi$
$272$	306.619	0.0683512
$273$	0	0
$274$	4502.62	0.992750
$275$	4294.23	0.941643
$276$	733.090	0.159880
$277$	7932.12	1.72056	0.860279	−	0.509823i	$-0.170289\pi$
$277$	7932.12	1.72056	0.860279	+	0.509823i	$0.170289\pi$
$278$	−1022.05	−0.220498
$279$	12332.1	2.64625
$280$	0	0
$281$	−2332.70	−0.495221	−0.247610	−	0.968860i	$-0.579645\pi$
$281$	−2332.70	−0.495221	−0.247610	+	0.968860i	$0.579645\pi$
$282$	−2427.55	−0.512619
$283$	5844.61	1.22765	0.613827	−	0.789441i	$-0.289629\pi$
$283$	5844.61	1.22765	0.613827	+	0.789441i	$0.289629\pi$
$284$	3698.90	0.772850
$285$	19.9933	0.00415544
$286$	−5370.10	−1.11028
$287$	0	0
$288$	1167.84	0.238943
$289$	−4545.75	−0.925250
$290$	−11.7289	−0.00237498
$291$	9872.83	1.98885
$292$	−2956.04	−0.592428
$293$	5575.83	1.11175	0.555876	−	0.831265i	$-0.312383\pi$
$293$	5575.83	1.11175	0.555876	+	0.831265i	$0.312383\pi$
$294$	0	0
$295$	−174.642	−0.0344680
$296$	−521.565	−0.102417
$297$	2601.21	0.508208
$298$	−60.8685	−0.0118323
$299$	−1796.24	−0.347422
$300$	3981.06	0.766156
$301$	0	0
$302$	3339.20	0.636256
$303$	−4450.53	−0.843816
$304$	−128.246	−0.0241955
$305$	155.922	0.0292724
$306$	1398.76	0.261312
$307$	−7939.06	−1.47592	−0.737958	−	0.674847i	$-0.764210\pi$
$307$	−7939.06	−1.47592	−0.737958	+	0.674847i	$0.764210\pi$
$308$	0	0
$309$	−4860.89	−0.894908
$310$	211.557	0.0387600
$311$	133.670	0.0243721	0.0121861	−	0.999926i	$-0.496121\pi$
$311$	133.670	0.0243721	0.0121861	+	0.999926i	$0.496121\pi$
$312$	−4978.47	−0.903366
$313$	8070.75	1.45746	0.728731	−	0.684800i	$-0.240111\pi$
$313$	8070.75	1.45746	0.728731	+	0.684800i	$0.240111\pi$
$314$	4724.40	0.849088
$315$	0	0
$316$	−10.9338	−0.00194644
$317$	3029.19	0.536708	0.268354	−	0.963320i	$-0.413520\pi$
$317$	3029.19	0.536708	0.268354	+	0.963320i	$0.413520\pi$
$318$	5087.55	0.897155
$319$	644.097	0.113049
$320$	20.0341	0.00349982
$321$	−11804.2	−2.05247
$322$	0	0
$323$	−153.604	−0.0264606
$324$	−1529.94	−0.262335
$325$	−9754.52	−1.66487
$326$	5533.66	0.940127
$327$	15464.3	2.61523
$328$	−2025.45	−0.340966
$329$	0	0
$330$	171.517	0.0286111
$331$	−1718.59	−0.285385	−0.142693	−	0.989767i	$-0.545576\pi$
$331$	−1718.59	−0.285385	−0.142693	+	0.989767i	$0.545576\pi$
$332$	5341.20	0.882941
$333$	−2379.31	−0.391548
$334$	−132.561	−0.0217169
$335$	−237.965	−0.0388103
$336$	0	0
$337$	2456.53	0.397079	0.198539	−	0.980093i	$-0.436380\pi$
$337$	2456.53	0.397079	0.198539	+	0.980093i	$0.436380\pi$
$338$	7804.40	1.25593
$339$	−6374.41	−1.02127
$340$	23.9955	0.00382747
$341$	−11617.7	−1.84497
$342$	−585.041	−0.0925012
$343$	0	0
$344$	967.721	0.151674
$345$	57.3704	0.00895281
$346$	487.026	0.0756726
$347$	−634.776	−0.0982034	−0.0491017	−	0.998794i	$-0.515636\pi$
$347$	−634.776	−0.0982034	−0.0491017	+	0.998794i	$0.515636\pi$
$348$	597.124	0.0919805
$349$	−7354.37	−1.12800	−0.563998	−	0.825776i	$-0.690737\pi$
$349$	−7354.37	−1.12800	−0.563998	+	0.825776i	$0.690737\pi$
$350$	0	0
$351$	−5908.77	−0.898537
$352$	−1100.19	−0.166591
$353$	−3217.22	−0.485086	−0.242543	−	0.970141i	$-0.577981\pi$
$353$	−3217.22	−0.485086	−0.242543	+	0.970141i	$0.577981\pi$
$354$	8891.14	1.33491
$355$	289.470	0.0432774
$356$	−4051.40	−0.603157
$357$	0	0
$358$	−1097.88	−0.162081
$359$	2938.32	0.431973	0.215987	−	0.976396i	$-0.430703\pi$
$359$	2938.32	0.431973	0.215987	+	0.976396i	$0.430703\pi$
$360$	91.3931	0.0133801
$361$	−6794.75	−0.990633
$362$	−2972.58	−0.431589
$363$	1186.98	0.171626
$364$	0	0
$365$	−231.335	−0.0331743
$366$	−7938.11	−1.13369
$367$	−2943.33	−0.418639	−0.209320	−	0.977847i	$-0.567125\pi$
$367$	−2943.33	−0.418639	−0.209320	+	0.977847i	$0.567125\pi$
$368$	−368.000	−0.0521286
$369$	−9239.82	−1.30354
$370$	−40.8169	−0.00573505
$371$	0	0
$372$	−10770.5	−1.50114
$373$	47.6624	0.00661626	0.00330813	−	0.999995i	$-0.498947\pi$
$373$	47.6624	0.00661626	0.00330813	+	0.999995i	$0.498947\pi$
$374$	−1317.73	−0.182187
$375$	623.347	0.0858387
$376$	1218.59	0.167139
$377$	−1463.09	−0.199875
$378$	0	0
$379$	523.434	0.0709419	0.0354710	−	0.999371i	$-0.488707\pi$
$379$	523.434	0.0709419	0.0354710	+	0.999371i	$0.488707\pi$
$380$	−10.0363	−0.00135488
$381$	4567.24	0.614139
$382$	8967.65	1.20111
$383$	−2203.59	−0.293990	−0.146995	−	0.989137i	$-0.546960\pi$
$383$	−2203.59	−0.293990	−0.146995	+	0.989137i	$0.546960\pi$
$384$	−1019.95	−0.135545
$385$	0	0
$386$	−6272.60	−0.827116
$387$	4414.61	0.579864
$388$	−4956.01	−0.648462
$389$	9847.66	1.28354	0.641769	−	0.766898i	$-0.278201\pi$
$389$	9847.66	1.28354	0.641769	+	0.766898i	$0.278201\pi$
$390$	−389.607	−0.0505859
$391$	−440.765	−0.0570088
$392$	0	0
$393$	−10559.4	−1.35535
$394$	−2487.63	−0.318084
$395$	−0.855664	−0.000108995 0
$396$	−5018.90	−0.636891
$397$	8879.05	1.12249	0.561243	−	0.827651i	$-0.310323\pi$
$397$	8879.05	1.12249	0.561243	+	0.827651i	$0.310323\pi$
$398$	6833.88	0.860682
$399$	0	0
$400$	−1998.43	−0.249804
$401$	14797.3	1.84275	0.921374	−	0.388678i	$-0.127068\pi$
$401$	14797.3	1.84275	0.921374	+	0.388678i	$0.127068\pi$
$402$	12115.0	1.50308
$403$	26390.2	3.26200
$404$	2234.10	0.275125
$405$	−119.730	−0.0146900
$406$	0	0
$407$	2241.48	0.272988
$408$	−1221.63	−0.148234
$409$	−1741.95	−0.210597	−0.105298	−	0.994441i	$-0.533580\pi$
$409$	−1741.95	−0.210597	−0.105298	+	0.994441i	$0.533580\pi$
$410$	−158.508	−0.0190931
$411$	−17939.3	−2.15299
$412$	2440.10	0.291784
$413$	0	0
$414$	−1678.77	−0.199292
$415$	417.994	0.0494422
$416$	2499.12	0.294541
$417$	4072.04	0.478198
$418$	551.150	0.0644920
$419$	12637.5	1.47347	0.736733	−	0.676183i	$-0.236367\pi$
$419$	12637.5	1.47347	0.736733	+	0.676183i	$0.236367\pi$
$420$	0	0
$421$	469.615	0.0543649	0.0271824	−	0.999630i	$-0.491346\pi$
$421$	469.615	0.0543649	0.0271824	+	0.999630i	$0.491346\pi$
$422$	−2846.94	−0.328405
$423$	5559.06	0.638985
$424$	−2553.87	−0.292516
$425$	−2393.59	−0.273190
$426$	−14737.1	−1.67609
$427$	0	0
$428$	5925.51	0.669206
$429$	21395.5	2.40789
$430$	75.7323	0.00849334
$431$	125.350	0.0140090	0.00700452	−	0.999975i	$-0.497770\pi$
$431$	125.350	0.0140090	0.00700452	+	0.999975i	$0.497770\pi$
$432$	−1210.54	−0.134820
$433$	1147.01	0.127302	0.0636508	−	0.997972i	$-0.479726\pi$
$433$	1147.01	0.127302	0.0636508	+	0.997972i	$0.479726\pi$
$434$	0	0
$435$	46.7300	0.00515065
$436$	−7762.86	−0.852692
$437$	184.354	0.0201804
$438$	11777.4	1.28481
$439$	348.110	0.0378460	0.0189230	−	0.999821i	$-0.493976\pi$
$439$	348.110	0.0378460	0.0189230	+	0.999821i	$0.493976\pi$
$440$	−86.0987	−0.00932863
$441$	0	0
$442$	2993.27	0.322116
$443$	−13219.2	−1.41775	−0.708875	−	0.705334i	$-0.750797\pi$
$443$	−13219.2	−1.41775	−0.708875	+	0.705334i	$0.750797\pi$
$444$	2078.01	0.222113
$445$	−317.056	−0.0337751
$446$	6894.97	0.732032
$447$	242.511	0.0256608
$448$	0	0
$449$	11766.4	1.23672	0.618362	−	0.785893i	$-0.287797\pi$
$449$	11766.4	1.23672	0.618362	+	0.785893i	$0.287797\pi$
$450$	−9116.58	−0.955021
$451$	8704.56	0.908829
$452$	3199.86	0.332984
$453$	−13304.0	−1.37986
$454$	6109.31	0.631551
$455$	0	0
$456$	510.956	0.0524731
$457$	6992.19	0.715713	0.357856	−	0.933777i	$-0.383508\pi$
$457$	6992.19	0.715713	0.357856	+	0.933777i	$0.383508\pi$
$458$	−5226.54	−0.533231
$459$	−1449.91	−0.147442
$460$	−28.7991	−0.00291905
$461$	17986.6	1.81718	0.908591	−	0.417686i	$-0.137159\pi$
$461$	17986.6	1.81718	0.908591	+	0.417686i	$0.137159\pi$
$462$	0	0
$463$	−13843.8	−1.38958	−0.694790	−	0.719212i	$-0.744503\pi$
$463$	−13843.8	−1.38958	−0.694790	+	0.719212i	$0.744503\pi$
$464$	−299.747	−0.0299901
$465$	−842.880	−0.0840594
$466$	5021.65	0.499192
$467$	10777.1	1.06789	0.533945	−	0.845519i	$-0.320709\pi$
$467$	10777.1	1.06789	0.533945	+	0.845519i	$0.320709\pi$
$468$	11400.6	1.12606
$469$	0	0
$470$	95.3652	0.00935929
$471$	−18822.9	−1.84143
$472$	−4463.22	−0.435246
$473$	−4158.88	−0.404282
$474$	43.5624	0.00422128
$475$	1001.14	0.0967060
$476$	0	0
$477$	−11650.4	−1.11831
$478$	4625.61	0.442616
$479$	8288.07	0.790588	0.395294	−	0.918555i	$-0.370643\pi$
$479$	8288.07	0.790588	0.395294	+	0.918555i	$0.370643\pi$
$480$	−79.8197	−0.00759012
$481$	−5091.61	−0.482656
$482$	1694.20	0.160101
$483$	0	0
$484$	−595.845	−0.0559584
$485$	−387.849	−0.0363120
$486$	10181.1	0.950258
$487$	−15294.3	−1.42311	−0.711553	−	0.702632i	$-0.752008\pi$
$487$	−15294.3	−1.42311	−0.711553	+	0.702632i	$0.752008\pi$
$488$	3984.81	0.369639
$489$	−22047.1	−2.03887
$490$	0	0
$491$	3480.01	0.319859	0.159930	−	0.987128i	$-0.448873\pi$
$491$	3480.01	0.319859	0.159930	+	0.987128i	$0.448873\pi$
$492$	8069.76	0.739457
$493$	−359.017	−0.0327978
$494$	−1251.96	−0.114025
$495$	−392.771	−0.0356641
$496$	5406.62	0.489444
$497$	0	0
$498$	−21280.3	−1.91485
$499$	2018.14	0.181051	0.0905256	−	0.995894i	$-0.471145\pi$
$499$	2018.14	0.181051	0.0905256	+	0.995894i	$0.471145\pi$
$500$	−312.911	−0.0279876
$501$	528.149	0.0470977
$502$	13229.4	1.17621
$503$	4293.58	0.380599	0.190300	−	0.981726i	$-0.439054\pi$
$503$	4293.58	0.380599	0.190300	+	0.981726i	$0.439054\pi$
$504$	0	0
$505$	174.837	0.0154062
$506$	1581.52	0.138947
$507$	−31094.2	−2.72375
$508$	−2292.69	−0.200239
$509$	7654.07	0.666524	0.333262	−	0.942834i	$-0.391851\pi$
$509$	7654.07	0.666524	0.333262	+	0.942834i	$0.391851\pi$
$510$	−95.6026	−0.00830070
$511$	0	0
$512$	512.000	0.0441942
$513$	606.436	0.0521926
$514$	4256.97	0.365305
$515$	190.958	0.0163391
$516$	−3855.58	−0.328939
$517$	−5237.03	−0.445501
$518$	0	0
$519$	−1940.40	−0.164112
$520$	195.577	0.0164935
$521$	10284.9	0.864856	0.432428	−	0.901669i	$-0.357657\pi$
$521$	10284.9	0.864856	0.432428	+	0.901669i	$0.357657\pi$
$522$	−1367.41	−0.114655
$523$	9011.36	0.753421	0.376710	−	0.926331i	$-0.377055\pi$
$523$	9011.36	0.753421	0.376710	+	0.926331i	$0.377055\pi$
$524$	5300.67	0.441910
$525$	0	0
$526$	5867.82	0.486406
$527$	6475.68	0.535265
$528$	4383.34	0.361289
$529$	529.000	0.0434783
$530$	−199.862	−0.0163801
$531$	−20360.6	−1.66398
$532$	0	0
$533$	−19772.8	−1.60686
$534$	16141.5	1.30808
$535$	463.721	0.0374736
$536$	−6081.53	−0.490079
$537$	4374.17	0.351507
$538$	9153.50	0.733523
$539$	0	0
$540$	−94.7352	−0.00754954
$541$	−19266.8	−1.53113	−0.765567	−	0.643356i	$-0.777541\pi$
$541$	−19266.8	−1.53113	−0.765567	+	0.643356i	$0.777541\pi$
$542$	−9398.33	−0.744821
$543$	11843.3	0.935994
$544$	613.239	0.0483316
$545$	−607.509	−0.0477483
$546$	0	0
$547$	13655.7	1.06742	0.533709	−	0.845668i	$-0.320798\pi$
$547$	13655.7	1.06742	0.533709	+	0.845668i	$0.320798\pi$
$548$	9005.25	0.701980
$549$	18178.2	1.41316
$550$	8588.46	0.665842
$551$	150.162	0.0116100
$552$	1466.18	0.113052
$553$	0	0
$554$	15864.2	1.21662
$555$	162.622	0.0124377
$556$	−2044.10	−0.155916
$557$	−2831.01	−0.215357	−0.107678	−	0.994186i	$-0.534342\pi$
$557$	−2831.01	−0.215357	−0.107678	+	0.994186i	$0.534342\pi$
$558$	24664.2	1.87118
$559$	9447.06	0.714791
$560$	0	0
$561$	5250.07	0.395112
$562$	−4665.39	−0.350174
$563$	−25606.4	−1.91684	−0.958420	−	0.285360i	$-0.907887\pi$
$563$	−25606.4	−1.91684	−0.958420	+	0.285360i	$0.907887\pi$
$564$	−4855.10	−0.362476
$565$	250.416	0.0186461
$566$	11689.2	0.868082
$567$	0	0
$568$	7397.81	0.546488
$569$	−22672.4	−1.67043	−0.835217	−	0.549920i	$-0.814658\pi$
$569$	−22672.4	−1.67043	−0.835217	+	0.549920i	$0.814658\pi$
$570$	39.9866	0.00293834
$571$	−4307.71	−0.315713	−0.157856	−	0.987462i	$-0.550458\pi$
$571$	−4307.71	−0.315713	−0.157856	+	0.987462i	$0.550458\pi$
$572$	−10740.2	−0.785088
$573$	−35728.8	−2.60487
$574$	0	0
$575$	2872.75	0.208351
$576$	2335.67	0.168958
$577$	−5659.36	−0.408323	−0.204161	−	0.978937i	$-0.565447\pi$
$577$	−5659.36	−0.408323	−0.204161	+	0.978937i	$0.565447\pi$
$578$	−9091.50	−0.654250
$579$	24991.2	1.79378
$580$	−23.4577	−0.00167936
$581$	0	0
$582$	19745.7	1.40633
$583$	10975.5	0.779690
$584$	−5912.07	−0.418910
$585$	892.195	0.0630559
$586$	11151.7	0.786128
$587$	−15466.8	−1.08753	−0.543766	−	0.839237i	$-0.683002\pi$
$587$	−15466.8	−1.08753	−0.543766	+	0.839237i	$0.683002\pi$
$588$	0	0
$589$	−2708.51	−0.189477
$590$	−349.284	−0.0243726
$591$	9911.19	0.689834
$592$	−1043.13	−0.0724196
$593$	26319.1	1.82259	0.911296	−	0.411751i	$-0.135083\pi$
$593$	26319.1	1.82259	0.911296	+	0.411751i	$0.135083\pi$
$594$	5202.43	0.359357
$595$	0	0
$596$	−121.737	−0.00836668
$597$	−27227.4	−1.86657
$598$	−3592.48	−0.245665
$599$	14268.0	0.973247	0.486624	−	0.873612i	$-0.338228\pi$
$599$	14268.0	0.973247	0.486624	+	0.873612i	$0.338228\pi$
$600$	7962.12	0.541754
$601$	−12068.3	−0.819092	−0.409546	−	0.912289i	$-0.634313\pi$
$601$	−12068.3	−0.819092	−0.409546	+	0.912289i	$0.634313\pi$
$602$	0	0
$603$	−27743.1	−1.87361
$604$	6678.40	0.449901
$605$	−46.6299	−0.00313351
$606$	−8901.06	−0.596668
$607$	−23440.5	−1.56741	−0.783707	−	0.621130i	$-0.786674\pi$
$607$	−23440.5	−1.56741	−0.783707	+	0.621130i	$0.786674\pi$
$608$	−256.492	−0.0171088
$609$	0	0
$610$	311.845	0.0206987
$611$	11896.1	0.787668
$612$	2797.51	0.184776
$613$	9242.96	0.609004	0.304502	−	0.952512i	$-0.401510\pi$
$613$	9242.96	0.609004	0.304502	+	0.952512i	$0.401510\pi$
$614$	−15878.1	−1.04363
$615$	631.527	0.0414075
$616$	0	0
$617$	13063.7	0.852391	0.426195	−	0.904631i	$-0.359854\pi$
$617$	13063.7	0.852391	0.426195	+	0.904631i	$0.359854\pi$
$618$	−9721.79	−0.632796
$619$	17653.8	1.14631	0.573154	−	0.819448i	$-0.305720\pi$
$619$	17653.8	1.14631	0.573154	+	0.819448i	$0.305720\pi$
$620$	423.113	0.0274075
$621$	1740.16	0.112448
$622$	267.340	0.0172337
$623$	0	0
$624$	−9956.94	−0.638777
$625$	15588.3	0.997649
$626$	16141.5	1.03058
$627$	−2195.89	−0.139865
$628$	9448.81	0.600396
$629$	−1249.39	−0.0791995
$630$	0	0
$631$	−27750.7	−1.75078	−0.875388	−	0.483421i	$-0.839394\pi$
$631$	−27750.7	−1.75078	−0.875388	+	0.483421i	$0.839394\pi$
$632$	−21.8677	−0.00137634
$633$	11342.7	0.712217
$634$	6058.39	0.379510
$635$	−179.422	−0.0112128
$636$	10175.1	0.634385
$637$	0	0
$638$	1288.19	0.0799374
$639$	33747.8	2.08927
$640$	40.0683	0.00247475
$641$	−22967.6	−1.41524	−0.707618	−	0.706595i	$-0.750230\pi$
$641$	−22967.6	−1.41524	−0.707618	+	0.706595i	$0.750230\pi$
$642$	−23608.3	−1.45132
$643$	16102.0	0.987557	0.493778	−	0.869588i	$-0.335615\pi$
$643$	16102.0	0.987557	0.493778	+	0.869588i	$0.335615\pi$
$644$	0	0
$645$	−301.731	−0.0184196
$646$	−307.209	−0.0187105
$647$	−29250.8	−1.77738	−0.888692	−	0.458505i	$-0.848385\pi$
$647$	−29250.8	−1.77738	−0.888692	+	0.458505i	$0.848385\pi$
$648$	−3059.87	−0.185499
$649$	19181.1	1.16013
$650$	−19509.0	−1.17724
$651$	0	0
$652$	11067.3	0.664770
$653$	−17144.4	−1.02743	−0.513716	−	0.857960i	$-0.671731\pi$
$653$	−17144.4	−1.02743	−0.513716	+	0.857960i	$0.671731\pi$
$654$	30928.7	1.84925
$655$	414.822	0.0247457
$656$	−4050.90	−0.241099
$657$	−26970.1	−1.60153
$658$	0	0
$659$	−28408.3	−1.67925	−0.839627	−	0.543163i	$-0.817227\pi$
$659$	−28408.3	−1.67925	−0.839627	+	0.543163i	$0.817227\pi$
$660$	343.033	0.0202311
$661$	18051.8	1.06223	0.531115	−	0.847300i	$-0.321773\pi$
$661$	18051.8	1.06223	0.531115	+	0.847300i	$0.321773\pi$
$662$	−3437.19	−0.201798
$663$	−11925.7	−0.698578
$664$	10682.4	0.624334
$665$	0	0
$666$	−4758.62	−0.276866
$667$	430.887	0.0250135
$668$	−265.123	−0.0153561
$669$	−27470.8	−1.58757
$670$	−475.931	−0.0274430
$671$	−17125.1	−0.985257
$672$	0	0
$673$	7811.47	0.447414	0.223707	−	0.974656i	$-0.428184\pi$
$673$	7811.47	0.447414	0.223707	+	0.974656i	$0.428184\pi$
$674$	4913.05	0.280777
$675$	9449.96	0.538858
$676$	15608.8	0.888074
$677$	−4649.11	−0.263929	−0.131964	−	0.991254i	$-0.542128\pi$
$677$	−4649.11	−0.263929	−0.131964	+	0.991254i	$0.542128\pi$
$678$	−12748.8	−0.722147
$679$	0	0
$680$	47.9911	0.00270643
$681$	−24340.6	−1.36965
$682$	−23235.5	−1.30459
$683$	572.789	0.0320896	0.0160448	−	0.999871i	$-0.494893\pi$
$683$	572.789	0.0320896	0.0160448	+	0.999871i	$0.494893\pi$
$684$	−1170.08	−0.0654082
$685$	704.736	0.0393089
$686$	0	0
$687$	20823.5	1.15643
$688$	1935.44	0.107250
$689$	−24931.3	−1.37853
$690$	114.741	0.00633060
$691$	−33351.4	−1.83610	−0.918051	−	0.396463i	$-0.870237\pi$
$691$	−33351.4	−1.83610	−0.918051	+	0.396463i	$0.870237\pi$
$692$	974.053	0.0535086
$693$	0	0
$694$	−1269.55	−0.0694403
$695$	−159.968	−0.00873084
$696$	1194.25	0.0650400
$697$	−4851.89	−0.263671
$698$	−14708.7	−0.797613
$699$	−20007.2	−1.08261
$700$	0	0
$701$	−2875.54	−0.154932	−0.0774662	−	0.996995i	$-0.524683\pi$
$701$	−2875.54	−0.154932	−0.0774662	+	0.996995i	$0.524683\pi$
$702$	−11817.5	−0.635362
$703$	522.568	0.0280356
$704$	−2200.37	−0.117798
$705$	−379.952	−0.0202976
$706$	−6434.44	−0.343007
$707$	0	0
$708$	17782.3	0.943925
$709$	−2491.57	−0.131979	−0.0659894	−	0.997820i	$-0.521020\pi$
$709$	−2491.57	−0.131979	−0.0659894	+	0.997820i	$0.521020\pi$
$710$	578.940	0.0306018
$711$	−99.7573	−0.00526187
$712$	−8102.81	−0.426497
$713$	−7772.01	−0.408225
$714$	0	0
$715$	−840.511	−0.0439627
$716$	−2195.77	−0.114608
$717$	−18429.3	−0.959908
$718$	5876.63	0.305451
$719$	−26670.1	−1.38335	−0.691673	−	0.722211i	$-0.743126\pi$
$719$	−26670.1	−1.38335	−0.691673	+	0.722211i	$0.743126\pi$
$720$	182.786	0.00946116
$721$	0	0
$722$	−13589.5	−0.700484
$723$	−6750.01	−0.347214
$724$	−5945.16	−0.305180
$725$	2339.94	0.119866
$726$	2373.96	0.121358
$727$	−30347.5	−1.54818	−0.774090	−	0.633076i	$-0.781792\pi$
$727$	−30347.5	−1.54818	−0.774090	+	0.633076i	$0.781792\pi$
$728$	0	0
$729$	−30236.4	−1.53617
$730$	−462.669	−0.0234578
$731$	2318.14	0.117291
$732$	−15876.2	−0.801642
$733$	35966.0	1.81233	0.906163	−	0.422929i	$-0.138998\pi$
$733$	35966.0	1.81233	0.906163	+	0.422929i	$0.138998\pi$
$734$	−5886.66	−0.296023
$735$	0	0
$736$	−736.000	−0.0368605
$737$	26136.0	1.30628
$738$	−18479.6	−0.921741
$739$	2709.69	0.134882	0.0674409	−	0.997723i	$-0.478517\pi$
$739$	2709.69	0.134882	0.0674409	+	0.997723i	$0.478517\pi$
$740$	−81.6337	−0.00405529
$741$	4988.04	0.247288
$742$	0	0
$743$	−882.352	−0.0435671	−0.0217836	−	0.999763i	$-0.506934\pi$
$743$	−882.352	−0.0435671	−0.0217836	+	0.999763i	$0.506934\pi$
$744$	−21541.0	−1.06146
$745$	−9.52694	−0.000468510 0
$746$	95.3248	0.00467840
$747$	48731.7	2.38688
$748$	−2635.45	−0.128826
$749$	0	0
$750$	1246.69	0.0606971
$751$	−4318.15	−0.209816	−0.104908	−	0.994482i	$-0.533455\pi$
$751$	−4318.15	−0.209816	−0.104908	+	0.994482i	$0.533455\pi$
$752$	2437.19	0.118185
$753$	−52708.4	−2.55086
$754$	−2926.18	−0.141333
$755$	522.641	0.0251932
$756$	0	0
$757$	−17626.7	−0.846303	−0.423152	−	0.906059i	$-0.639076\pi$
$757$	−17626.7	−0.846303	−0.423152	+	0.906059i	$0.639076\pi$
$758$	1046.87	0.0501635
$759$	−6301.05	−0.301336
$760$	−20.0727	−0.000958043 0
$761$	35179.4	1.67576	0.837880	−	0.545854i	$-0.183795\pi$
$761$	35179.4	1.67576	0.837880	+	0.545854i	$0.183795\pi$
$762$	9134.49	0.434262
$763$	0	0
$764$	17935.3	0.849314
$765$	218.929	0.0103469
$766$	−4407.18	−0.207882
$767$	−43570.7	−2.05117
$768$	−2039.90	−0.0958446
$769$	36542.9	1.71362	0.856808	−	0.515635i	$-0.172444\pi$
$769$	36542.9	1.71362	0.856808	+	0.515635i	$0.172444\pi$
$770$	0	0
$771$	−16960.5	−0.792242
$772$	−12545.2	−0.584859
$773$	9958.57	0.463370	0.231685	−	0.972791i	$-0.425576\pi$
$773$	9958.57	0.463370	0.231685	+	0.972791i	$0.425576\pi$
$774$	8829.22	0.410026
$775$	−42206.1	−1.95624
$776$	−9912.02	−0.458532
$777$	0	0
$778$	19695.3	0.907598
$779$	2029.34	0.0933360
$780$	−779.214	−0.0357697
$781$	−31792.8	−1.45664
$782$	−881.531	−0.0403113
$783$	1417.41	0.0646923
$784$	0	0
$785$	739.449	0.0336204
$786$	−21118.9	−0.958378
$787$	5915.84	0.267950	0.133975	−	0.990985i	$-0.457226\pi$
$787$	5915.84	0.267950	0.133975	+	0.990985i	$0.457226\pi$
$788$	−4975.27	−0.224919
$789$	−23378.5	−1.05488
$790$	−1.71133	−7.70713e−5 0
$791$	0	0
$792$	−10037.8	−0.450350
$793$	38900.4	1.74198
$794$	17758.1	0.793718
$795$	796.286	0.0355237
$796$	13667.8	0.608594
$797$	−4183.71	−0.185940	−0.0929702	−	0.995669i	$-0.529636\pi$
$797$	−4183.71	−0.185940	−0.0929702	+	0.995669i	$0.529636\pi$
$798$	0	0
$799$	2919.10	0.129249
$800$	−3996.86	−0.176638
$801$	−36963.9	−1.63053
$802$	29594.6	1.30302
$803$	25407.7	1.11659
$804$	24229.9	1.06284
$805$	0	0
$806$	52780.3	2.30658
$807$	−36469.2	−1.59080
$808$	4468.20	0.194543
$809$	6345.65	0.275774	0.137887	−	0.990448i	$-0.455969\pi$
$809$	6345.65	0.275774	0.137887	+	0.990448i	$0.455969\pi$
$810$	−239.461	−0.0103874
$811$	−26595.7	−1.15154	−0.575771	−	0.817611i	$-0.695298\pi$
$811$	−26595.7	−1.15154	−0.575771	+	0.817611i	$0.695298\pi$
$812$	0	0
$813$	37444.7	1.61530
$814$	4482.96	0.193031
$815$	866.111	0.0372252
$816$	−2443.26	−0.104817
$817$	−969.582	−0.0415194
$818$	−3483.91	−0.148914
$819$	0	0
$820$	−317.017	−0.0135009
$821$	−5939.20	−0.252472	−0.126236	−	0.992000i	$-0.540290\pi$
$821$	−5939.20	−0.252472	−0.126236	+	0.992000i	$0.540290\pi$
$822$	−35878.6	−1.52240
$823$	−746.431	−0.0316148	−0.0158074	−	0.999875i	$-0.505032\pi$
$823$	−746.431	−0.0316148	−0.0158074	+	0.999875i	$0.505032\pi$
$824$	4880.19	0.206322
$825$	−34218.0	−1.44402
$826$	0	0
$827$	−20368.5	−0.856449	−0.428225	−	0.903672i	$-0.640861\pi$
$827$	−20368.5	−0.856449	−0.428225	+	0.903672i	$0.640861\pi$
$828$	−3357.53	−0.140921
$829$	6739.18	0.282342	0.141171	−	0.989985i	$-0.454913\pi$
$829$	6739.18	0.282342	0.141171	+	0.989985i	$0.454913\pi$
$830$	835.988	0.0349609
$831$	−63206.0	−2.63850
$832$	4998.23	0.208272
$833$	0	0
$834$	8144.07	0.338137
$835$	−20.7481	−0.000859900 0
$836$	1102.30	0.0456027
$837$	−25566.2	−1.05579
$838$	25275.0	1.04190
$839$	40067.4	1.64872	0.824362	−	0.566064i	$-0.191534\pi$
$839$	40067.4	1.64872	0.824362	+	0.566064i	$0.191534\pi$
$840$	0	0
$841$	−24038.0	−0.985609
$842$	939.229	0.0384418
$843$	18587.8	0.759428
$844$	−5693.88	−0.232217
$845$	1221.52	0.0497296
$846$	11118.1	0.451831
$847$	0	0
$848$	−5107.74	−0.206840
$849$	−46572.0	−1.88262
$850$	−4787.17	−0.193175
$851$	1499.50	0.0604021
$852$	−29474.2	−1.18518
$853$	1865.79	0.0748925	0.0374462	−	0.999299i	$-0.488078\pi$
$853$	1865.79	0.0748925	0.0374462	+	0.999299i	$0.488078\pi$
$854$	0	0
$855$	−91.5688	−0.00366268
$856$	11851.0	0.473200
$857$	−44608.4	−1.77805	−0.889027	−	0.457855i	$-0.848618\pi$
$857$	−44608.4	−1.77805	−0.889027	+	0.457855i	$0.848618\pi$
$858$	42790.9	1.70263
$859$	−42366.2	−1.68279	−0.841395	−	0.540421i	$-0.818265\pi$
$859$	−42366.2	−1.68279	−0.841395	+	0.540421i	$0.818265\pi$
$860$	151.465	0.00600570
$861$	0	0
$862$	250.700	0.00990589
$863$	9724.26	0.383566	0.191783	−	0.981437i	$-0.438573\pi$
$863$	9724.26	0.383566	0.191783	+	0.981437i	$0.438573\pi$
$864$	−2421.09	−0.0953322
$865$	76.2278	0.00299633
$866$	2294.01	0.0900159
$867$	36222.2	1.41888
$868$	0	0
$869$	93.9785	0.00366859
$870$	93.4599	0.00364206
$871$	−59368.9	−2.30957
$872$	−15525.7	−0.602944
$873$	−45217.3	−1.75300
$874$	368.708	0.0142697
$875$	0	0
$876$	23554.8	0.908496
$877$	10094.3	0.388667	0.194333	−	0.980936i	$-0.437746\pi$
$877$	10094.3	0.388667	0.194333	+	0.980936i	$0.437746\pi$
$878$	696.220	0.0267611
$879$	−44430.3	−1.70489
$880$	−172.197	−0.00659634
$881$	305.924	0.0116990	0.00584952	−	0.999983i	$-0.498138\pi$
$881$	305.924	0.0116990	0.00584952	+	0.999983i	$0.498138\pi$
$882$	0	0
$883$	21381.8	0.814899	0.407449	−	0.913228i	$-0.366418\pi$
$883$	21381.8	0.814899	0.407449	+	0.913228i	$0.366418\pi$
$884$	5986.54	0.227771
$885$	1391.61	0.0528571
$886$	−26438.4	−1.00250
$887$	−28514.5	−1.07940	−0.539698	−	0.841859i	$-0.681461\pi$
$887$	−28514.5	−1.07940	−0.539698	+	0.841859i	$0.681461\pi$
$888$	4156.02	0.157058
$889$	0	0
$890$	−634.113	−0.0238826
$891$	13150.1	0.494439
$892$	13789.9	0.517625
$893$	−1220.94	−0.0457526
$894$	485.023	0.0181450
$895$	−171.837	−0.00641774
$896$	0	0
$897$	14313.1	0.532776
$898$	23532.7	0.874496
$899$	−6330.54	−0.234856
$900$	−18233.2	−0.675302
$901$	−6117.70	−0.226204
$902$	17409.1	0.642639
$903$	0	0
$904$	6399.71	0.235455
$905$	−465.258	−0.0170892
$906$	−26608.0	−0.975707
$907$	39454.6	1.44440	0.722198	−	0.691686i	$-0.243132\pi$
$907$	39454.6	1.44440	0.722198	+	0.691686i	$0.243132\pi$
$908$	12218.6	0.446574
$909$	20383.3	0.743753
$910$	0	0
$911$	36089.0	1.31250	0.656248	−	0.754546i	$-0.272143\pi$
$911$	36089.0	1.31250	0.656248	+	0.754546i	$0.272143\pi$
$912$	1021.91	0.0371041
$913$	−45908.7	−1.66414
$914$	13984.4	0.506085
$915$	−1242.45	−0.0448896
$916$	−10453.1	−0.377052
$917$	0	0
$918$	−2899.81	−0.104257
$919$	−14857.2	−0.533289	−0.266644	−	0.963795i	$-0.585915\pi$
$919$	−14857.2	−0.533289	−0.266644	+	0.963795i	$0.585915\pi$
$920$	−57.5982	−0.00206408
$921$	63261.3	2.26334
$922$	35973.3	1.28494
$923$	72218.6	2.57541
$924$	0	0
$925$	8143.07	0.289451
$926$	−27687.6	−0.982582
$927$	22262.8	0.788787
$928$	−599.494	−0.0212062
$929$	−10769.2	−0.380330	−0.190165	−	0.981752i	$-0.560902\pi$
$929$	−10769.2	−0.380330	−0.190165	+	0.981752i	$0.560902\pi$
$930$	−1685.76	−0.0594390
$931$	0	0
$932$	10043.3	0.352982
$933$	−1065.13	−0.0373750
$934$	21554.2	0.755112
$935$	−206.246	−0.00721388
$936$	22801.3	0.796242
$937$	37240.3	1.29839	0.649194	−	0.760623i	$-0.275106\pi$
$937$	37240.3	1.29839	0.649194	+	0.760623i	$0.275106\pi$
$938$	0	0
$939$	−64310.7	−2.23504
$940$	190.730	0.00661802
$941$	5280.84	0.182944	0.0914721	−	0.995808i	$-0.470843\pi$
$941$	5280.84	0.182944	0.0914721	+	0.995808i	$0.470843\pi$
$942$	−37645.8	−1.30209
$943$	5823.16	0.201090
$944$	−8926.44	−0.307766
$945$	0	0
$946$	−8317.75	−0.285870
$947$	−1242.51	−0.0426360	−0.0213180	−	0.999773i	$-0.506786\pi$
$947$	−1242.51	−0.0426360	−0.0213180	+	0.999773i	$0.506786\pi$
$948$	87.1248	0.00298490
$949$	−57714.7	−1.97418
$950$	2002.28	0.0683815
$951$	−24137.7	−0.823049
$952$	0	0
$953$	13076.5	0.444481	0.222240	−	0.974992i	$-0.428663\pi$
$953$	13076.5	0.444481	0.222240	+	0.974992i	$0.428663\pi$
$954$	−23300.8	−0.790767
$955$	1403.59	0.0475592
$956$	9251.21	0.312977
$957$	−5132.40	−0.173361
$958$	16576.1	0.559030
$959$	0	0
$960$	−159.639	−0.00536702
$961$	84394.6	2.83289
$962$	−10183.2	−0.341289
$963$	54062.7	1.80908
$964$	3388.40	0.113209
$965$	−981.767	−0.0327505
$966$	0	0
$967$	32434.3	1.07861	0.539305	−	0.842110i	$-0.318687\pi$
$967$	32434.3	1.07861	0.539305	+	0.842110i	$0.318687\pi$
$968$	−1191.69	−0.0395686
$969$	1223.98	0.0405777
$970$	−775.698	−0.0256765
$971$	−18890.5	−0.624331	−0.312165	−	0.950028i	$-0.601054\pi$
$971$	−18890.5	−0.624331	−0.312165	+	0.950028i	$0.601054\pi$
$972$	20362.3	0.671934
$973$	0	0
$974$	−30588.7	−1.00629
$975$	77727.6	2.55310
$976$	7969.62	0.261374
$977$	−47115.0	−1.54283	−0.771413	−	0.636334i	$-0.780450\pi$
$977$	−47115.0	−1.54283	−0.771413	+	0.636334i	$0.780450\pi$
$978$	−44094.3	−1.44170
$979$	34822.6	1.13681
$980$	0	0
$981$	−70826.3	−2.30511
$982$	6960.03	0.226174
$983$	34041.4	1.10453	0.552264	−	0.833669i	$-0.313764\pi$
$983$	34041.4	1.10453	0.552264	+	0.833669i	$0.313764\pi$
$984$	16139.5	0.522875
$985$	−389.356	−0.0125948
$986$	−718.033	−0.0231915
$987$	0	0
$988$	−2503.92	−0.0806279
$989$	−2782.20	−0.0894527
$990$	−785.541	−0.0252183
$991$	45731.9	1.46591	0.732957	−	0.680275i	$-0.238140\pi$
$991$	45731.9	1.46591	0.732957	+	0.680275i	$0.238140\pi$
$992$	10813.2	0.346089
$993$	13694.4	0.437642
$994$	0	0
$995$	1069.62	0.0340795
$996$	−42560.7	−1.35400
$997$	−50455.4	−1.60275	−0.801374	−	0.598164i	$-0.795897\pi$
$997$	−50455.4	−1.60275	−0.801374	+	0.598164i	$0.795897\pi$
$998$	4036.29	0.128022
$999$	4932.64	0.156218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.x.1.1		11
7.3	odd	6		322.4.e.b.93.1	✓	22
7.5	odd	6		322.4.e.b.277.1	yes	22
7.6	odd	2		2254.4.a.w.1.11		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.b.93.1	✓	22	7.3	odd	6
322.4.e.b.277.1	yes	22	7.5	odd	6
2254.4.a.w.1.11		11	7.6	odd	2
2254.4.a.x.1.1		11	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -7.96837 q^{3} +4.00000 q^{4} +0.313034 q^{5} -15.9367 q^{6} +8.00000 q^{8} +36.4949 q^{9} +0.626067 q^{10} -34.3808 q^{11} -31.8735 q^{12} +78.0974 q^{13} -2.49437 q^{15} +16.0000 q^{16} +19.1637 q^{17} +72.9898 q^{18} -8.01538 q^{19} +1.25213 q^{20} -68.7616 q^{22} -23.0000 q^{23} -63.7470 q^{24} -124.902 q^{25} +156.195 q^{26} -75.6590 q^{27} -18.7342 q^{29} -4.98873 q^{30} +337.914 q^{31} +32.0000 q^{32} +273.959 q^{33} +38.3274 q^{34} +145.980 q^{36} -65.1957 q^{37} -16.0308 q^{38} -622.309 q^{39} +2.50427 q^{40} -253.181 q^{41} +120.965 q^{43} -137.523 q^{44} +11.4241 q^{45} -46.0000 q^{46} +152.324 q^{47} -127.494 q^{48} -249.804 q^{50} -152.704 q^{51} +312.389 q^{52} -319.234 q^{53} -151.318 q^{54} -10.7623 q^{55} +63.8695 q^{57} -37.4684 q^{58} -557.902 q^{59} -9.97747 q^{60} +498.101 q^{61} +675.827 q^{62} +64.0000 q^{64} +24.4471 q^{65} +547.918 q^{66} -760.191 q^{67} +76.6548 q^{68} +183.272 q^{69} +924.726 q^{71} +291.959 q^{72} -739.009 q^{73} -130.391 q^{74} +995.265 q^{75} -32.0615 q^{76} -1244.62 q^{78} -2.73346 q^{79} +5.00854 q^{80} -382.484 q^{81} -506.362 q^{82} +1335.30 q^{83} +5.99888 q^{85} +241.930 q^{86} +149.281 q^{87} -275.046 q^{88} -1012.85 q^{89} +22.8483 q^{90} -92.0000 q^{92} -2692.62 q^{93} +304.648 q^{94} -2.50908 q^{95} -254.988 q^{96} -1239.00 q^{97} -1254.72 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 6 * q^3 + 44 * q^4 + 27 * q^5 + 12 * q^6 + 88 * q^8 + 59 * q^9 + 54 * q^10 + 56 * q^11 + 24 * q^12 + 103 * q^13 + 62 * q^15 + 176 * q^16 + 157 * q^17 + 118 * q^18 + 266 * q^19 + 108 * q^20 + 112 * q^22 - 253 * q^23 + 48 * q^24 + 142 * q^25 + 206 * q^26 + 531 * q^27 - 195 * q^29 + 124 * q^30 + 316 * q^31 + 352 * q^32 + 682 * q^33 + 314 * q^34 + 236 * q^36 + 347 * q^37 + 532 * q^38 - 99 * q^39 + 216 * q^40 + 540 * q^41 + 193 * q^43 + 224 * q^44 + 609 * q^45 - 506 * q^46 + 731 * q^47 + 96 * q^48 + 284 * q^50 - 303 * q^51 + 412 * q^52 + 413 * q^53 + 1062 * q^54 + 1809 * q^55 + 343 * q^57 - 390 * q^58 + 1061 * q^59 + 248 * q^60 + 1157 * q^61 + 632 * q^62 + 704 * q^64 - 37 * q^65 + 1364 * q^66 - 841 * q^67 + 628 * q^68 - 138 * q^69 + 869 * q^71 + 472 * q^72 + 409 * q^73 + 694 * q^74 + 2024 * q^75 + 1064 * q^76 - 198 * q^78 + 247 * q^79 + 432 * q^80 + 1019 * q^81 + 1080 * q^82 + 2863 * q^83 + 785 * q^85 + 386 * q^86 + 670 * q^87 + 448 * q^88 + 1610 * q^89 + 1218 * q^90 - 1012 * q^92 - 1290 * q^93 + 1462 * q^94 - 209 * q^95 + 192 * q^96 + 1783 * q^97 + 101 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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