LMFDB - Embedded newform 1911.4.a.bd.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1911,4,Mod(1,1911)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1911, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1911.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1911.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,-6,-42,50,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$112.752650021$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 63 x^{12} + 408 x^{11} + 1393 x^{10} - 10374 x^{9} - 12229 x^{8} + 122556 x^{7} + \cdots - 43904 $ x^14 - 6x^13 - 63x^12 + 408x^11 + 1393x^10 - 10374x^9 - 12229x^8 + 122556x^7 + 27054x^6 - 691944x^5 + 111012x^4 + 1785120x^3 - 380480x^2 - 1650432*x - 43904
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.30041$ of defining polynomial
Character	$\chi$	$=$	1911.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.30041 q^{2} -3.00000 q^{3} -2.70812 q^{4} +6.49112 q^{5} +6.90123 q^{6} +24.6331 q^{8} +9.00000 q^{9} -14.9322 q^{10} +18.3314 q^{11} +8.12436 q^{12} +13.0000 q^{13} -19.4734 q^{15} -35.0011 q^{16} +70.9629 q^{17} -20.7037 q^{18} -29.2042 q^{19} -17.5787 q^{20} -42.1698 q^{22} +14.8530 q^{23} -73.8992 q^{24} -82.8653 q^{25} -29.9053 q^{26} -27.0000 q^{27} +90.1767 q^{29} +44.7967 q^{30} -296.323 q^{31} -116.547 q^{32} -54.9943 q^{33} -163.244 q^{34} -24.3731 q^{36} -388.677 q^{37} +67.1815 q^{38} -39.0000 q^{39} +159.896 q^{40} +82.0539 q^{41} +318.156 q^{43} -49.6437 q^{44} +58.4201 q^{45} -34.1680 q^{46} +124.964 q^{47} +105.003 q^{48} +190.624 q^{50} -212.889 q^{51} -35.2055 q^{52} -27.0205 q^{53} +62.1110 q^{54} +118.992 q^{55} +87.6125 q^{57} -207.443 q^{58} -45.7534 q^{59} +52.7362 q^{60} -133.759 q^{61} +681.664 q^{62} +548.116 q^{64} +84.3846 q^{65} +126.509 q^{66} -713.284 q^{67} -192.176 q^{68} -44.5591 q^{69} -95.8064 q^{71} +221.697 q^{72} -211.205 q^{73} +894.117 q^{74} +248.596 q^{75} +79.0884 q^{76} +89.7159 q^{78} +262.652 q^{79} -227.197 q^{80} +81.0000 q^{81} -188.757 q^{82} -465.366 q^{83} +460.629 q^{85} -731.890 q^{86} -270.530 q^{87} +451.559 q^{88} +663.830 q^{89} -134.390 q^{90} -40.2238 q^{92} +888.968 q^{93} -287.468 q^{94} -189.568 q^{95} +349.642 q^{96} -1707.38 q^{97} +164.983 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 6 q^{2} - 42 q^{3} + 50 q^{4} - 4 q^{5} + 18 q^{6} - 30 q^{8} + 126 q^{9} + 32 q^{10} - 68 q^{11} - 150 q^{12} + 182 q^{13} + 12 q^{15} - 50 q^{16} - 54 q^{18} + 24 q^{19} + 96 q^{20} - 300 q^{22}+ \cdots - 612 q^{99}+O(q^{100}) $ 14 * q - 6 * q^2 - 42 * q^3 + 50 * q^4 - 4 * q^5 + 18 * q^6 - 30 * q^8 + 126 * q^9 + 32 * q^10 - 68 * q^11 - 150 * q^12 + 182 * q^13 + 12 * q^15 - 50 * q^16 - 54 * q^18 + 24 * q^19 + 96 * q^20 - 300 * q^22 - 64 * q^23 + 90 * q^24 - 118 * q^25 - 78 * q^26 - 378 * q^27 - 792 * q^29 - 96 * q^30 + 524 * q^31 - 126 * q^32 + 204 * q^33 - 88 * q^34 + 450 * q^36 - 344 * q^37 + 436 * q^38 - 546 * q^39 + 704 * q^40 + 44 * q^41 + 144 * q^43 - 1248 * q^44 - 36 * q^45 - 972 * q^46 - 236 * q^47 + 150 * q^48 - 714 * q^50 + 650 * q^52 - 1556 * q^53 + 162 * q^54 + 396 * q^55 - 72 * q^57 + 92 * q^58 + 1244 * q^59 - 288 * q^60 + 984 * q^61 + 236 * q^62 - 714 * q^64 - 52 * q^65 + 900 * q^66 - 1396 * q^67 + 960 * q^68 + 192 * q^69 - 1216 * q^71 - 270 * q^72 + 1768 * q^73 - 1212 * q^74 + 354 * q^75 + 496 * q^76 + 234 * q^78 - 1888 * q^79 + 1300 * q^80 + 1134 * q^81 + 1920 * q^82 - 1008 * q^83 + 56 * q^85 - 2128 * q^86 + 2376 * q^87 - 3148 * q^88 - 864 * q^89 + 288 * q^90 + 2856 * q^92 - 1572 * q^93 + 4860 * q^94 - 4984 * q^95 + 378 * q^96 + 1368 * q^97 - 612 * 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 $

$2$	−2.30041	−0.813317	−0.406659	−	0.913580i	$-0.633306\pi$
$2$	−2.30041	−0.813317	−0.406659	+	0.913580i	$0.633306\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.70812	−0.338515
$5$	6.49112	0.580584	0.290292	−	0.956938i	$-0.406248\pi$
$5$	6.49112	0.580584	0.290292	+	0.956938i	$0.406248\pi$
$6$	6.90123	0.469569
$7$	0	0
$7$	0	0
$8$	24.6331	1.08864
$9$	9.00000	0.333333
$10$	−14.9322	−0.472199
$11$	18.3314	0.502467	0.251234	−	0.967927i	$-0.419164\pi$
$11$	18.3314	0.502467	0.251234	+	0.967927i	$0.419164\pi$
$12$	8.12436	0.195442
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−19.4734	−0.335200
$16$	−35.0011	−0.546893
$17$	70.9629	1.01241	0.506207	−	0.862412i	$-0.331047\pi$
$17$	70.9629	1.01241	0.506207	+	0.862412i	$0.331047\pi$
$18$	−20.7037	−0.271106
$19$	−29.2042	−0.352626	−0.176313	−	0.984334i	$-0.556417\pi$
$19$	−29.2042	−0.352626	−0.176313	+	0.984334i	$0.556417\pi$
$20$	−17.5787	−0.196536
$21$	0	0
$22$	−42.1698	−0.408665
$23$	14.8530	0.134655	0.0673276	−	0.997731i	$-0.478553\pi$
$23$	14.8530	0.134655	0.0673276	+	0.997731i	$0.478553\pi$
$24$	−73.8992	−0.628525
$25$	−82.8653	−0.662923
$26$	−29.9053	−0.225574
$27$	−27.0000	−0.192450
$28$	0	0
$29$	90.1767	0.577427	0.288714	−	0.957416i	$-0.406772\pi$
$29$	90.1767	0.577427	0.288714	+	0.957416i	$0.406772\pi$
$30$	44.7967	0.272624
$31$	−296.323	−1.71681	−0.858405	−	0.512972i	$-0.828545\pi$
$31$	−296.323	−1.71681	−0.858405	+	0.512972i	$0.828545\pi$
$32$	−116.547	−0.643840
$33$	−54.9943	−0.290099
$34$	−163.244	−0.823413
$35$	0	0
$36$	−24.3731	−0.112838
$37$	−388.677	−1.72698	−0.863489	−	0.504368i	$-0.831726\pi$
$37$	−388.677	−1.72698	−0.863489	+	0.504368i	$0.831726\pi$
$38$	67.1815	0.286797
$39$	−39.0000	−0.160128
$40$	159.896	0.632045
$41$	82.0539	0.312553	0.156276	−	0.987713i	$-0.450051\pi$
$41$	82.0539	0.312553	0.156276	+	0.987713i	$0.450051\pi$
$42$	0	0
$43$	318.156	1.12833	0.564167	−	0.825661i	$-0.309197\pi$
$43$	318.156	1.12833	0.564167	+	0.825661i	$0.309197\pi$
$44$	−49.6437	−0.170093
$45$	58.4201	0.193528
$46$	−34.1680	−0.109517
$47$	124.964	0.387826	0.193913	−	0.981019i	$-0.437882\pi$
$47$	124.964	0.387826	0.193913	+	0.981019i	$0.437882\pi$
$48$	105.003	0.315749
$49$	0	0
$50$	190.624	0.539167
$51$	−212.889	−0.584517
$52$	−35.2055	−0.0938871
$53$	−27.0205	−0.0700292	−0.0350146	−	0.999387i	$-0.511148\pi$
$53$	−27.0205	−0.0700292	−0.0350146	+	0.999387i	$0.511148\pi$
$54$	62.1110	0.156523
$55$	118.992	0.291724
$56$	0	0
$57$	87.6125	0.203589
$58$	−207.443	−0.469631
$59$	−45.7534	−0.100959	−0.0504796	−	0.998725i	$-0.516075\pi$
$59$	−45.7534	−0.100959	−0.0504796	+	0.998725i	$0.516075\pi$
$60$	52.7362	0.113470
$61$	−133.759	−0.280755	−0.140378	−	0.990098i	$-0.544832\pi$
$61$	−133.759	−0.280755	−0.140378	+	0.990098i	$0.544832\pi$
$62$	681.664	1.39631
$63$	0	0
$64$	548.116	1.07054
$65$	84.3846	0.161025
$66$	126.509	0.235943
$67$	−713.284	−1.30062	−0.650309	−	0.759669i	$-0.725361\pi$
$67$	−713.284	−1.30062	−0.650309	+	0.759669i	$0.725361\pi$
$68$	−192.176	−0.342717
$69$	−44.5591	−0.0777432
$70$	0	0
$71$	−95.8064	−0.160143	−0.0800714	−	0.996789i	$-0.525515\pi$
$71$	−95.8064	−0.160143	−0.0800714	+	0.996789i	$0.525515\pi$
$72$	221.697	0.362879
$73$	−211.205	−0.338625	−0.169313	−	0.985562i	$-0.554155\pi$
$73$	−211.205	−0.338625	−0.169313	+	0.985562i	$0.554155\pi$
$74$	894.117	1.40458
$75$	248.596	0.382739
$76$	79.0884	0.119369
$77$	0	0
$78$	89.7159	0.130235
$79$	262.652	0.374058	0.187029	−	0.982354i	$-0.440114\pi$
$79$	262.652	0.374058	0.187029	+	0.982354i	$0.440114\pi$
$80$	−227.197	−0.317517
$81$	81.0000	0.111111
$82$	−188.757	−0.254205
$83$	−465.366	−0.615429	−0.307714	−	0.951479i	$-0.599564\pi$
$83$	−465.366	−0.615429	−0.307714	+	0.951479i	$0.599564\pi$
$84$	0	0
$85$	460.629	0.587791
$86$	−731.890	−0.917694
$87$	−270.530	−0.333378
$88$	451.559	0.547004
$89$	663.830	0.790627	0.395314	−	0.918546i	$-0.370636\pi$
$89$	663.830	0.790627	0.395314	+	0.918546i	$0.370636\pi$
$90$	−134.390	−0.157400
$91$	0	0
$92$	−40.2238	−0.0455828
$93$	888.968	0.991201
$94$	−287.468	−0.315426
$95$	−189.568	−0.204729
$96$	349.642	0.371721
$97$	−1707.38	−1.78720	−0.893600	−	0.448865i	$-0.851828\pi$
$97$	−1707.38	−1.78720	−0.893600	+	0.448865i	$0.851828\pi$
$98$	0	0
$99$	164.983	0.167489
$100$	224.409	0.224409
$101$	−684.010	−0.673876	−0.336938	−	0.941527i	$-0.609391\pi$
$101$	−684.010	−0.673876	−0.336938	+	0.941527i	$0.609391\pi$
$102$	489.731	0.475398
$103$	702.550	0.672081	0.336040	−	0.941848i	$-0.390912\pi$
$103$	702.550	0.672081	0.336040	+	0.941848i	$0.390912\pi$
$104$	320.230	0.301934
$105$	0	0
$106$	62.1581	0.0569560
$107$	1565.49	1.41441	0.707205	−	0.707009i	$-0.249956\pi$
$107$	1565.49	1.41441	0.707205	+	0.707009i	$0.249956\pi$
$108$	73.1192	0.0651472
$109$	409.492	0.359837	0.179919	−	0.983682i	$-0.442417\pi$
$109$	409.492	0.359837	0.179919	+	0.983682i	$0.442417\pi$
$110$	−273.729	−0.237264
$111$	1166.03	0.997071
$112$	0	0
$113$	1197.97	0.997307	0.498654	−	0.866801i	$-0.333828\pi$
$113$	1197.97	0.997307	0.498654	+	0.866801i	$0.333828\pi$
$114$	−201.545	−0.165582
$115$	96.4128	0.0781786
$116$	−244.209	−0.195468
$117$	117.000	0.0924500
$118$	105.252	0.0821118
$119$	0	0
$120$	−479.688	−0.364911
$121$	−994.958	−0.747527
$122$	307.700	0.228343
$123$	−246.162	−0.180452
$124$	802.477	0.581166
$125$	−1349.28	−0.965466
$126$	0	0
$127$	886.966	0.619729	0.309864	−	0.950781i	$-0.399716\pi$
$127$	886.966	0.619729	0.309864	+	0.950781i	$0.399716\pi$
$128$	−328.511	−0.226848
$129$	−954.469	−0.651444
$130$	−194.119	−0.130964
$131$	2695.42	1.79771	0.898853	−	0.438250i	$-0.144402\pi$
$131$	2695.42	1.79771	0.898853	+	0.438250i	$0.144402\pi$
$132$	148.931	0.0982030
$133$	0	0
$134$	1640.84	1.05782
$135$	−175.260	−0.111733
$136$	1748.03	1.10215
$137$	−121.610	−0.0758383	−0.0379192	−	0.999281i	$-0.512073\pi$
$137$	−121.610	−0.0758383	−0.0379192	+	0.999281i	$0.512073\pi$
$138$	102.504	0.0632299
$139$	1361.08	0.830544	0.415272	−	0.909697i	$-0.363686\pi$
$139$	1361.08	0.830544	0.415272	+	0.909697i	$0.363686\pi$
$140$	0	0
$141$	−374.891	−0.223912
$142$	220.394	0.130247
$143$	238.309	0.139359
$144$	−315.010	−0.182298
$145$	585.348	0.335245
$146$	485.857	0.275410
$147$	0	0
$148$	1052.58	0.584608
$149$	−1058.67	−0.582077	−0.291038	−	0.956711i	$-0.594001\pi$
$149$	−1058.67	−0.582077	−0.291038	+	0.956711i	$0.594001\pi$
$150$	−571.872	−0.311288
$151$	−3151.67	−1.69854	−0.849268	−	0.527962i	$-0.822956\pi$
$151$	−3151.67	−1.69854	−0.849268	+	0.527962i	$0.822956\pi$
$152$	−719.388	−0.383882
$153$	638.666	0.337471
$154$	0	0
$155$	−1923.47	−0.996752
$156$	105.617	0.0542058
$157$	−324.356	−0.164882	−0.0824409	−	0.996596i	$-0.526272\pi$
$157$	−324.356	−0.164882	−0.0824409	+	0.996596i	$0.526272\pi$
$158$	−604.206	−0.304228
$159$	81.0614	0.0404314
$160$	−756.524	−0.373803
$161$	0	0
$162$	−186.333	−0.0903686
$163$	−3156.15	−1.51662	−0.758309	−	0.651895i	$-0.773974\pi$
$163$	−3156.15	−1.51662	−0.758309	+	0.651895i	$0.773974\pi$
$164$	−222.212	−0.105804
$165$	−356.975	−0.168427
$166$	1070.53	0.500539
$167$	1566.64	0.725931	0.362965	−	0.931803i	$-0.381764\pi$
$167$	1566.64	0.725931	0.362965	+	0.931803i	$0.381764\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−1059.63	−0.478060
$171$	−262.838	−0.117542
$172$	−861.605	−0.381958
$173$	1330.13	0.584556	0.292278	−	0.956333i	$-0.405587\pi$
$173$	1330.13	0.584556	0.292278	+	0.956333i	$0.405587\pi$
$174$	622.330	0.271142
$175$	0	0
$176$	−641.621	−0.274796
$177$	137.260	0.0582888
$178$	−1527.08	−0.643031
$179$	1934.84	0.807913	0.403957	−	0.914778i	$-0.367635\pi$
$179$	1934.84	0.807913	0.403957	+	0.914778i	$0.367635\pi$
$180$	−158.209	−0.0655121
$181$	−560.218	−0.230059	−0.115029	−	0.993362i	$-0.536696\pi$
$181$	−560.218	−0.230059	−0.115029	+	0.993362i	$0.536696\pi$
$182$	0	0
$183$	401.277	0.162094
$184$	365.875	0.146591
$185$	−2522.95	−1.00265
$186$	−2044.99	−0.806161
$187$	1300.85	0.508704
$188$	−338.417	−0.131285
$189$	0	0
$190$	436.084	0.166510
$191$	563.500	0.213473	0.106737	−	0.994287i	$-0.465960\pi$
$191$	563.500	0.213473	0.106737	+	0.994287i	$0.465960\pi$
$192$	−1644.35	−0.618076
$193$	1315.51	0.490633	0.245317	−	0.969443i	$-0.421108\pi$
$193$	1315.51	0.490633	0.245317	+	0.969443i	$0.421108\pi$
$194$	3927.68	1.45356
$195$	−253.154	−0.0929678
$196$	0	0
$197$	166.769	0.0603138	0.0301569	−	0.999545i	$-0.490399\pi$
$197$	166.769	0.0603138	0.0301569	+	0.999545i	$0.490399\pi$
$198$	−379.528	−0.136222
$199$	−793.489	−0.282658	−0.141329	−	0.989963i	$-0.545138\pi$
$199$	−793.489	−0.282658	−0.141329	+	0.989963i	$0.545138\pi$
$200$	−2041.23	−0.721682
$201$	2139.85	0.750913
$202$	1573.50	0.548075
$203$	0	0
$204$	576.528	0.197868
$205$	532.622	0.181463
$206$	−1616.15	−0.546615
$207$	133.677	0.0448851
$208$	−455.015	−0.151681
$209$	−535.354	−0.177183
$210$	0	0
$211$	−3863.28	−1.26047	−0.630235	−	0.776404i	$-0.717041\pi$
$211$	−3863.28	−1.26047	−0.630235	+	0.776404i	$0.717041\pi$
$212$	73.1746	0.0237059
$213$	287.419	0.0924584
$214$	−3601.27	−1.15036
$215$	2065.19	0.655092
$216$	−665.092	−0.209508
$217$	0	0
$218$	−942.000	−0.292662
$219$	633.614	0.195505
$220$	−322.243	−0.0987529
$221$	922.517	0.280793
$222$	−2682.35	−0.810935
$223$	1525.79	0.458181	0.229090	−	0.973405i	$-0.426425\pi$
$223$	1525.79	0.458181	0.229090	+	0.973405i	$0.426425\pi$
$224$	0	0
$225$	−745.788	−0.220974
$226$	−2755.82	−0.811127
$227$	−4398.69	−1.28613	−0.643065	−	0.765812i	$-0.722338\pi$
$227$	−4398.69	−1.28613	−0.643065	+	0.765812i	$0.722338\pi$
$228$	−237.265	−0.0689178
$229$	4751.83	1.37122	0.685611	−	0.727968i	$-0.259535\pi$
$229$	4751.83	1.37122	0.685611	+	0.727968i	$0.259535\pi$
$230$	−221.789	−0.0635840
$231$	0	0
$232$	2221.33	0.628609
$233$	−4346.80	−1.22218	−0.611091	−	0.791561i	$-0.709269\pi$
$233$	−4346.80	−1.22218	−0.611091	+	0.791561i	$0.709269\pi$
$234$	−269.148	−0.0751912
$235$	811.155	0.225166
$236$	123.906	0.0341762
$237$	−787.955	−0.215963
$238$	0	0
$239$	−4303.53	−1.16474	−0.582369	−	0.812925i	$-0.697874\pi$
$239$	−4303.53	−1.16474	−0.582369	+	0.812925i	$0.697874\pi$
$240$	681.590	0.183319
$241$	2695.06	0.720350	0.360175	−	0.932885i	$-0.382717\pi$
$241$	2695.06	0.720350	0.360175	+	0.932885i	$0.382717\pi$
$242$	2288.81	0.607977
$243$	−243.000	−0.0641500
$244$	362.235	0.0950399
$245$	0	0
$246$	566.272	0.146765
$247$	−379.654	−0.0978009
$248$	−7299.33	−1.86898
$249$	1396.10	0.355318
$250$	3103.89	0.785230
$251$	4829.84	1.21457	0.607284	−	0.794485i	$-0.292259\pi$
$251$	4829.84	1.21457	0.607284	+	0.794485i	$0.292259\pi$
$252$	0	0
$253$	272.277	0.0676598
$254$	−2040.38	−0.504036
$255$	−1381.89	−0.339361
$256$	−3629.22	−0.886040
$257$	−4279.27	−1.03865	−0.519326	−	0.854576i	$-0.673817\pi$
$257$	−4279.27	−1.03865	−0.519326	+	0.854576i	$0.673817\pi$
$258$	2195.67	0.529831
$259$	0	0
$260$	−228.523	−0.0545093
$261$	811.590	0.192476
$262$	−6200.56	−1.46211
$263$	−2923.33	−0.685401	−0.342700	−	0.939445i	$-0.611342\pi$
$263$	−2923.33	−0.685401	−0.342700	+	0.939445i	$0.611342\pi$
$264$	−1354.68	−0.315813
$265$	−175.393	−0.0406578
$266$	0	0
$267$	−1991.49	−0.456469
$268$	1931.66	0.440279
$269$	−6329.23	−1.43457	−0.717286	−	0.696779i	$-0.754616\pi$
$269$	−6329.23	−1.43457	−0.717286	+	0.696779i	$0.754616\pi$
$270$	403.170	0.0908747
$271$	822.584	0.184385	0.0921926	−	0.995741i	$-0.470612\pi$
$271$	822.584	0.184385	0.0921926	+	0.995741i	$0.470612\pi$
$272$	−2483.78	−0.553682
$273$	0	0
$274$	279.753	0.0616806
$275$	−1519.04	−0.333097
$276$	120.671	0.0263172
$277$	−8358.12	−1.81296	−0.906482	−	0.422244i	$-0.861242\pi$
$277$	−8358.12	−1.81296	−0.906482	+	0.422244i	$0.861242\pi$
$278$	−3131.05	−0.675496
$279$	−2666.90	−0.572270
$280$	0	0
$281$	−1498.18	−0.318057	−0.159028	−	0.987274i	$-0.550836\pi$
$281$	−1498.18	−0.318057	−0.159028	+	0.987274i	$0.550836\pi$
$282$	862.403	0.182111
$283$	−2486.81	−0.522352	−0.261176	−	0.965291i	$-0.584110\pi$
$283$	−2486.81	−0.522352	−0.261176	+	0.965291i	$0.584110\pi$
$284$	259.455	0.0542107
$285$	568.703	0.118200
$286$	−548.208	−0.113343
$287$	0	0
$288$	−1048.93	−0.214613
$289$	122.729	0.0249805
$290$	−1346.54	−0.272660
$291$	5122.14	1.03184
$292$	571.967	0.114630
$293$	−8676.96	−1.73008	−0.865040	−	0.501703i	$-0.832707\pi$
$293$	−8676.96	−1.73008	−0.865040	+	0.501703i	$0.832707\pi$
$294$	0	0
$295$	−296.991	−0.0586152
$296$	−9574.31	−1.88005
$297$	−494.949	−0.0966998
$298$	2435.37	0.473413
$299$	193.089	0.0373466
$300$	−673.227	−0.129563
$301$	0	0
$302$	7250.12	1.38145
$303$	2052.03	0.389063
$304$	1022.18	0.192849
$305$	−868.246	−0.163002
$306$	−1469.19	−0.274471
$307$	5158.24	0.958946	0.479473	−	0.877557i	$-0.340828\pi$
$307$	5158.24	0.958946	0.479473	+	0.877557i	$0.340828\pi$
$308$	0	0
$309$	−2107.65	−0.388026
$310$	4424.76	0.810676
$311$	8589.06	1.56605	0.783024	−	0.621991i	$-0.213676\pi$
$311$	8589.06	1.56605	0.783024	+	0.621991i	$0.213676\pi$
$312$	−960.689	−0.174321
$313$	−231.476	−0.0418013	−0.0209006	−	0.999782i	$-0.506653\pi$
$313$	−231.476	−0.0418013	−0.0209006	+	0.999782i	$0.506653\pi$
$314$	746.152	0.134101
$315$	0	0
$316$	−711.292	−0.126624
$317$	254.028	0.0450084	0.0225042	−	0.999747i	$-0.492836\pi$
$317$	254.028	0.0450084	0.0225042	+	0.999747i	$0.492836\pi$
$318$	−186.474	−0.0328835
$319$	1653.07	0.290138
$320$	3557.89	0.621537
$321$	−4696.47	−0.816610
$322$	0	0
$323$	−2072.41	−0.357003
$324$	−219.358	−0.0376128
$325$	−1077.25	−0.183862
$326$	7260.43	1.23349
$327$	−1228.48	−0.207752
$328$	2021.24	0.340257
$329$	0	0
$330$	821.188	0.136985
$331$	−2434.59	−0.404281	−0.202141	−	0.979357i	$-0.564790\pi$
$331$	−2434.59	−0.404281	−0.202141	+	0.979357i	$0.564790\pi$
$332$	1260.27	0.208332
$333$	−3498.10	−0.575659
$334$	−3603.92	−0.590412
$335$	−4630.01	−0.755118
$336$	0	0
$337$	−5431.74	−0.878000	−0.439000	−	0.898487i	$-0.644667\pi$
$337$	−5431.74	−0.878000	−0.439000	+	0.898487i	$0.644667\pi$
$338$	−388.769	−0.0625629
$339$	−3593.92	−0.575795
$340$	−1247.44	−0.198976
$341$	−5432.02	−0.862641
$342$	604.634	0.0955990
$343$	0	0
$344$	7837.16	1.22835
$345$	−289.238	−0.0451364
$346$	−3059.85	−0.475430
$347$	−3877.63	−0.599891	−0.299946	−	0.953956i	$-0.596969\pi$
$347$	−3877.63	−0.599891	−0.299946	+	0.953956i	$0.596969\pi$
$348$	732.627	0.112853
$349$	10225.6	1.56838	0.784189	−	0.620522i	$-0.213079\pi$
$349$	10225.6	1.56838	0.784189	+	0.620522i	$0.213079\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	−2136.48	−0.323508
$353$	−3730.56	−0.562486	−0.281243	−	0.959637i	$-0.590747\pi$
$353$	−3730.56	−0.562486	−0.281243	+	0.959637i	$0.590747\pi$
$354$	−315.755	−0.0474073
$355$	−621.891	−0.0929762
$356$	−1797.73	−0.267639
$357$	0	0
$358$	−4450.91	−0.657090
$359$	−7412.63	−1.08976	−0.544880	−	0.838514i	$-0.683425\pi$
$359$	−7412.63	−1.08976	−0.544880	+	0.838514i	$0.683425\pi$
$360$	1439.07	0.210682
$361$	−6006.12	−0.875655
$362$	1288.73	0.187111
$363$	2984.87	0.431585
$364$	0	0
$365$	−1370.96	−0.196600
$366$	−923.101	−0.131834
$367$	11777.8	1.67519	0.837595	−	0.546291i	$-0.183961\pi$
$367$	11777.8	1.67519	0.837595	+	0.546291i	$0.183961\pi$
$368$	−519.873	−0.0736420
$369$	738.485	0.104184
$370$	5803.82	0.815477
$371$	0	0
$372$	−2407.43	−0.335536
$373$	−10844.5	−1.50538	−0.752689	−	0.658376i	$-0.771244\pi$
$373$	−10844.5	−1.50538	−0.752689	+	0.658376i	$0.771244\pi$
$374$	−2992.49	−0.413738
$375$	4047.84	0.557412
$376$	3078.24	0.422202
$377$	1172.30	0.160149
$378$	0	0
$379$	1521.26	0.206180	0.103090	−	0.994672i	$-0.467127\pi$
$379$	1521.26	0.206180	0.103090	+	0.994672i	$0.467127\pi$
$380$	513.372	0.0693038
$381$	−2660.90	−0.357800
$382$	−1296.28	−0.173622
$383$	−13373.9	−1.78427	−0.892137	−	0.451766i	$-0.850794\pi$
$383$	−13373.9	−1.78427	−0.892137	+	0.451766i	$0.850794\pi$
$384$	985.534	0.130971
$385$	0	0
$386$	−3026.20	−0.399041
$387$	2863.41	0.376111
$388$	4623.79	0.604993
$389$	−1238.58	−0.161436	−0.0807181	−	0.996737i	$-0.525721\pi$
$389$	−1238.58	−0.161436	−0.0807181	+	0.996737i	$0.525721\pi$
$390$	582.357	0.0756123
$391$	1054.01	0.136327
$392$	0	0
$393$	−8086.25	−1.03791
$394$	−383.638	−0.0490543
$395$	1704.90	0.217172
$396$	−446.793	−0.0566975
$397$	3857.21	0.487627	0.243813	−	0.969822i	$-0.421602\pi$
$397$	3857.21	0.487627	0.243813	+	0.969822i	$0.421602\pi$
$398$	1825.35	0.229891
$399$	0	0
$400$	2900.38	0.362548
$401$	−10143.6	−1.26321	−0.631604	−	0.775291i	$-0.717603\pi$
$401$	−10143.6	−1.26321	−0.631604	+	0.775291i	$0.717603\pi$
$402$	−4922.53	−0.610730
$403$	−3852.20	−0.476158
$404$	1852.38	0.228117
$405$	525.781	0.0645093
$406$	0	0
$407$	−7125.02	−0.867749
$408$	−5244.10	−0.636327
$409$	−5742.38	−0.694236	−0.347118	−	0.937822i	$-0.612840\pi$
$409$	−5742.38	−0.694236	−0.347118	+	0.937822i	$0.612840\pi$
$410$	−1225.25	−0.147587
$411$	364.830	0.0437853
$412$	−1902.59	−0.227509
$413$	0	0
$414$	−307.512	−0.0365058
$415$	−3020.75	−0.357308
$416$	−1515.12	−0.178569
$417$	−4083.25	−0.479515
$418$	1231.53	0.144106
$419$	−5033.18	−0.586842	−0.293421	−	0.955983i	$-0.594794\pi$
$419$	−5033.18	−0.586842	−0.293421	+	0.955983i	$0.594794\pi$
$420$	0	0
$421$	4747.80	0.549628	0.274814	−	0.961497i	$-0.411384\pi$
$421$	4747.80	0.549628	0.274814	+	0.961497i	$0.411384\pi$
$422$	8887.12	1.02516
$423$	1124.67	0.129275
$424$	−665.597	−0.0762364
$425$	−5880.36	−0.671152
$426$	−661.182	−0.0751981
$427$	0	0
$428$	−4239.54	−0.478798
$429$	−714.926	−0.0804591
$430$	−4750.78	−0.532798
$431$	14372.3	1.60624	0.803120	−	0.595818i	$-0.203172\pi$
$431$	14372.3	1.60624	0.803120	+	0.595818i	$0.203172\pi$
$432$	945.031	0.105250
$433$	6796.64	0.754332	0.377166	−	0.926146i	$-0.376899\pi$
$433$	6796.64	0.754332	0.377166	+	0.926146i	$0.376899\pi$
$434$	0	0
$435$	−1756.04	−0.193554
$436$	−1108.95	−0.121810
$437$	−433.770	−0.0474829
$438$	−1457.57	−0.159008
$439$	−84.0065	−0.00913305	−0.00456653	−	0.999990i	$-0.501454\pi$
$439$	−84.0065	−0.00913305	−0.00456653	+	0.999990i	$0.501454\pi$
$440$	2931.13	0.317582
$441$	0	0
$442$	−2122.17	−0.228374
$443$	−6394.05	−0.685757	−0.342878	−	0.939380i	$-0.611402\pi$
$443$	−6394.05	−0.685757	−0.342878	+	0.939380i	$0.611402\pi$
$444$	−3157.75	−0.337523
$445$	4309.00	0.459025
$446$	−3509.94	−0.372646
$447$	3176.00	0.336062
$448$	0	0
$449$	−8684.68	−0.912818	−0.456409	−	0.889770i	$-0.650865\pi$
$449$	−8684.68	−0.912818	−0.456409	+	0.889770i	$0.650865\pi$
$450$	1715.62	0.179722
$451$	1504.17	0.157047
$452$	−3244.25	−0.337603
$453$	9455.00	0.980650
$454$	10118.8	1.04603
$455$	0	0
$456$	2158.16	0.221634
$457$	3705.07	0.379247	0.189623	−	0.981857i	$-0.439273\pi$
$457$	3705.07	0.379247	0.189623	+	0.981857i	$0.439273\pi$
$458$	−10931.2	−1.11524
$459$	−1916.00	−0.194839
$460$	−261.097	−0.0264646
$461$	−1109.11	−0.112053	−0.0560264	−	0.998429i	$-0.517843\pi$
$461$	−1109.11	−0.112053	−0.0560264	+	0.998429i	$0.517843\pi$
$462$	0	0
$463$	16318.8	1.63801	0.819004	−	0.573788i	$-0.194527\pi$
$463$	16318.8	1.63801	0.819004	+	0.573788i	$0.194527\pi$
$464$	−3156.29	−0.315791
$465$	5770.40	0.575475
$466$	9999.41	0.994021
$467$	10345.0	1.02507	0.512536	−	0.858665i	$-0.328706\pi$
$467$	10345.0	1.02507	0.512536	+	0.858665i	$0.328706\pi$
$468$	−316.850	−0.0312957
$469$	0	0
$470$	−1865.99	−0.183131
$471$	973.069	0.0951945
$472$	−1127.05	−0.109908
$473$	5832.26	0.566951
$474$	1812.62	0.175646
$475$	2420.01	0.233764
$476$	0	0
$477$	−243.184	−0.0233431
$478$	9899.88	0.947301
$479$	−2635.22	−0.251370	−0.125685	−	0.992070i	$-0.540113\pi$
$479$	−2635.22	−0.251370	−0.125685	+	0.992070i	$0.540113\pi$
$480$	2269.57	0.215815
$481$	−5052.81	−0.478977
$482$	−6199.75	−0.585873
$483$	0	0
$484$	2694.46	0.253049
$485$	−11082.8	−1.03762
$486$	558.999	0.0521743
$487$	−15546.4	−1.44656	−0.723278	−	0.690557i	$-0.757365\pi$
$487$	−15546.4	−1.44656	−0.723278	+	0.690557i	$0.757365\pi$
$488$	−3294.89	−0.305641
$489$	9468.45	0.875620
$490$	0	0
$491$	−15347.6	−1.41064	−0.705321	−	0.708888i	$-0.749197\pi$
$491$	−15347.6	−1.41064	−0.705321	+	0.708888i	$0.749197\pi$
$492$	666.635	0.0610858
$493$	6399.19	0.584595
$494$	873.360	0.0795432
$495$	1070.92	0.0972414
$496$	10371.6	0.938912
$497$	0	0
$498$	−3211.60	−0.288986
$499$	−2660.47	−0.238675	−0.119337	−	0.992854i	$-0.538077\pi$
$499$	−2660.47	−0.238675	−0.119337	+	0.992854i	$0.538077\pi$
$500$	3654.01	0.326824
$501$	−4699.93	−0.419116
$502$	−11110.6	−0.987830
$503$	4459.34	0.395293	0.197646	−	0.980273i	$-0.436670\pi$
$503$	4459.34	0.395293	0.197646	+	0.980273i	$0.436670\pi$
$504$	0	0
$505$	−4439.99	−0.391242
$506$	−626.349	−0.0550289
$507$	−507.000	−0.0444116
$508$	−2402.01	−0.209787
$509$	2022.07	0.176083	0.0880417	−	0.996117i	$-0.471939\pi$
$509$	2022.07	0.176083	0.0880417	+	0.996117i	$0.471939\pi$
$510$	3178.90	0.276008
$511$	0	0
$512$	10976.8	0.947480
$513$	788.513	0.0678629
$514$	9844.07	0.844753
$515$	4560.34	0.390199
$516$	2584.81	0.220524
$517$	2290.77	0.194870
$518$	0	0
$519$	−3990.40	−0.337494
$520$	2078.65	0.175298
$521$	5329.77	0.448180	0.224090	−	0.974568i	$-0.428059\pi$
$521$	5329.77	0.448180	0.224090	+	0.974568i	$0.428059\pi$
$522$	−1866.99	−0.156544
$523$	272.713	0.0228010	0.0114005	−	0.999935i	$-0.496371\pi$
$523$	272.713	0.0228010	0.0114005	+	0.999935i	$0.496371\pi$
$524$	−7299.50	−0.608550
$525$	0	0
$526$	6724.86	0.557448
$527$	−21027.9	−1.73812
$528$	1924.86	0.158653
$529$	−11946.4	−0.981868
$530$	403.476	0.0330677
$531$	−411.781	−0.0336530
$532$	0	0
$533$	1066.70	0.0866865
$534$	4581.24	0.371254
$535$	10161.8	0.821183
$536$	−17570.4	−1.41590
$537$	−5804.51	−0.466449
$538$	14559.8	1.16676
$539$	0	0
$540$	474.626	0.0378234
$541$	−12728.4	−1.01153	−0.505763	−	0.862672i	$-0.668789\pi$
$541$	−12728.4	−1.01153	−0.505763	+	0.862672i	$0.668789\pi$
$542$	−1892.28	−0.149964
$543$	1680.65	0.132825
$544$	−8270.54	−0.651832
$545$	2658.06	0.208916
$546$	0	0
$547$	−6159.68	−0.481479	−0.240739	−	0.970590i	$-0.577390\pi$
$547$	−6159.68	−0.481479	−0.240739	+	0.970590i	$0.577390\pi$
$548$	329.335	0.0256724
$549$	−1203.83	−0.0935851
$550$	3494.42	0.270913
$551$	−2633.53	−0.203616
$552$	−1097.63	−0.0846342
$553$	0	0
$554$	19227.1	1.47452
$555$	7568.86	0.578883
$556$	−3685.98	−0.281152
$557$	8289.50	0.630588	0.315294	−	0.948994i	$-0.397897\pi$
$557$	8289.50	0.630588	0.315294	+	0.948994i	$0.397897\pi$
$558$	6134.97	0.465437
$559$	4136.03	0.312944
$560$	0	0
$561$	−3902.56	−0.293701
$562$	3446.43	0.258681
$563$	2500.87	0.187210	0.0936049	−	0.995609i	$-0.470161\pi$
$563$	2500.87	0.187210	0.0936049	+	0.995609i	$0.470161\pi$
$564$	1015.25	0.0757974
$565$	7776.18	0.579020
$566$	5720.68	0.424838
$567$	0	0
$568$	−2360.01	−0.174337
$569$	1122.48	0.0827013	0.0413506	−	0.999145i	$-0.486834\pi$
$569$	1122.48	0.0827013	0.0413506	+	0.999145i	$0.486834\pi$
$570$	−1308.25	−0.0961344
$571$	−3755.95	−0.275274	−0.137637	−	0.990483i	$-0.543951\pi$
$571$	−3755.95	−0.275274	−0.137637	+	0.990483i	$0.543951\pi$
$572$	−645.368	−0.0471752
$573$	−1690.50	−0.123249
$574$	0	0
$575$	−1230.80	−0.0892660
$576$	4933.04	0.356846
$577$	9761.33	0.704280	0.352140	−	0.935947i	$-0.385454\pi$
$577$	9761.33	0.704280	0.352140	+	0.935947i	$0.385454\pi$
$578$	−282.328	−0.0203171
$579$	−3946.52	−0.283267
$580$	−1585.19	−0.113485
$581$	0	0
$582$	−11783.0	−0.839213
$583$	−495.324	−0.0351874
$584$	−5202.62	−0.368640
$585$	759.461	0.0536750
$586$	19960.6	1.40710
$587$	−15036.2	−1.05726	−0.528629	−	0.848853i	$-0.677294\pi$
$587$	−15036.2	−1.05726	−0.528629	+	0.848853i	$0.677294\pi$
$588$	0	0
$589$	8653.86	0.605392
$590$	683.201	0.0476728
$591$	−500.308	−0.0348222
$592$	13604.2	0.944472
$593$	−12879.5	−0.891901	−0.445950	−	0.895058i	$-0.647134\pi$
$593$	−12879.5	−0.891901	−0.445950	+	0.895058i	$0.647134\pi$
$594$	1138.58	0.0786476
$595$	0	0
$596$	2867.00	0.197042
$597$	2380.47	0.163193
$598$	−444.184	−0.0303747
$599$	−22540.7	−1.53754	−0.768771	−	0.639525i	$-0.779131\pi$
$599$	−22540.7	−1.53754	−0.768771	+	0.639525i	$0.779131\pi$
$600$	6123.68	0.416664
$601$	13338.7	0.905320	0.452660	−	0.891683i	$-0.350475\pi$
$601$	13338.7	0.905320	0.452660	+	0.891683i	$0.350475\pi$
$602$	0	0
$603$	−6419.55	−0.433540
$604$	8535.08	0.574980
$605$	−6458.40	−0.434002
$606$	−4720.51	−0.316431
$607$	−11916.6	−0.796837	−0.398418	−	0.917204i	$-0.630441\pi$
$607$	−11916.6	−0.796837	−0.398418	+	0.917204i	$0.630441\pi$
$608$	3403.67	0.227035
$609$	0	0
$610$	1997.32	0.132572
$611$	1624.53	0.107564
$612$	−1729.58	−0.114239
$613$	−7011.21	−0.461958	−0.230979	−	0.972959i	$-0.574193\pi$
$613$	−7011.21	−0.461958	−0.230979	+	0.972959i	$0.574193\pi$
$614$	−11866.1	−0.779927
$615$	−1597.86	−0.104768
$616$	0	0
$617$	8178.61	0.533644	0.266822	−	0.963746i	$-0.414026\pi$
$617$	8178.61	0.533644	0.266822	+	0.963746i	$0.414026\pi$
$618$	4848.46	0.315588
$619$	−7287.01	−0.473166	−0.236583	−	0.971611i	$-0.576027\pi$
$619$	−7287.01	−0.473166	−0.236583	+	0.971611i	$0.576027\pi$
$620$	5208.98	0.337415
$621$	−401.032	−0.0259144
$622$	−19758.4	−1.27369
$623$	0	0
$624$	1365.04	0.0875729
$625$	1599.83	0.102389
$626$	532.490	0.0339977
$627$	1606.06	0.102297
$628$	878.395	0.0558149
$629$	−27581.7	−1.74841
$630$	0	0
$631$	−6587.01	−0.415570	−0.207785	−	0.978174i	$-0.566625\pi$
$631$	−6587.01	−0.415570	−0.207785	+	0.978174i	$0.566625\pi$
$632$	6469.91	0.407214
$633$	11589.8	0.727733
$634$	−584.369	−0.0366061
$635$	5757.41	0.359804
$636$	−219.524	−0.0136866
$637$	0	0
$638$	−3802.73	−0.235974
$639$	−862.258	−0.0533809
$640$	−2132.41	−0.131704
$641$	−2112.68	−0.130181	−0.0650904	−	0.997879i	$-0.520734\pi$
$641$	−2112.68	−0.130181	−0.0650904	+	0.997879i	$0.520734\pi$
$642$	10803.8	0.664163
$643$	−17906.3	−1.09822	−0.549109	−	0.835751i	$-0.685033\pi$
$643$	−17906.3	−1.09822	−0.549109	+	0.835751i	$0.685033\pi$
$644$	0	0
$645$	−6195.57	−0.378218
$646$	4767.39	0.290357
$647$	25737.0	1.56387	0.781936	−	0.623358i	$-0.214232\pi$
$647$	25737.0	1.56387	0.781936	+	0.623358i	$0.214232\pi$
$648$	1995.28	0.120960
$649$	−838.726	−0.0507286
$650$	2478.11	0.149538
$651$	0	0
$652$	8547.23	0.513398
$653$	30506.6	1.82820	0.914100	−	0.405488i	$-0.132899\pi$
$653$	30506.6	1.82820	0.914100	+	0.405488i	$0.132899\pi$
$654$	2826.00	0.168968
$655$	17496.3	1.04372
$656$	−2871.98	−0.170933
$657$	−1900.84	−0.112875
$658$	0	0
$659$	1244.16	0.0735440	0.0367720	−	0.999324i	$-0.488292\pi$
$659$	1244.16	0.0735440	0.0367720	+	0.999324i	$0.488292\pi$
$660$	966.730	0.0570150
$661$	6621.65	0.389641	0.194820	−	0.980839i	$-0.437588\pi$
$661$	6621.65	0.389641	0.194820	+	0.980839i	$0.437588\pi$
$662$	5600.55	0.328809
$663$	−2767.55	−0.162116
$664$	−11463.4	−0.669979
$665$	0	0
$666$	8047.05	0.468194
$667$	1339.40	0.0777536
$668$	−4242.65	−0.245738
$669$	−4577.36	−0.264531
$670$	10650.9	0.614151
$671$	−2451.99	−0.141070
$672$	0	0
$673$	−185.685	−0.0106354	−0.00531770	−	0.999986i	$-0.501693\pi$
$673$	−185.685	−0.0106354	−0.00531770	+	0.999986i	$0.501693\pi$
$674$	12495.2	0.714093
$675$	2237.36	0.127580
$676$	−457.672	−0.0260396
$677$	7200.43	0.408767	0.204383	−	0.978891i	$-0.434481\pi$
$677$	7200.43	0.408767	0.204383	+	0.978891i	$0.434481\pi$
$678$	8267.47	0.468304
$679$	0	0
$680$	11346.7	0.639891
$681$	13196.1	0.742548
$682$	12495.9	0.701601
$683$	16462.5	0.922283	0.461141	−	0.887327i	$-0.347440\pi$
$683$	16462.5	0.922283	0.461141	+	0.887327i	$0.347440\pi$
$684$	711.795	0.0397897
$685$	−789.386	−0.0440305
$686$	0	0
$687$	−14255.5	−0.791675
$688$	−11135.8	−0.617078
$689$	−351.266	−0.0194226
$690$	665.367	0.0367102
$691$	578.872	0.0318688	0.0159344	−	0.999873i	$-0.494928\pi$
$691$	578.872	0.0318688	0.0159344	+	0.999873i	$0.494928\pi$
$692$	−3602.16	−0.197881
$693$	0	0
$694$	8920.14	0.487902
$695$	8834.96	0.482200
$696$	−6663.98	−0.362927
$697$	5822.78	0.316432
$698$	−23523.1	−1.27559
$699$	13040.4	0.705627
$700$	0	0
$701$	9222.05	0.496879	0.248439	−	0.968647i	$-0.420082\pi$
$701$	9222.05	0.496879	0.248439	+	0.968647i	$0.420082\pi$
$702$	807.444	0.0434117
$703$	11351.0	0.608977
$704$	10047.8	0.537911
$705$	−2433.46	−0.129999
$706$	8581.81	0.457480
$707$	0	0
$708$	−371.717	−0.0197316
$709$	−1204.23	−0.0637882	−0.0318941	−	0.999491i	$-0.510154\pi$
$709$	−1204.23	−0.0637882	−0.0318941	+	0.999491i	$0.510154\pi$
$710$	1430.60	0.0756192
$711$	2363.86	0.124686
$712$	16352.2	0.860706
$713$	−4401.29	−0.231178
$714$	0	0
$715$	1546.89	0.0809097
$716$	−5239.77	−0.273491
$717$	12910.6	0.672461
$718$	17052.1	0.886320
$719$	−11557.0	−0.599446	−0.299723	−	0.954026i	$-0.596894\pi$
$719$	−11557.0	−0.599446	−0.299723	+	0.954026i	$0.596894\pi$
$720$	−2044.77	−0.105839
$721$	0	0
$722$	13816.5	0.712185
$723$	−8085.19	−0.415894
$724$	1517.14	0.0778783
$725$	−7472.52	−0.382789
$726$	−6866.43	−0.351015
$727$	15374.8	0.784348	0.392174	−	0.919891i	$-0.371723\pi$
$727$	15374.8	0.784348	0.392174	+	0.919891i	$0.371723\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	3153.76	0.159898
$731$	22577.3	1.14234
$732$	−1086.71	−0.0548713
$733$	−21765.1	−1.09674	−0.548371	−	0.836235i	$-0.684752\pi$
$733$	−21765.1	−1.09674	−0.548371	+	0.836235i	$0.684752\pi$
$734$	−27093.7	−1.36246
$735$	0	0
$736$	−1731.08	−0.0866964
$737$	−13075.5	−0.653518
$738$	−1698.82	−0.0847348
$739$	−35547.1	−1.76945	−0.884723	−	0.466116i	$-0.845653\pi$
$739$	−35547.1	−1.76945	−0.884723	+	0.466116i	$0.845653\pi$
$740$	6832.45	0.339414
$741$	1138.96	0.0564654
$742$	0	0
$743$	27378.3	1.35183	0.675917	−	0.736978i	$-0.263748\pi$
$743$	27378.3	1.35183	0.675917	+	0.736978i	$0.263748\pi$
$744$	21898.0	1.07906
$745$	−6871.94	−0.337944
$746$	24946.7	1.22435
$747$	−4188.30	−0.205143
$748$	−3522.86	−0.172204
$749$	0	0
$750$	−9311.68	−0.453353
$751$	22165.5	1.07700	0.538501	−	0.842625i	$-0.318991\pi$
$751$	22165.5	1.07700	0.538501	+	0.842625i	$0.318991\pi$
$752$	−4373.87	−0.212099
$753$	−14489.5	−0.701232
$754$	−2696.76	−0.130252
$755$	−20457.8	−0.986142
$756$	0	0
$757$	11078.4	0.531906	0.265953	−	0.963986i	$-0.414313\pi$
$757$	11078.4	0.531906	0.265953	+	0.963986i	$0.414313\pi$
$758$	−3499.53	−0.167690
$759$	−816.832	−0.0390634
$760$	−4669.63	−0.222876
$761$	−36505.5	−1.73893	−0.869464	−	0.493996i	$-0.835536\pi$
$761$	−36505.5	−1.73893	−0.869464	+	0.493996i	$0.835536\pi$
$762$	6121.15	0.291005
$763$	0	0
$764$	−1526.02	−0.0722639
$765$	4145.66	0.195930
$766$	30765.5	1.45118
$767$	−594.795	−0.0280010
$768$	10887.7	0.511555
$769$	−25846.6	−1.21203	−0.606015	−	0.795453i	$-0.707233\pi$
$769$	−25846.6	−1.21203	−0.606015	+	0.795453i	$0.707233\pi$
$770$	0	0
$771$	12837.8	0.599666
$772$	−3562.55	−0.166087
$773$	15995.0	0.744245	0.372123	−	0.928184i	$-0.378630\pi$
$773$	15995.0	0.744245	0.372123	+	0.928184i	$0.378630\pi$
$774$	−6587.01	−0.305898
$775$	24554.9	1.13811
$776$	−42058.0	−1.94561
$777$	0	0
$778$	2849.25	0.131299
$779$	−2396.31	−0.110214
$780$	685.570	0.0314710
$781$	−1756.27	−0.0804664
$782$	−2424.66	−0.110877
$783$	−2434.77	−0.111126
$784$	0	0
$785$	−2105.44	−0.0957277
$786$	18601.7	0.844147
$787$	16809.8	0.761380	0.380690	−	0.924703i	$-0.375687\pi$
$787$	16809.8	0.761380	0.380690	+	0.924703i	$0.375687\pi$
$788$	−451.631	−0.0204171
$789$	8770.00	0.395716
$790$	−3921.98	−0.176630
$791$	0	0
$792$	4064.03	0.182335
$793$	−1738.87	−0.0778675
$794$	−8873.16	−0.396595
$795$	526.180	0.0234738
$796$	2148.86	0.0956839
$797$	−21593.1	−0.959681	−0.479841	−	0.877356i	$-0.659306\pi$
$797$	−21593.1	−0.959681	−0.479841	+	0.877356i	$0.659306\pi$
$798$	0	0
$799$	8867.79	0.392640
$800$	9657.74	0.426816
$801$	5974.47	0.263542
$802$	23334.4	1.02739
$803$	−3871.69	−0.170148
$804$	−5794.97	−0.254195
$805$	0	0
$806$	8861.63	0.387267
$807$	18987.7	0.828251
$808$	−16849.2	−0.733607
$809$	25072.5	1.08962	0.544810	−	0.838559i	$-0.316602\pi$
$809$	25072.5	1.08962	0.544810	+	0.838559i	$0.316602\pi$
$810$	−1209.51	−0.0524665
$811$	−608.030	−0.0263265	−0.0131633	−	0.999913i	$-0.504190\pi$
$811$	−608.030	−0.0263265	−0.0131633	+	0.999913i	$0.504190\pi$
$812$	0	0
$813$	−2467.75	−0.106455
$814$	16390.5	0.705756
$815$	−20486.9	−0.880523
$816$	7451.35	0.319668
$817$	−9291.49	−0.397880
$818$	13209.8	0.564634
$819$	0	0
$820$	−1442.40	−0.0614279
$821$	−24468.5	−1.04014	−0.520072	−	0.854123i	$-0.674095\pi$
$821$	−24468.5	−1.04014	−0.520072	+	0.854123i	$0.674095\pi$
$822$	−839.259	−0.0356113
$823$	15609.0	0.661113	0.330556	−	0.943786i	$-0.392764\pi$
$823$	15609.0	0.661113	0.330556	+	0.943786i	$0.392764\pi$
$824$	17306.0	0.731652
$825$	4557.12	0.192314
$826$	0	0
$827$	652.098	0.0274192	0.0137096	−	0.999906i	$-0.495636\pi$
$827$	652.098	0.0274192	0.0137096	+	0.999906i	$0.495636\pi$
$828$	−362.014	−0.0151943
$829$	2364.96	0.0990812	0.0495406	−	0.998772i	$-0.484224\pi$
$829$	2364.96	0.0990812	0.0495406	+	0.998772i	$0.484224\pi$
$830$	6948.96	0.290605
$831$	25074.4	1.04672
$832$	7125.51	0.296914
$833$	0	0
$834$	9393.15	0.389998
$835$	10169.3	0.421464
$836$	1449.80	0.0599791
$837$	8000.71	0.330400
$838$	11578.4	0.477289
$839$	16664.6	0.685730	0.342865	−	0.939385i	$-0.388603\pi$
$839$	16664.6	0.685730	0.342865	+	0.939385i	$0.388603\pi$
$840$	0	0
$841$	−16257.2	−0.666578
$842$	−10921.9	−0.447022
$843$	4494.54	0.183630
$844$	10462.2	0.426688
$845$	1097.00	0.0446603
$846$	−2587.21	−0.105142
$847$	0	0
$848$	945.748	0.0382985
$849$	7460.44	0.301580
$850$	13527.2	0.545859
$851$	−5773.03	−0.232547
$852$	−778.366	−0.0312986
$853$	−8267.76	−0.331867	−0.165934	−	0.986137i	$-0.553064\pi$
$853$	−8267.76	−0.331867	−0.165934	+	0.986137i	$0.553064\pi$
$854$	0	0
$855$	−1706.11	−0.0682430
$856$	38562.8	1.53978
$857$	−10868.8	−0.433223	−0.216612	−	0.976258i	$-0.569501\pi$
$857$	−10868.8	−0.433223	−0.216612	+	0.976258i	$0.569501\pi$
$858$	1644.62	0.0654388
$859$	−1400.80	−0.0556398	−0.0278199	−	0.999613i	$-0.508856\pi$
$859$	−1400.80	−0.0556398	−0.0278199	+	0.999613i	$0.508856\pi$
$860$	−5592.78	−0.221759
$861$	0	0
$862$	−33062.2	−1.30638
$863$	−38078.9	−1.50199	−0.750997	−	0.660305i	$-0.770427\pi$
$863$	−38078.9	−1.50199	−0.750997	+	0.660305i	$0.770427\pi$
$864$	3146.78	0.123907
$865$	8634.06	0.339384
$866$	−15635.0	−0.613511
$867$	−368.188	−0.0144225
$868$	0	0
$869$	4814.78	0.187952
$870$	4039.62	0.157421
$871$	−9272.69	−0.360727
$872$	10087.0	0.391732
$873$	−15366.4	−0.595733
$874$	997.849	0.0386187
$875$	0	0
$876$	−1715.90	−0.0661815
$877$	25837.4	0.994832	0.497416	−	0.867512i	$-0.334282\pi$
$877$	25837.4	0.994832	0.497416	+	0.867512i	$0.334282\pi$
$878$	193.249	0.00742807
$879$	26030.9	0.998862
$880$	−4164.84	−0.159542
$881$	−16123.6	−0.616591	−0.308296	−	0.951291i	$-0.599759\pi$
$881$	−16123.6	−0.616591	−0.308296	+	0.951291i	$0.599759\pi$
$882$	0	0
$883$	−45405.2	−1.73047	−0.865236	−	0.501365i	$-0.832831\pi$
$883$	−45405.2	−1.73047	−0.865236	+	0.501365i	$0.832831\pi$
$884$	−2498.29	−0.0950526
$885$	890.973	0.0338415
$886$	14708.9	0.557738
$887$	−20269.7	−0.767293	−0.383647	−	0.923480i	$-0.625332\pi$
$887$	−20269.7	−0.767293	−0.383647	+	0.923480i	$0.625332\pi$
$888$	28722.9	1.08545
$889$	0	0
$890$	−9912.46	−0.373333
$891$	1484.85	0.0558297
$892$	−4132.02	−0.155101
$893$	−3649.46	−0.136758
$894$	−7306.10	−0.273325
$895$	12559.3	0.469061
$896$	0	0
$897$	−579.268	−0.0215621
$898$	19978.3	0.742411
$899$	−26721.4	−0.991333
$900$	2019.68	0.0748030
$901$	−1917.45	−0.0708985
$902$	−3460.20	−0.127729
$903$	0	0
$904$	29509.7	1.08571
$905$	−3636.44	−0.133568
$906$	−21750.4	−0.797580
$907$	−12300.2	−0.450301	−0.225150	−	0.974324i	$-0.572287\pi$
$907$	−12300.2	−0.450301	−0.225150	+	0.974324i	$0.572287\pi$
$908$	11912.2	0.435374
$909$	−6156.09	−0.224625
$910$	0	0
$911$	32418.4	1.17900	0.589500	−	0.807769i	$-0.299325\pi$
$911$	32418.4	1.17900	0.589500	+	0.807769i	$0.299325\pi$
$912$	−3066.54	−0.111341
$913$	−8530.83	−0.309233
$914$	−8523.17	−0.308448
$915$	2604.74	0.0941093
$916$	−12868.5	−0.464179
$917$	0	0
$918$	4407.58	0.158466
$919$	−2034.41	−0.0730239	−0.0365119	−	0.999333i	$-0.511625\pi$
$919$	−2034.41	−0.0730239	−0.0365119	+	0.999333i	$0.511625\pi$
$920$	2374.94	0.0851081
$921$	−15474.7	−0.553648
$922$	2551.40	0.0911345
$923$	−1245.48	−0.0444156
$924$	0	0
$925$	32207.9	1.14485
$926$	−37539.8	−1.33222
$927$	6322.95	0.224027
$928$	−10509.9	−0.371771
$929$	44854.2	1.58409	0.792045	−	0.610462i	$-0.209016\pi$
$929$	44854.2	1.58409	0.792045	+	0.610462i	$0.209016\pi$
$930$	−13274.3	−0.468044
$931$	0	0
$932$	11771.6	0.413726
$933$	−25767.2	−0.904159
$934$	−23797.7	−0.833710
$935$	8443.99	0.295345
$936$	2882.07	0.100645
$937$	7037.56	0.245365	0.122683	−	0.992446i	$-0.460850\pi$
$937$	7037.56	0.245365	0.122683	+	0.992446i	$0.460850\pi$
$938$	0	0
$939$	694.428	0.0241340
$940$	−2196.70	−0.0762219
$941$	31954.6	1.10700	0.553502	−	0.832848i	$-0.313291\pi$
$941$	31954.6	1.10700	0.553502	+	0.832848i	$0.313291\pi$
$942$	−2238.46	−0.0774234
$943$	1218.75	0.0420868
$944$	1601.42	0.0552138
$945$	0	0
$946$	−13416.6	−0.461111
$947$	32079.0	1.10077	0.550385	−	0.834911i	$-0.314481\pi$
$947$	32079.0	1.10077	0.550385	+	0.834911i	$0.314481\pi$
$948$	2133.87	0.0731066
$949$	−2745.66	−0.0939177
$950$	−5567.02	−0.190124
$951$	−762.085	−0.0259856
$952$	0	0
$953$	−33462.1	−1.13740	−0.568700	−	0.822545i	$-0.692554\pi$
$953$	−33462.1	−1.13740	−0.568700	+	0.822545i	$0.692554\pi$
$954$	559.423	0.0189853
$955$	3657.75	0.123939
$956$	11654.5	0.394281
$957$	−4959.20	−0.167511
$958$	6062.08	0.204443
$959$	0	0
$960$	−10673.7	−0.358845
$961$	58016.2	1.94744
$962$	11623.5	0.389561
$963$	14089.4	0.471470
$964$	−7298.55	−0.243849
$965$	8539.12	0.284854
$966$	0	0
$967$	−18051.2	−0.600298	−0.300149	−	0.953892i	$-0.597036\pi$
$967$	−18051.2	−0.600298	−0.300149	+	0.953892i	$0.597036\pi$
$968$	−24508.9	−0.813786
$969$	6217.24	0.206116
$970$	25495.0	0.843913
$971$	−16678.4	−0.551221	−0.275611	−	0.961269i	$-0.588880\pi$
$971$	−16678.4	−0.551221	−0.275611	+	0.961269i	$0.588880\pi$
$972$	658.073	0.0217157
$973$	0	0
$974$	35763.0	1.17651
$975$	3231.75	0.106153
$976$	4681.72	0.153543
$977$	35548.7	1.16408	0.582038	−	0.813161i	$-0.302255\pi$
$977$	35548.7	1.16408	0.582038	+	0.813161i	$0.302255\pi$
$978$	−21781.3	−0.712157
$979$	12169.0	0.397264
$980$	0	0
$981$	3685.43	0.119946
$982$	35305.7	1.14730
$983$	−11404.2	−0.370029	−0.185015	−	0.982736i	$-0.559233\pi$
$983$	−11404.2	−0.370029	−0.185015	+	0.982736i	$0.559233\pi$
$984$	−6063.71	−0.196447
$985$	1082.52	0.0350172
$986$	−14720.8	−0.475461
$987$	0	0
$988$	1028.15	0.0331070
$989$	4725.58	0.151936
$990$	−2463.56	−0.0790881
$991$	−12945.2	−0.414953	−0.207477	−	0.978240i	$-0.566525\pi$
$991$	−12945.2	−0.414953	−0.207477	+	0.978240i	$0.566525\pi$
$992$	34535.7	1.10535
$993$	7303.76	0.233412
$994$	0	0
$995$	−5150.63	−0.164107
$996$	−3780.80	−0.120280
$997$	8623.30	0.273924	0.136962	−	0.990576i	$-0.456266\pi$
$997$	8623.30	0.273924	0.136962	+	0.990576i	$0.456266\pi$
$998$	6120.16	0.194119
$999$	10494.3	0.332357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.4.a.bd.1.5	✓	14
7.6	odd	2		1911.4.a.be.1.5	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1911.4.a.bd.1.5	✓	14	1.1	even	1	trivial
1911.4.a.be.1.5	yes	14	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1911.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30041 q^{2} -3.00000 q^{3} -2.70812 q^{4} +6.49112 q^{5} +6.90123 q^{6} +24.6331 q^{8} +9.00000 q^{9} -14.9322 q^{10} +18.3314 q^{11} +8.12436 q^{12} +13.0000 q^{13} -19.4734 q^{15} -35.0011 q^{16} +70.9629 q^{17} -20.7037 q^{18} -29.2042 q^{19} -17.5787 q^{20} -42.1698 q^{22} +14.8530 q^{23} -73.8992 q^{24} -82.8653 q^{25} -29.9053 q^{26} -27.0000 q^{27} +90.1767 q^{29} +44.7967 q^{30} -296.323 q^{31} -116.547 q^{32} -54.9943 q^{33} -163.244 q^{34} -24.3731 q^{36} -388.677 q^{37} +67.1815 q^{38} -39.0000 q^{39} +159.896 q^{40} +82.0539 q^{41} +318.156 q^{43} -49.6437 q^{44} +58.4201 q^{45} -34.1680 q^{46} +124.964 q^{47} +105.003 q^{48} +190.624 q^{50} -212.889 q^{51} -35.2055 q^{52} -27.0205 q^{53} +62.1110 q^{54} +118.992 q^{55} +87.6125 q^{57} -207.443 q^{58} -45.7534 q^{59} +52.7362 q^{60} -133.759 q^{61} +681.664 q^{62} +548.116 q^{64} +84.3846 q^{65} +126.509 q^{66} -713.284 q^{67} -192.176 q^{68} -44.5591 q^{69} -95.8064 q^{71} +221.697 q^{72} -211.205 q^{73} +894.117 q^{74} +248.596 q^{75} +79.0884 q^{76} +89.7159 q^{78} +262.652 q^{79} -227.197 q^{80} +81.0000 q^{81} -188.757 q^{82} -465.366 q^{83} +460.629 q^{85} -731.890 q^{86} -270.530 q^{87} +451.559 q^{88} +663.830 q^{89} -134.390 q^{90} -40.2238 q^{92} +888.968 q^{93} -287.468 q^{94} -189.568 q^{95} +349.642 q^{96} -1707.38 q^{97} +164.983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 6 q^{2} - 42 q^{3} + 50 q^{4} - 4 q^{5} + 18 q^{6} - 30 q^{8} + 126 q^{9} + 32 q^{10} - 68 q^{11} - 150 q^{12} + 182 q^{13} + 12 q^{15} - 50 q^{16} - 54 q^{18} + 24 q^{19} + 96 q^{20} - 300 q^{22}+ \cdots - 612 q^{99}+O(q^{100}) \) 14 * q - 6 * q^2 - 42 * q^3 + 50 * q^4 - 4 * q^5 + 18 * q^6 - 30 * q^8 + 126 * q^9 + 32 * q^10 - 68 * q^11 - 150 * q^12 + 182 * q^13 + 12 * q^15 - 50 * q^16 - 54 * q^18 + 24 * q^19 + 96 * q^20 - 300 * q^22 - 64 * q^23 + 90 * q^24 - 118 * q^25 - 78 * q^26 - 378 * q^27 - 792 * q^29 - 96 * q^30 + 524 * q^31 - 126 * q^32 + 204 * q^33 - 88 * q^34 + 450 * q^36 - 344 * q^37 + 436 * q^38 - 546 * q^39 + 704 * q^40 + 44 * q^41 + 144 * q^43 - 1248 * q^44 - 36 * q^45 - 972 * q^46 - 236 * q^47 + 150 * q^48 - 714 * q^50 + 650 * q^52 - 1556 * q^53 + 162 * q^54 + 396 * q^55 - 72 * q^57 + 92 * q^58 + 1244 * q^59 - 288 * q^60 + 984 * q^61 + 236 * q^62 - 714 * q^64 - 52 * q^65 + 900 * q^66 - 1396 * q^67 + 960 * q^68 + 192 * q^69 - 1216 * q^71 - 270 * q^72 + 1768 * q^73 - 1212 * q^74 + 354 * q^75 + 496 * q^76 + 234 * q^78 - 1888 * q^79 + 1300 * q^80 + 1134 * q^81 + 1920 * q^82 - 1008 * q^83 + 56 * q^85 - 2128 * q^86 + 2376 * q^87 - 3148 * q^88 - 864 * q^89 + 288 * q^90 + 2856 * q^92 - 1572 * q^93 + 4860 * q^94 - 4984 * q^95 + 378 * q^96 + 1368 * q^97 - 612 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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