LMFDB - Embedded newform 7865.2.a.w.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7865,2,Mod(1,7865)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7865 = 5 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7865.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,-1,2,15,9,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.8023411897$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 16x^{7} + 14x^{6} + 86x^{5} - 57x^{4} - 179x^{3} + 64x^{2} + 118x - 14 $ x^9 - x^8 - 16x^7 + 14x^6 + 86x^5 - 57x^4 - 179x^3 + 64x^2 + 118*x - 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 715)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.13118$ of defining polynomial
Character	$\chi$	$=$	7865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.13118 q^{2} -3.33877 q^{3} -0.720430 q^{4} +1.00000 q^{5} +3.77675 q^{6} +4.42443 q^{7} +3.07730 q^{8} +8.14739 q^{9} -1.13118 q^{10} +2.40535 q^{12} +1.00000 q^{13} -5.00483 q^{14} -3.33877 q^{15} -2.04012 q^{16} -2.59407 q^{17} -9.21618 q^{18} -5.87796 q^{19} -0.720430 q^{20} -14.7722 q^{21} +2.18940 q^{23} -10.2744 q^{24} +1.00000 q^{25} -1.13118 q^{26} -17.1860 q^{27} -3.18749 q^{28} -3.69729 q^{29} +3.77675 q^{30} +0.431901 q^{31} -3.84685 q^{32} +2.93436 q^{34} +4.42443 q^{35} -5.86962 q^{36} -6.49884 q^{37} +6.64904 q^{38} -3.33877 q^{39} +3.07730 q^{40} +6.74312 q^{41} +16.7100 q^{42} -7.75955 q^{43} +8.14739 q^{45} -2.47661 q^{46} +8.34232 q^{47} +6.81150 q^{48} +12.5756 q^{49} -1.13118 q^{50} +8.66101 q^{51} -0.720430 q^{52} +3.37912 q^{53} +19.4404 q^{54} +13.6153 q^{56} +19.6252 q^{57} +4.18230 q^{58} +6.43744 q^{59} +2.40535 q^{60} -14.9657 q^{61} -0.488559 q^{62} +36.0476 q^{63} +8.43173 q^{64} +1.00000 q^{65} -3.21119 q^{67} +1.86884 q^{68} -7.30990 q^{69} -5.00483 q^{70} -15.1767 q^{71} +25.0720 q^{72} -9.79960 q^{73} +7.35136 q^{74} -3.33877 q^{75} +4.23466 q^{76} +3.77675 q^{78} +12.1235 q^{79} -2.04012 q^{80} +32.9378 q^{81} -7.62768 q^{82} +10.2593 q^{83} +10.6423 q^{84} -2.59407 q^{85} +8.77745 q^{86} +12.3444 q^{87} -3.80313 q^{89} -9.21618 q^{90} +4.42443 q^{91} -1.57731 q^{92} -1.44202 q^{93} -9.43667 q^{94} -5.87796 q^{95} +12.8438 q^{96} +0.336593 q^{97} -14.2253 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} + 2 q^{3} + 15 q^{4} + 9 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} - q^{10} + 6 q^{12} + 9 q^{13} + 16 q^{14} + 2 q^{15} + 15 q^{16} - 13 q^{17} - 3 q^{18} + 3 q^{19} + 15 q^{20} - 14 q^{21}+ \cdots - 24 q^{98}+O(q^{100}) $ 9 * q - q^2 + 2 * q^3 + 15 * q^4 + 9 * q^5 + q^6 + 4 * q^7 - 3 * q^8 + 23 * q^9 - q^10 + 6 * q^12 + 9 * q^13 + 16 * q^14 + 2 * q^15 + 15 * q^16 - 13 * q^17 - 3 * q^18 + 3 * q^19 + 15 * q^20 - 14 * q^21 - 2 * q^24 + 9 * q^25 - q^26 + 8 * q^27 + 27 * q^28 - 16 * q^29 + q^30 + 12 * q^31 + 18 * q^32 + 14 * q^34 + 4 * q^35 - 4 * q^36 + q^37 - 9 * q^38 + 2 * q^39 - 3 * q^40 - 5 * q^41 - 11 * q^42 + 3 * q^43 + 23 * q^45 - 4 * q^46 + 27 * q^47 + q^48 + 31 * q^49 - q^50 + 15 * q^52 + 34 * q^53 + 49 * q^54 + 73 * q^56 + 16 * q^57 - 38 * q^58 - 4 * q^59 + 6 * q^60 - 30 * q^61 + 8 * q^62 + 30 * q^63 + 3 * q^64 + 9 * q^65 - 11 * q^67 - 52 * q^68 + 52 * q^69 + 16 * q^70 - 4 * q^71 + 8 * q^72 + 12 * q^73 + 16 * q^74 + 2 * q^75 + 16 * q^76 + q^78 + 16 * q^79 + 15 * q^80 + 61 * q^81 - 42 * q^82 - 34 * q^83 + 79 * q^84 - 13 * q^85 + 52 * q^86 + 64 * q^87 - 10 * q^89 - 3 * q^90 + 4 * q^91 + 7 * q^92 - 22 * q^93 + 18 * q^94 + 3 * q^95 - 15 * q^97 - 24 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.13118	−0.799866	−0.399933	−	0.916544i	$-0.630967\pi$
$2$	−1.13118	−0.799866	−0.399933	+	0.916544i	$0.630967\pi$
$3$	−3.33877	−1.92764	−0.963820	−	0.266553i	$-0.914115\pi$
$3$	−3.33877	−1.92764	−0.963820	+	0.266553i	$0.914115\pi$
$4$	−0.720430	−0.360215
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	3.77675	1.54185
$7$	4.42443	1.67228	0.836139	−	0.548518i	$-0.184808\pi$
$7$	4.42443	1.67228	0.836139	+	0.548518i	$0.184808\pi$
$8$	3.07730	1.08799
$9$	8.14739	2.71580
$10$	−1.13118	−0.357711
$11$	0	0
$11$	0	0
$12$	2.40535	0.694365
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−5.00483	−1.33760
$15$	−3.33877	−0.862067
$16$	−2.04012	−0.510030
$17$	−2.59407	−0.629154	−0.314577	−	0.949232i	$-0.601863\pi$
$17$	−2.59407	−0.629154	−0.314577	+	0.949232i	$0.601863\pi$
$18$	−9.21618	−2.17227
$19$	−5.87796	−1.34850	−0.674248	−	0.738505i	$-0.735532\pi$
$19$	−5.87796	−1.34850	−0.674248	+	0.738505i	$0.735532\pi$
$20$	−0.720430	−0.161093
$21$	−14.7722	−3.22355
$22$	0	0
$23$	2.18940	0.456521	0.228261	−	0.973600i	$-0.426696\pi$
$23$	2.18940	0.456521	0.228261	+	0.973600i	$0.426696\pi$
$24$	−10.2744	−2.09725
$25$	1.00000	0.200000
$26$	−1.13118	−0.221843
$27$	−17.1860	−3.30744
$28$	−3.18749	−0.602379
$29$	−3.69729	−0.686569	−0.343284	−	0.939232i	$-0.611539\pi$
$29$	−3.69729	−0.686569	−0.343284	+	0.939232i	$0.611539\pi$
$30$	3.77675	0.689538
$31$	0.431901	0.0775718	0.0387859	−	0.999248i	$-0.487651\pi$
$31$	0.431901	0.0775718	0.0387859	+	0.999248i	$0.487651\pi$
$32$	−3.84685	−0.680033
$33$	0	0
$34$	2.93436	0.503239
$35$	4.42443	0.747865
$36$	−5.86962	−0.978271
$37$	−6.49884	−1.06840	−0.534201	−	0.845357i	$-0.679387\pi$
$37$	−6.49884	−1.06840	−0.534201	+	0.845357i	$0.679387\pi$
$38$	6.64904	1.07862
$39$	−3.33877	−0.534631
$40$	3.07730	0.486564
$41$	6.74312	1.05310	0.526549	−	0.850145i	$-0.323486\pi$
$41$	6.74312	1.05310	0.526549	+	0.850145i	$0.323486\pi$
$42$	16.7100	2.57841
$43$	−7.75955	−1.18332	−0.591660	−	0.806188i	$-0.701527\pi$
$43$	−7.75955	−1.18332	−0.591660	+	0.806188i	$0.701527\pi$
$44$	0	0
$45$	8.14739	1.21454
$46$	−2.47661	−0.365156
$47$	8.34232	1.21685	0.608426	−	0.793611i	$-0.291801\pi$
$47$	8.34232	1.21685	0.608426	+	0.793611i	$0.291801\pi$
$48$	6.81150	0.983155
$49$	12.5756	1.79651
$50$	−1.13118	−0.159973
$51$	8.66101	1.21278
$52$	−0.720430	−0.0999056
$53$	3.37912	0.464158	0.232079	−	0.972697i	$-0.425447\pi$
$53$	3.37912	0.464158	0.232079	+	0.972697i	$0.425447\pi$
$54$	19.4404	2.64551
$55$	0	0
$56$	13.6153	1.81942
$57$	19.6252	2.59942
$58$	4.18230	0.549163
$59$	6.43744	0.838084	0.419042	−	0.907967i	$-0.362366\pi$
$59$	6.43744	0.838084	0.419042	+	0.907967i	$0.362366\pi$
$60$	2.40535	0.310529
$61$	−14.9657	−1.91616	−0.958079	−	0.286505i	$-0.907506\pi$
$61$	−14.9657	−1.91616	−0.958079	+	0.286505i	$0.907506\pi$
$62$	−0.488559	−0.0620470
$63$	36.0476	4.54157
$64$	8.43173	1.05397
$65$	1.00000	0.124035
$66$	0	0
$67$	−3.21119	−0.392309	−0.196155	−	0.980573i	$-0.562845\pi$
$67$	−3.21119	−0.392309	−0.196155	+	0.980573i	$0.562845\pi$
$68$	1.86884	0.226631
$69$	−7.30990	−0.880009
$70$	−5.00483	−0.598192
$71$	−15.1767	−1.80114	−0.900572	−	0.434707i	$-0.856852\pi$
$71$	−15.1767	−1.80114	−0.900572	+	0.434707i	$0.856852\pi$
$72$	25.0720	2.95476
$73$	−9.79960	−1.14696	−0.573478	−	0.819221i	$-0.694406\pi$
$73$	−9.79960	−1.14696	−0.573478	+	0.819221i	$0.694406\pi$
$74$	7.35136	0.854578
$75$	−3.33877	−0.385528
$76$	4.23466	0.485749
$77$	0	0
$78$	3.77675	0.427633
$79$	12.1235	1.36400	0.681999	−	0.731354i	$-0.261111\pi$
$79$	12.1235	1.36400	0.681999	+	0.731354i	$0.261111\pi$
$80$	−2.04012	−0.228093
$81$	32.9378	3.65976
$82$	−7.62768	−0.842337
$83$	10.2593	1.12610	0.563050	−	0.826423i	$-0.309628\pi$
$83$	10.2593	1.12610	0.563050	+	0.826423i	$0.309628\pi$
$84$	10.6423	1.16117
$85$	−2.59407	−0.281366
$86$	8.77745	0.946497
$87$	12.3444	1.32346
$88$	0	0
$89$	−3.80313	−0.403131	−0.201565	−	0.979475i	$-0.564603\pi$
$89$	−3.80313	−0.403131	−0.201565	+	0.979475i	$0.564603\pi$
$90$	−9.21618	−0.971470
$91$	4.42443	0.463806
$92$	−1.57731	−0.164446
$93$	−1.44202	−0.149531
$94$	−9.43667	−0.973318
$95$	−5.87796	−0.603066
$96$	12.8438	1.31086
$97$	0.336593	0.0341759	0.0170879	−	0.999854i	$-0.494560\pi$
$97$	0.336593	0.0341759	0.0170879	+	0.999854i	$0.494560\pi$
$98$	−14.2253	−1.43697
$99$	0	0
$100$	−0.720430	−0.0720430
$101$	5.29201	0.526575	0.263288	−	0.964717i	$-0.415193\pi$
$101$	5.29201	0.526575	0.263288	+	0.964717i	$0.415193\pi$
$102$	−9.79716	−0.970064
$103$	12.3861	1.22044	0.610218	−	0.792234i	$-0.291082\pi$
$103$	12.3861	1.22044	0.610218	+	0.792234i	$0.291082\pi$
$104$	3.07730	0.301754
$105$	−14.7722	−1.44162
$106$	−3.82240	−0.371264
$107$	3.29272	0.318319	0.159159	−	0.987253i	$-0.449122\pi$
$107$	3.29272	0.318319	0.159159	+	0.987253i	$0.449122\pi$
$108$	12.3813	1.19139
$109$	2.22914	0.213513	0.106757	−	0.994285i	$-0.465953\pi$
$109$	2.22914	0.213513	0.106757	+	0.994285i	$0.465953\pi$
$110$	0	0
$111$	21.6981	2.05950
$112$	−9.02637	−0.852912
$113$	8.35938	0.786384	0.393192	−	0.919456i	$-0.371371\pi$
$113$	8.35938	0.786384	0.393192	+	0.919456i	$0.371371\pi$
$114$	−22.1996	−2.07918
$115$	2.18940	0.204163
$116$	2.66363	0.247312
$117$	8.14739	0.753227
$118$	−7.28191	−0.670354
$119$	−11.4773	−1.05212
$120$	−10.2744	−0.937920
$121$	0	0
$122$	16.9289	1.53267
$123$	−22.5137	−2.02999
$124$	−0.311155	−0.0279425
$125$	1.00000	0.0894427
$126$	−40.7763	−3.63264
$127$	−6.08018	−0.539529	−0.269764	−	0.962926i	$-0.586946\pi$
$127$	−6.08018	−0.539529	−0.269764	+	0.962926i	$0.586946\pi$
$128$	−1.84411	−0.162998
$129$	25.9073	2.28101
$130$	−1.13118	−0.0992111
$131$	6.07608	0.530870	0.265435	−	0.964129i	$-0.414484\pi$
$131$	6.07608	0.530870	0.265435	+	0.964129i	$0.414484\pi$
$132$	0	0
$133$	−26.0066	−2.25506
$134$	3.63244	0.313795
$135$	−17.1860	−1.47913
$136$	−7.98273	−0.684513
$137$	−11.7293	−1.00210	−0.501052	−	0.865417i	$-0.667053\pi$
$137$	−11.7293	−1.00210	−0.501052	+	0.865417i	$0.667053\pi$
$138$	8.26882	0.703889
$139$	1.95677	0.165971	0.0829856	−	0.996551i	$-0.473554\pi$
$139$	1.95677	0.165971	0.0829856	+	0.996551i	$0.473554\pi$
$140$	−3.18749	−0.269392
$141$	−27.8531	−2.34565
$142$	17.1676	1.44067
$143$	0	0
$144$	−16.6217	−1.38514
$145$	−3.69729	−0.307043
$146$	11.0851	0.917411
$147$	−41.9870	−3.46303
$148$	4.68196	0.384854
$149$	−12.7341	−1.04322	−0.521609	−	0.853185i	$-0.674668\pi$
$149$	−12.7341	−1.04322	−0.521609	+	0.853185i	$0.674668\pi$
$150$	3.77675	0.308371
$151$	3.51593	0.286123	0.143061	−	0.989714i	$-0.454305\pi$
$151$	3.51593	0.286123	0.143061	+	0.989714i	$0.454305\pi$
$152$	−18.0882	−1.46715
$153$	−21.1349	−1.70866
$154$	0	0
$155$	0.431901	0.0346912
$156$	2.40535	0.192582
$157$	4.43087	0.353622	0.176811	−	0.984245i	$-0.443422\pi$
$157$	4.43087	0.353622	0.176811	+	0.984245i	$0.443422\pi$
$158$	−13.7138	−1.09101
$159$	−11.2821	−0.894731
$160$	−3.84685	−0.304120
$161$	9.68684	0.763430
$162$	−37.2587	−2.92732
$163$	18.7596	1.46936	0.734681	−	0.678413i	$-0.237332\pi$
$163$	18.7596	1.46936	0.734681	+	0.678413i	$0.237332\pi$
$164$	−4.85794	−0.379342
$165$	0	0
$166$	−11.6051	−0.900729
$167$	−21.2041	−1.64082	−0.820409	−	0.571776i	$-0.806254\pi$
$167$	−21.2041	−1.64082	−0.820409	+	0.571776i	$0.806254\pi$
$168$	−45.4583	−3.50719
$169$	1.00000	0.0769231
$170$	2.93436	0.225055
$171$	−47.8901	−3.66224
$172$	5.59021	0.426249
$173$	−13.6030	−1.03422	−0.517110	−	0.855919i	$-0.672992\pi$
$173$	−13.6030	−1.03422	−0.517110	+	0.855919i	$0.672992\pi$
$174$	−13.9637	−1.05859
$175$	4.42443	0.334455
$176$	0	0
$177$	−21.4932	−1.61552
$178$	4.30203	0.322451
$179$	12.6133	0.942760	0.471380	−	0.881930i	$-0.343756\pi$
$179$	12.6133	0.942760	0.471380	+	0.881930i	$0.343756\pi$
$180$	−5.86962	−0.437496
$181$	14.8932	1.10701	0.553503	−	0.832847i	$-0.313291\pi$
$181$	14.8932	1.10701	0.553503	+	0.832847i	$0.313291\pi$
$182$	−5.00483	−0.370983
$183$	49.9669	3.69366
$184$	6.73744	0.496690
$185$	−6.49884	−0.477804
$186$	1.63119	0.119604
$187$	0	0
$188$	−6.01005	−0.438328
$189$	−76.0381	−5.53096
$190$	6.64904	0.482372
$191$	19.3365	1.39914	0.699570	−	0.714564i	$-0.253375\pi$
$191$	19.3365	1.39914	0.699570	+	0.714564i	$0.253375\pi$
$192$	−28.1516	−2.03167
$193$	15.2407	1.09705	0.548523	−	0.836135i	$-0.315190\pi$
$193$	15.2407	1.09705	0.548523	+	0.836135i	$0.315190\pi$
$194$	−0.380748	−0.0273361
$195$	−3.33877	−0.239094
$196$	−9.05982	−0.647130
$197$	−17.5628	−1.25130	−0.625648	−	0.780106i	$-0.715165\pi$
$197$	−17.5628	−1.25130	−0.625648	+	0.780106i	$0.715165\pi$
$198$	0	0
$199$	26.1437	1.85328	0.926639	−	0.375953i	$-0.122685\pi$
$199$	26.1437	1.85328	0.926639	+	0.375953i	$0.122685\pi$
$200$	3.07730	0.217598
$201$	10.7214	0.756231
$202$	−5.98623	−0.421189
$203$	−16.3584	−1.14813
$204$	−6.23965	−0.436863
$205$	6.74312	0.470960
$206$	−14.0109	−0.976184
$207$	17.8379	1.23982
$208$	−2.04012	−0.141457
$209$	0	0
$210$	16.7100	1.15310
$211$	9.73935	0.670485	0.335242	−	0.942132i	$-0.391182\pi$
$211$	9.73935	0.670485	0.335242	+	0.942132i	$0.391182\pi$
$212$	−2.43442	−0.167197
$213$	50.6716	3.47196
$214$	−3.72466	−0.254612
$215$	−7.75955	−0.529197
$216$	−52.8864	−3.59846
$217$	1.91092	0.129722
$218$	−2.52157	−0.170782
$219$	32.7186	2.21092
$220$	0	0
$221$	−2.59407	−0.174496
$222$	−24.5445	−1.64732
$223$	−0.349873	−0.0234292	−0.0117146	−	0.999931i	$-0.503729\pi$
$223$	−0.349873	−0.0234292	−0.0117146	+	0.999931i	$0.503729\pi$
$224$	−17.0201	−1.13720
$225$	8.14739	0.543160
$226$	−9.45597	−0.629002
$227$	−21.5022	−1.42715	−0.713574	−	0.700580i	$-0.752925\pi$
$227$	−21.5022	−1.42715	−0.713574	+	0.700580i	$0.752925\pi$
$228$	−14.1386	−0.936349
$229$	25.6393	1.69429	0.847146	−	0.531360i	$-0.178319\pi$
$229$	25.6393	1.69429	0.847146	+	0.531360i	$0.178319\pi$
$230$	−2.47661	−0.163303
$231$	0	0
$232$	−11.3776	−0.746979
$233$	2.33138	0.152733	0.0763667	−	0.997080i	$-0.475668\pi$
$233$	2.33138	0.152733	0.0763667	+	0.997080i	$0.475668\pi$
$234$	−9.21618	−0.602480
$235$	8.34232	0.544193
$236$	−4.63773	−0.301890
$237$	−40.4775	−2.62930
$238$	12.9829	0.841555
$239$	−10.8582	−0.702359	−0.351179	−	0.936308i	$-0.614219\pi$
$239$	−10.8582	−0.702359	−0.351179	+	0.936308i	$0.614219\pi$
$240$	6.81150	0.439680
$241$	20.2866	1.30678	0.653388	−	0.757023i	$-0.273347\pi$
$241$	20.2866	1.30678	0.653388	+	0.757023i	$0.273347\pi$
$242$	0	0
$243$	−58.4140	−3.74726
$244$	10.7817	0.690228
$245$	12.5756	0.803424
$246$	25.4671	1.62372
$247$	−5.87796	−0.374006
$248$	1.32909	0.0843973
$249$	−34.2533	−2.17072
$250$	−1.13118	−0.0715422
$251$	12.2027	0.770229	0.385115	−	0.922869i	$-0.374162\pi$
$251$	12.2027	0.770229	0.385115	+	0.922869i	$0.374162\pi$
$252$	−25.9697	−1.63594
$253$	0	0
$254$	6.87779	0.431551
$255$	8.66101	0.542373
$256$	−14.7774	−0.923590
$257$	−8.21285	−0.512303	−0.256152	−	0.966637i	$-0.582455\pi$
$257$	−8.21285	−0.512303	−0.256152	+	0.966637i	$0.582455\pi$
$258$	−29.3059	−1.82451
$259$	−28.7536	−1.78666
$260$	−0.720430	−0.0446792
$261$	−30.1232	−1.86458
$262$	−6.87315	−0.424625
$263$	0.260793	0.0160812	0.00804060	−	0.999968i	$-0.497441\pi$
$263$	0.260793	0.0160812	0.00804060	+	0.999968i	$0.497441\pi$
$264$	0	0
$265$	3.37912	0.207578
$266$	29.4182	1.80375
$267$	12.6978	0.777091
$268$	2.31344	0.141316
$269$	10.9411	0.667094	0.333547	−	0.942734i	$-0.391754\pi$
$269$	10.9411	0.667094	0.333547	+	0.942734i	$0.391754\pi$
$270$	19.4404	1.18311
$271$	18.6262	1.13146	0.565731	−	0.824590i	$-0.308594\pi$
$271$	18.6262	1.13146	0.565731	+	0.824590i	$0.308594\pi$
$272$	5.29222	0.320888
$273$	−14.7722	−0.894052
$274$	13.2680	0.801548
$275$	0	0
$276$	5.26627	0.316992
$277$	12.9722	0.779426	0.389713	−	0.920936i	$-0.372574\pi$
$277$	12.9722	0.779426	0.389713	+	0.920936i	$0.372574\pi$
$278$	−2.21346	−0.132755
$279$	3.51887	0.210669
$280$	13.6153	0.813669
$281$	12.9630	0.773310	0.386655	−	0.922224i	$-0.373630\pi$
$281$	12.9630	0.773310	0.386655	+	0.922224i	$0.373630\pi$
$282$	31.5069	1.87621
$283$	13.4951	0.802198	0.401099	−	0.916035i	$-0.368628\pi$
$283$	13.4951	0.802198	0.401099	+	0.916035i	$0.368628\pi$
$284$	10.9338	0.648799
$285$	19.6252	1.16249
$286$	0	0
$287$	29.8344	1.76107
$288$	−31.3418	−1.84683
$289$	−10.2708	−0.604165
$290$	4.18230	0.245593
$291$	−1.12381	−0.0658788
$292$	7.05992	0.413151
$293$	12.0529	0.704139	0.352070	−	0.935974i	$-0.385478\pi$
$293$	12.0529	0.704139	0.352070	+	0.935974i	$0.385478\pi$
$294$	47.4949	2.76996
$295$	6.43744	0.374802
$296$	−19.9989	−1.16241
$297$	0	0
$298$	14.4046	0.834434
$299$	2.18940	0.126616
$300$	2.40535	0.138873
$301$	−34.3316	−1.97884
$302$	−3.97716	−0.228860
$303$	−17.6688	−1.01505
$304$	11.9918	0.687774
$305$	−14.9657	−0.856932
$306$	23.9074	1.36670
$307$	−20.2564	−1.15610	−0.578048	−	0.816003i	$-0.696185\pi$
$307$	−20.2564	−1.15610	−0.578048	+	0.816003i	$0.696185\pi$
$308$	0	0
$309$	−41.3542	−2.35256
$310$	−0.488559	−0.0277483
$311$	16.7105	0.947566	0.473783	−	0.880642i	$-0.342888\pi$
$311$	16.7105	0.947566	0.473783	+	0.880642i	$0.342888\pi$
$312$	−10.2744	−0.581673
$313$	−15.2117	−0.859818	−0.429909	−	0.902872i	$-0.641454\pi$
$313$	−15.2117	−0.859818	−0.429909	+	0.902872i	$0.641454\pi$
$314$	−5.01211	−0.282850
$315$	36.0476	2.03105
$316$	−8.73411	−0.491332
$317$	−3.33665	−0.187405	−0.0937025	−	0.995600i	$-0.529870\pi$
$317$	−3.33665	−0.187405	−0.0937025	+	0.995600i	$0.529870\pi$
$318$	12.7621	0.715664
$319$	0	0
$320$	8.43173	0.471348
$321$	−10.9936	−0.613604
$322$	−10.9576	−0.610642
$323$	15.2478	0.848413
$324$	−23.7294	−1.31830
$325$	1.00000	0.0554700
$326$	−21.2205	−1.17529
$327$	−7.44260	−0.411577
$328$	20.7506	1.14576
$329$	36.9100	2.03491
$330$	0	0
$331$	−23.8002	−1.30818	−0.654089	−	0.756417i	$-0.726948\pi$
$331$	−23.8002	−1.30818	−0.654089	+	0.756417i	$0.726948\pi$
$332$	−7.39107	−0.405638
$333$	−52.9486	−2.90156
$334$	23.9856	1.31243
$335$	−3.21119	−0.175446
$336$	30.1370	1.64411
$337$	−8.96215	−0.488199	−0.244100	−	0.969750i	$-0.578492\pi$
$337$	−8.96215	−0.488199	−0.244100	+	0.969750i	$0.578492\pi$
$338$	−1.13118	−0.0615281
$339$	−27.9101	−1.51587
$340$	1.86884	0.101352
$341$	0	0
$342$	54.1723	2.92930
$343$	24.6688	1.33199
$344$	−23.8784	−1.28744
$345$	−7.30990	−0.393552
$346$	15.3875	0.827236
$347$	9.88257	0.530524	0.265262	−	0.964176i	$-0.414542\pi$
$347$	9.88257	0.530524	0.265262	+	0.964176i	$0.414542\pi$
$348$	−8.89327	−0.476729
$349$	−22.1831	−1.18744	−0.593718	−	0.804674i	$-0.702340\pi$
$349$	−22.1831	−1.18744	−0.593718	+	0.804674i	$0.702340\pi$
$350$	−5.00483	−0.267519
$351$	−17.1860	−0.917319
$352$	0	0
$353$	−2.47522	−0.131743	−0.0658714	−	0.997828i	$-0.520983\pi$
$353$	−2.47522	−0.131743	−0.0658714	+	0.997828i	$0.520983\pi$
$354$	24.3126	1.29220
$355$	−15.1767	−0.805496
$356$	2.73989	0.145214
$357$	38.3200	2.02811
$358$	−14.2679	−0.754081
$359$	7.35173	0.388009	0.194005	−	0.981001i	$-0.437852\pi$
$359$	7.35173	0.388009	0.194005	+	0.981001i	$0.437852\pi$
$360$	25.0720	1.32141
$361$	15.5504	0.818444
$362$	−16.8469	−0.885455
$363$	0	0
$364$	−3.18749	−0.167070
$365$	−9.79960	−0.512934
$366$	−56.5216	−2.95443
$367$	−18.9963	−0.991600	−0.495800	−	0.868437i	$-0.665125\pi$
$367$	−18.9963	−0.991600	−0.495800	+	0.868437i	$0.665125\pi$
$368$	−4.46664	−0.232840
$369$	54.9388	2.86000
$370$	7.35136	0.382179
$371$	14.9507	0.776202
$372$	1.03887	0.0538631
$373$	28.3356	1.46716	0.733582	−	0.679601i	$-0.237847\pi$
$373$	28.3356	1.46716	0.733582	+	0.679601i	$0.237847\pi$
$374$	0	0
$375$	−3.33877	−0.172413
$376$	25.6718	1.32392
$377$	−3.69729	−0.190420
$378$	86.0129	4.42402
$379$	8.72609	0.448229	0.224114	−	0.974563i	$-0.428051\pi$
$379$	8.72609	0.448229	0.224114	+	0.974563i	$0.428051\pi$
$380$	4.23466	0.217233
$381$	20.3003	1.04002
$382$	−21.8731	−1.11912
$383$	19.7049	1.00687	0.503436	−	0.864032i	$-0.332069\pi$
$383$	19.7049	1.00687	0.503436	+	0.864032i	$0.332069\pi$
$384$	6.15705	0.314201
$385$	0	0
$386$	−17.2399	−0.877490
$387$	−63.2201	−3.21366
$388$	−0.242492	−0.0123107
$389$	15.9784	0.810135	0.405067	−	0.914287i	$-0.367248\pi$
$389$	15.9784	0.810135	0.405067	+	0.914287i	$0.367248\pi$
$390$	3.77675	0.191243
$391$	−5.67945	−0.287222
$392$	38.6988	1.95458
$393$	−20.2867	−1.02333
$394$	19.8667	1.00087
$395$	12.1235	0.609998
$396$	0	0
$397$	12.5518	0.629956	0.314978	−	0.949099i	$-0.398003\pi$
$397$	12.5518	0.629956	0.314978	+	0.949099i	$0.398003\pi$
$398$	−29.5732	−1.48237
$399$	86.8302	4.34695
$400$	−2.04012	−0.102006
$401$	11.6966	0.584102	0.292051	−	0.956403i	$-0.405662\pi$
$401$	11.6966	0.584102	0.292051	+	0.956403i	$0.405662\pi$
$402$	−12.1279	−0.604883
$403$	0.431901	0.0215145
$404$	−3.81252	−0.189680
$405$	32.9378	1.63669
$406$	18.5043	0.918352
$407$	0	0
$408$	26.6525	1.31950
$409$	6.97614	0.344948	0.172474	−	0.985014i	$-0.444824\pi$
$409$	6.97614	0.344948	0.172474	+	0.985014i	$0.444824\pi$
$410$	−7.62768	−0.376704
$411$	39.1615	1.93169
$412$	−8.92329	−0.439619
$413$	28.4820	1.40151
$414$	−20.1779	−0.991689
$415$	10.2593	0.503607
$416$	−3.84685	−0.188607
$417$	−6.53322	−0.319933
$418$	0	0
$419$	40.4814	1.97765	0.988824	−	0.149087i	$-0.0476335\pi$
$419$	40.4814	1.97765	0.988824	+	0.149087i	$0.0476335\pi$
$420$	10.6423	0.519291
$421$	−21.6528	−1.05529	−0.527647	−	0.849464i	$-0.676926\pi$
$421$	−21.6528	−1.05529	−0.527647	+	0.849464i	$0.676926\pi$
$422$	−11.0170	−0.536298
$423$	67.9682	3.30472
$424$	10.3986	0.504999
$425$	−2.59407	−0.125831
$426$	−57.3187	−2.77710
$427$	−66.2145	−3.20435
$428$	−2.37217	−0.114663
$429$	0	0
$430$	8.77745	0.423286
$431$	11.0342	0.531499	0.265749	−	0.964042i	$-0.414381\pi$
$431$	11.0342	0.531499	0.265749	+	0.964042i	$0.414381\pi$
$432$	35.0615	1.68690
$433$	−23.7361	−1.14068	−0.570342	−	0.821407i	$-0.693189\pi$
$433$	−23.7361	−1.14068	−0.570342	+	0.821407i	$0.693189\pi$
$434$	−2.16159	−0.103760
$435$	12.3444	0.591868
$436$	−1.60594	−0.0769107
$437$	−12.8692	−0.615618
$438$	−37.0107	−1.76844
$439$	11.7722	0.561857	0.280929	−	0.959729i	$-0.409358\pi$
$439$	11.7722	0.561857	0.280929	+	0.959729i	$0.409358\pi$
$440$	0	0
$441$	102.458	4.87896
$442$	2.93436	0.139573
$443$	1.63575	0.0777167	0.0388583	−	0.999245i	$-0.487628\pi$
$443$	1.63575	0.0777167	0.0388583	+	0.999245i	$0.487628\pi$
$444$	−15.6320	−0.741861
$445$	−3.80313	−0.180286
$446$	0.395770	0.0187402
$447$	42.5162	2.01095
$448$	37.3056	1.76252
$449$	−34.4135	−1.62407	−0.812036	−	0.583608i	$-0.801641\pi$
$449$	−34.4135	−1.62407	−0.812036	+	0.583608i	$0.801641\pi$
$450$	−9.21618	−0.434455
$451$	0	0
$452$	−6.02235	−0.283267
$453$	−11.7389	−0.551541
$454$	24.3228	1.14153
$455$	4.42443	0.207420
$456$	60.3925	2.82814
$457$	32.2262	1.50748	0.753739	−	0.657173i	$-0.228248\pi$
$457$	32.2262	1.50748	0.753739	+	0.657173i	$0.228248\pi$
$458$	−29.0027	−1.35521
$459$	44.5816	2.08089
$460$	−1.57731	−0.0735424
$461$	0.849990	0.0395880	0.0197940	−	0.999804i	$-0.493699\pi$
$461$	0.849990	0.0395880	0.0197940	+	0.999804i	$0.493699\pi$
$462$	0	0
$463$	18.7174	0.869872	0.434936	−	0.900461i	$-0.356771\pi$
$463$	18.7174	0.869872	0.434936	+	0.900461i	$0.356771\pi$
$464$	7.54291	0.350171
$465$	−1.44202	−0.0668721
$466$	−2.63721	−0.122166
$467$	−28.8534	−1.33518	−0.667589	−	0.744530i	$-0.732674\pi$
$467$	−28.8534	−1.33518	−0.667589	+	0.744530i	$0.732674\pi$
$468$	−5.86962	−0.271324
$469$	−14.2077	−0.656050
$470$	−9.43667	−0.435281
$471$	−14.7937	−0.681656
$472$	19.8099	0.911826
$473$	0	0
$474$	45.7874	2.10308
$475$	−5.87796	−0.269699
$476$	8.26857	0.378989
$477$	27.5311	1.26056
$478$	12.2826	0.561793
$479$	11.3098	0.516760	0.258380	−	0.966043i	$-0.416811\pi$
$479$	11.3098	0.516760	0.258380	+	0.966043i	$0.416811\pi$
$480$	12.8438	0.586234
$481$	−6.49884	−0.296321
$482$	−22.9478	−1.04525
$483$	−32.3422	−1.47162
$484$	0	0
$485$	0.336593	0.0152839
$486$	66.0768	2.99731
$487$	27.3173	1.23786	0.618932	−	0.785444i	$-0.287566\pi$
$487$	27.3173	1.23786	0.618932	+	0.785444i	$0.287566\pi$
$488$	−46.0538	−2.08476
$489$	−62.6339	−2.83240
$490$	−14.2253	−0.642632
$491$	1.94926	0.0879689	0.0439845	−	0.999032i	$-0.485995\pi$
$491$	1.94926	0.0879689	0.0439845	+	0.999032i	$0.485995\pi$
$492$	16.2196	0.731234
$493$	9.59102	0.431958
$494$	6.64904	0.299154
$495$	0	0
$496$	−0.881131	−0.0395640
$497$	−67.1483	−3.01201
$498$	38.7467	1.73628
$499$	25.9632	1.16227	0.581136	−	0.813806i	$-0.302608\pi$
$499$	25.9632	1.16227	0.581136	+	0.813806i	$0.302608\pi$
$500$	−0.720430	−0.0322186
$501$	70.7955	3.16291
$502$	−13.8035	−0.616080
$503$	−4.78726	−0.213453	−0.106727	−	0.994288i	$-0.534037\pi$
$503$	−4.78726	−0.213453	−0.106727	+	0.994288i	$0.534037\pi$
$504$	110.929	4.94118
$505$	5.29201	0.235492
$506$	0	0
$507$	−3.33877	−0.148280
$508$	4.38034	0.194346
$509$	−0.644478	−0.0285660	−0.0142830	−	0.999898i	$-0.504547\pi$
$509$	−0.644478	−0.0285660	−0.0142830	+	0.999898i	$0.504547\pi$
$510$	−9.79716	−0.433826
$511$	−43.3576	−1.91803
$512$	20.4042	0.901745
$513$	101.018	4.46007
$514$	9.29022	0.409774
$515$	12.3861	0.545795
$516$	−18.6644	−0.821656
$517$	0	0
$518$	32.5256	1.42909
$519$	45.4174	1.99360
$520$	3.07730	0.134948
$521$	−21.5446	−0.943887	−0.471944	−	0.881629i	$-0.656447\pi$
$521$	−21.5446	−0.943887	−0.471944	+	0.881629i	$0.656447\pi$
$522$	34.0748	1.49141
$523$	1.31658	0.0575700	0.0287850	−	0.999586i	$-0.490836\pi$
$523$	1.31658	0.0575700	0.0287850	+	0.999586i	$0.490836\pi$
$524$	−4.37739	−0.191227
$525$	−14.7722	−0.644710
$526$	−0.295004	−0.0128628
$527$	−1.12038	−0.0488046
$528$	0	0
$529$	−18.2065	−0.791588
$530$	−3.82240	−0.166034
$531$	52.4484	2.27607
$532$	18.7359	0.812306
$533$	6.74312	0.292077
$534$	−14.3635	−0.621569
$535$	3.29272	0.142357
$536$	−9.88179	−0.426828
$537$	−42.1128	−1.81730
$538$	−12.3764	−0.533585
$539$	0	0
$540$	12.3813	0.532806
$541$	1.14587	0.0492650	0.0246325	−	0.999697i	$-0.492158\pi$
$541$	1.14587	0.0492650	0.0246325	+	0.999697i	$0.492158\pi$
$542$	−21.0696	−0.905018
$543$	−49.7251	−2.13391
$544$	9.97900	0.427846
$545$	2.22914	0.0954860
$546$	16.7100	0.715121
$547$	−10.9095	−0.466455	−0.233228	−	0.972422i	$-0.574929\pi$
$547$	−10.9095	−0.466455	−0.233228	+	0.972422i	$0.574929\pi$
$548$	8.45015	0.360972
$549$	−121.931	−5.20390
$550$	0	0
$551$	21.7325	0.925836
$552$	−22.4948	−0.957440
$553$	53.6394	2.28098
$554$	−14.6739	−0.623436
$555$	21.6981	0.921034
$556$	−1.40972	−0.0597853
$557$	−30.7645	−1.30353	−0.651766	−	0.758420i	$-0.725972\pi$
$557$	−30.7645	−1.30353	−0.651766	+	0.758420i	$0.725972\pi$
$558$	−3.98048	−0.168507
$559$	−7.75955	−0.328194
$560$	−9.02637	−0.381434
$561$	0	0
$562$	−14.6635	−0.618544
$563$	−19.1181	−0.805732	−0.402866	−	0.915259i	$-0.631986\pi$
$563$	−19.1181	−0.805732	−0.402866	+	0.915259i	$0.631986\pi$
$564$	20.0662	0.844939
$565$	8.35938	0.351682
$566$	−15.2653	−0.641650
$567$	145.731	6.12013
$568$	−46.7033	−1.95963
$569$	12.5432	0.525839	0.262920	−	0.964818i	$-0.415315\pi$
$569$	12.5432	0.525839	0.262920	+	0.964818i	$0.415315\pi$
$570$	−22.1996	−0.929840
$571$	34.0574	1.42526	0.712630	−	0.701540i	$-0.247504\pi$
$571$	34.0574	1.42526	0.712630	+	0.701540i	$0.247504\pi$
$572$	0	0
$573$	−64.5601	−2.69704
$574$	−33.7482	−1.40862
$575$	2.18940	0.0913043
$576$	68.6966	2.86236
$577$	26.9491	1.12191	0.560954	−	0.827847i	$-0.310435\pi$
$577$	26.9491	1.12191	0.560954	+	0.827847i	$0.310435\pi$
$578$	11.6181	0.483251
$579$	−50.8851	−2.11471
$580$	2.66363	0.110601
$581$	45.3914	1.88315
$582$	1.27123	0.0526942
$583$	0	0
$584$	−30.1563	−1.24788
$585$	8.14739	0.336853
$586$	−13.6340	−0.563217
$587$	−39.2837	−1.62141	−0.810706	−	0.585454i	$-0.800916\pi$
$587$	−39.2837	−1.62141	−0.810706	+	0.585454i	$0.800916\pi$
$588$	30.2487	1.24743
$589$	−2.53870	−0.104605
$590$	−7.28191	−0.299792
$591$	58.6381	2.41205
$592$	13.2584	0.544918
$593$	−19.5180	−0.801509	−0.400754	−	0.916185i	$-0.631252\pi$
$593$	−19.5180	−0.801509	−0.400754	+	0.916185i	$0.631252\pi$
$594$	0	0
$595$	−11.4773	−0.470523
$596$	9.17402	0.375783
$597$	−87.2878	−3.57245
$598$	−2.47661	−0.101276
$599$	10.4430	0.426690	0.213345	−	0.976977i	$-0.431564\pi$
$599$	10.4430	0.426690	0.213345	+	0.976977i	$0.431564\pi$
$600$	−10.2744	−0.419450
$601$	21.4314	0.874203	0.437102	−	0.899412i	$-0.356005\pi$
$601$	21.4314	0.874203	0.437102	+	0.899412i	$0.356005\pi$
$602$	38.8352	1.58281
$603$	−26.1628	−1.06543
$604$	−2.53298	−0.103066
$605$	0	0
$606$	19.9866	0.811902
$607$	−23.5700	−0.956675	−0.478338	−	0.878176i	$-0.658760\pi$
$607$	−23.5700	−0.956675	−0.478338	+	0.878176i	$0.658760\pi$
$608$	22.6116	0.917023
$609$	54.6169	2.21319
$610$	16.9289	0.685430
$611$	8.34232	0.337494
$612$	15.2262	0.615483
$613$	−3.81144	−0.153942	−0.0769712	−	0.997033i	$-0.524525\pi$
$613$	−3.81144	−0.153942	−0.0769712	+	0.997033i	$0.524525\pi$
$614$	22.9137	0.924721
$615$	−22.5137	−0.907841
$616$	0	0
$617$	0.502452	0.0202280	0.0101140	−	0.999949i	$-0.496781\pi$
$617$	0.502452	0.0202280	0.0101140	+	0.999949i	$0.496781\pi$
$618$	46.7791	1.88173
$619$	45.1440	1.81449	0.907246	−	0.420601i	$-0.138181\pi$
$619$	45.1440	1.81449	0.907246	+	0.420601i	$0.138181\pi$
$620$	−0.311155	−0.0124963
$621$	−37.6270	−1.50992
$622$	−18.9026	−0.757926
$623$	−16.8267	−0.674147
$624$	6.81150	0.272678
$625$	1.00000	0.0400000
$626$	17.2072	0.687739
$627$	0	0
$628$	−3.19213	−0.127380
$629$	16.8584	0.672190
$630$	−40.7763	−1.62457
$631$	−19.6367	−0.781723	−0.390862	−	0.920449i	$-0.627823\pi$
$631$	−19.6367	−0.781723	−0.390862	+	0.920449i	$0.627823\pi$
$632$	37.3075	1.48401
$633$	−32.5175	−1.29245
$634$	3.77436	0.149899
$635$	−6.08018	−0.241285
$636$	8.12798	0.322295
$637$	12.5756	0.498263
$638$	0	0
$639$	−123.651	−4.89154
$640$	−1.84411	−0.0728948
$641$	25.4251	1.00423	0.502115	−	0.864801i	$-0.332556\pi$
$641$	25.4251	1.00423	0.502115	+	0.864801i	$0.332556\pi$
$642$	12.4358	0.490801
$643$	8.01245	0.315980	0.157990	−	0.987441i	$-0.449499\pi$
$643$	8.01245	0.315980	0.157990	+	0.987441i	$0.449499\pi$
$644$	−6.97869	−0.274999
$645$	25.9073	1.02010
$646$	−17.2481	−0.678616
$647$	20.4630	0.804485	0.402242	−	0.915533i	$-0.368231\pi$
$647$	20.4630	0.804485	0.402242	+	0.915533i	$0.368231\pi$
$648$	101.360	3.98178
$649$	0	0
$650$	−1.13118	−0.0443686
$651$	−6.38012	−0.250056
$652$	−13.5149	−0.529286
$653$	−4.67131	−0.182803	−0.0914013	−	0.995814i	$-0.529135\pi$
$653$	−4.67131	−0.182803	−0.0914013	+	0.995814i	$0.529135\pi$
$654$	8.41893	0.329206
$655$	6.07608	0.237412
$656$	−13.7568	−0.537112
$657$	−79.8412	−3.11490
$658$	−41.7519	−1.62766
$659$	−8.34828	−0.325203	−0.162601	−	0.986692i	$-0.551988\pi$
$659$	−8.34828	−0.325203	−0.162601	+	0.986692i	$0.551988\pi$
$660$	0	0
$661$	33.9259	1.31957	0.659783	−	0.751457i	$-0.270648\pi$
$661$	33.9259	1.31957	0.659783	+	0.751457i	$0.270648\pi$
$662$	26.9224	1.04637
$663$	8.66101	0.336366
$664$	31.5708	1.22518
$665$	−26.0066	−1.00849
$666$	59.8944	2.32086
$667$	−8.09483	−0.313433
$668$	15.2760	0.591047
$669$	1.16815	0.0451632
$670$	3.63244	0.140333
$671$	0	0
$672$	56.8263	2.19212
$673$	9.48362	0.365567	0.182783	−	0.983153i	$-0.441489\pi$
$673$	9.48362	0.365567	0.182783	+	0.983153i	$0.441489\pi$
$674$	10.1378	0.390494
$675$	−17.1860	−0.661488
$676$	−0.720430	−0.0277088
$677$	−30.3886	−1.16793	−0.583964	−	0.811779i	$-0.698499\pi$
$677$	−30.3886	−1.16793	−0.583964	+	0.811779i	$0.698499\pi$
$678$	31.5713	1.21249
$679$	1.48923	0.0571515
$680$	−7.98273	−0.306124
$681$	71.7908	2.75103
$682$	0	0
$683$	−33.6635	−1.28810	−0.644049	−	0.764984i	$-0.722747\pi$
$683$	−33.6635	−1.28810	−0.644049	+	0.764984i	$0.722747\pi$
$684$	34.5014	1.31920
$685$	−11.7293	−0.448154
$686$	−27.9048	−1.06541
$687$	−85.6037	−3.26599
$688$	15.8304	0.603529
$689$	3.37912	0.128734
$690$	8.26882	0.314789
$691$	35.5742	1.35331	0.676653	−	0.736302i	$-0.263430\pi$
$691$	35.5742	1.35331	0.676653	+	0.736302i	$0.263430\pi$
$692$	9.80003	0.372541
$693$	0	0
$694$	−11.1790	−0.424348
$695$	1.95677	0.0742246
$696$	37.9874	1.43991
$697$	−17.4921	−0.662561
$698$	25.0931	0.949789
$699$	−7.78393	−0.294415
$700$	−3.18749	−0.120476
$701$	−13.7246	−0.518370	−0.259185	−	0.965828i	$-0.583454\pi$
$701$	−13.7246	−0.518370	−0.259185	+	0.965828i	$0.583454\pi$
$702$	19.4404	0.733732
$703$	38.1999	1.44074
$704$	0	0
$705$	−27.8531	−1.04901
$706$	2.79992	0.105376
$707$	23.4141	0.880580
$708$	15.4843	0.581936
$709$	−47.8364	−1.79653	−0.898267	−	0.439451i	$-0.855173\pi$
$709$	−47.8364	−1.79653	−0.898267	+	0.439451i	$0.855173\pi$
$710$	17.1676	0.644289
$711$	98.7747	3.70434
$712$	−11.7034	−0.438602
$713$	0.945605	0.0354132
$714$	−43.3469	−1.62222
$715$	0	0
$716$	−9.08697	−0.339596
$717$	36.2531	1.35390
$718$	−8.31614	−0.310355
$719$	10.6872	0.398566	0.199283	−	0.979942i	$-0.436139\pi$
$719$	10.6872	0.398566	0.199283	+	0.979942i	$0.436139\pi$
$720$	−16.6217	−0.619453
$721$	54.8013	2.04091
$722$	−17.5903	−0.654645
$723$	−67.7324	−2.51899
$724$	−10.7295	−0.398760
$725$	−3.69729	−0.137314
$726$	0	0
$727$	−26.2267	−0.972694	−0.486347	−	0.873766i	$-0.661671\pi$
$727$	−26.2267	−0.972694	−0.486347	+	0.873766i	$0.661671\pi$
$728$	13.6153	0.504616
$729$	96.2175	3.56361
$730$	11.0851	0.410279
$731$	20.1288	0.744491
$732$	−35.9977	−1.33051
$733$	12.4094	0.458351	0.229175	−	0.973385i	$-0.426397\pi$
$733$	12.4094	0.458351	0.229175	+	0.973385i	$0.426397\pi$
$734$	21.4883	0.793147
$735$	−41.9870	−1.54871
$736$	−8.42229	−0.310450
$737$	0	0
$738$	−62.1457	−2.28762
$739$	14.8018	0.544492	0.272246	−	0.962228i	$-0.412234\pi$
$739$	14.8018	0.544492	0.272246	+	0.962228i	$0.412234\pi$
$740$	4.68196	0.172112
$741$	19.6252	0.720949
$742$	−16.9119	−0.620857
$743$	−34.4730	−1.26469	−0.632345	−	0.774687i	$-0.717907\pi$
$743$	−34.4730	−1.26469	−0.632345	+	0.774687i	$0.717907\pi$
$744$	−4.43753	−0.162688
$745$	−12.7341	−0.466541
$746$	−32.0527	−1.17353
$747$	83.5862	3.05826
$748$	0	0
$749$	14.5684	0.532317
$750$	3.77675	0.137908
$751$	44.3549	1.61853	0.809266	−	0.587443i	$-0.199865\pi$
$751$	44.3549	1.61853	0.809266	+	0.587443i	$0.199865\pi$
$752$	−17.0193	−0.620632
$753$	−40.7421	−1.48472
$754$	4.18230	0.152310
$755$	3.51593	0.127958
$756$	54.7801	1.99233
$757$	−0.643538	−0.0233898	−0.0116949	−	0.999932i	$-0.503723\pi$
$757$	−0.643538	−0.0233898	−0.0116949	+	0.999932i	$0.503723\pi$
$758$	−9.87078	−0.358523
$759$	0	0
$760$	−18.0882	−0.656129
$761$	−16.5759	−0.600878	−0.300439	−	0.953801i	$-0.597133\pi$
$761$	−16.5759	−0.600878	−0.300439	+	0.953801i	$0.597133\pi$
$762$	−22.9634	−0.831875
$763$	9.86269	0.357053
$764$	−13.9306	−0.503991
$765$	−21.1349	−0.764134
$766$	−22.2898	−0.805363
$767$	6.43744	0.232443
$768$	49.3385	1.78035
$769$	33.4701	1.20696	0.603481	−	0.797377i	$-0.293780\pi$
$769$	33.4701	1.20696	0.603481	+	0.797377i	$0.293780\pi$
$770$	0	0
$771$	27.4208	0.987537
$772$	−10.9798	−0.395172
$773$	44.8029	1.61145	0.805725	−	0.592290i	$-0.201776\pi$
$773$	44.8029	1.61145	0.805725	+	0.592290i	$0.201776\pi$
$774$	71.5133	2.57049
$775$	0.431901	0.0155144
$776$	1.03580	0.0371830
$777$	96.0019	3.44405
$778$	−18.0744	−0.647999
$779$	−39.6358	−1.42010
$780$	2.40535	0.0861254
$781$	0	0
$782$	6.42449	0.229739
$783$	63.5414	2.27079
$784$	−25.6557	−0.916275
$785$	4.43087	0.158144
$786$	22.9479	0.818524
$787$	−11.8111	−0.421020	−0.210510	−	0.977592i	$-0.567513\pi$
$787$	−11.8111	−0.421020	−0.210510	+	0.977592i	$0.567513\pi$
$788$	12.6527	0.450735
$789$	−0.870729	−0.0309988
$790$	−13.7138	−0.487916
$791$	36.9855	1.31505
$792$	0	0
$793$	−14.9657	−0.531446
$794$	−14.1983	−0.503881
$795$	−11.2821	−0.400136
$796$	−18.8347	−0.667578
$797$	24.2331	0.858380	0.429190	−	0.903214i	$-0.358799\pi$
$797$	24.2331	0.858380	0.429190	+	0.903214i	$0.358799\pi$
$798$	−98.2206	−3.47697
$799$	−21.6406	−0.765588
$800$	−3.84685	−0.136007
$801$	−30.9856	−1.09482
$802$	−13.2310	−0.467203
$803$	0	0
$804$	−7.72404	−0.272406
$805$	9.68684	0.341416
$806$	−0.488559	−0.0172087
$807$	−36.5300	−1.28592
$808$	16.2851	0.572908
$809$	−31.1027	−1.09351	−0.546757	−	0.837291i	$-0.684138\pi$
$809$	−31.1027	−1.09351	−0.546757	+	0.837291i	$0.684138\pi$
$810$	−37.2587	−1.30914
$811$	−3.04310	−0.106858	−0.0534289	−	0.998572i	$-0.517015\pi$
$811$	−3.04310	−0.106858	−0.0534289	+	0.998572i	$0.517015\pi$
$812$	11.7851	0.413575
$813$	−62.1887	−2.18105
$814$	0	0
$815$	18.7596	0.657119
$816$	−17.6695	−0.618556
$817$	45.6103	1.59570
$818$	−7.89127	−0.275912
$819$	36.0476	1.25960
$820$	−4.85794	−0.169647
$821$	7.99479	0.279020	0.139510	−	0.990221i	$-0.455447\pi$
$821$	7.99479	0.279020	0.139510	+	0.990221i	$0.455447\pi$
$822$	−44.2987	−1.54510
$823$	−1.13399	−0.0395285	−0.0197642	−	0.999805i	$-0.506292\pi$
$823$	−1.13399	−0.0395285	−0.0197642	+	0.999805i	$0.506292\pi$
$824$	38.1156	1.32782
$825$	0	0
$826$	−32.2183	−1.12102
$827$	33.9938	1.18208	0.591040	−	0.806642i	$-0.298717\pi$
$827$	33.9938	1.18208	0.591040	+	0.806642i	$0.298717\pi$
$828$	−12.8510	−0.446602
$829$	−35.3174	−1.22662	−0.613312	−	0.789841i	$-0.710163\pi$
$829$	−35.3174	−1.22662	−0.613312	+	0.789841i	$0.710163\pi$
$830$	−11.6051	−0.402818
$831$	−43.3113	−1.50245
$832$	8.43173	0.292318
$833$	−32.6219	−1.13028
$834$	7.39025	0.255903
$835$	−21.2041	−0.733797
$836$	0	0
$837$	−7.42264	−0.256564
$838$	−45.7918	−1.58185
$839$	−52.6759	−1.81857	−0.909286	−	0.416171i	$-0.863372\pi$
$839$	−52.6759	−1.81857	−0.909286	+	0.416171i	$0.863372\pi$
$840$	−45.4583	−1.56846
$841$	−15.3301	−0.528623
$842$	24.4933	0.844094
$843$	−43.2806	−1.49066
$844$	−7.01652	−0.241519
$845$	1.00000	0.0344010
$846$	−76.8843	−2.64334
$847$	0	0
$848$	−6.89382	−0.236735
$849$	−45.0569	−1.54635
$850$	2.93436	0.100648
$851$	−14.2286	−0.487748
$852$	−36.5053	−1.25065
$853$	29.7612	1.01900	0.509502	−	0.860470i	$-0.329830\pi$
$853$	29.7612	1.01900	0.509502	+	0.860470i	$0.329830\pi$
$854$	74.9006	2.56305
$855$	−47.8901	−1.63781
$856$	10.1327	0.346327
$857$	28.1349	0.961070	0.480535	−	0.876976i	$-0.340443\pi$
$857$	28.1349	0.961070	0.480535	+	0.876976i	$0.340443\pi$
$858$	0	0
$859$	−8.95665	−0.305597	−0.152798	−	0.988257i	$-0.548829\pi$
$859$	−8.95665	−0.305597	−0.152798	+	0.988257i	$0.548829\pi$
$860$	5.59021	0.190625
$861$	−99.6104	−3.39471
$862$	−12.4817	−0.425128
$863$	41.4051	1.40945	0.704723	−	0.709483i	$-0.251071\pi$
$863$	41.4051	1.40945	0.704723	+	0.709483i	$0.251071\pi$
$864$	66.1119	2.24917
$865$	−13.6030	−0.462517
$866$	26.8498	0.912394
$867$	34.2919	1.16461
$868$	−1.37668	−0.0467276
$869$	0	0
$870$	−13.9637	−0.473415
$871$	−3.21119	−0.108807
$872$	6.85974	0.232300
$873$	2.74236	0.0928147
$874$	14.5574	0.492411
$875$	4.42443	0.149573
$876$	−23.5715	−0.796406
$877$	−21.4266	−0.723524	−0.361762	−	0.932271i	$-0.617825\pi$
$877$	−21.4266	−0.723524	−0.361762	+	0.932271i	$0.617825\pi$
$878$	−13.3165	−0.449410
$879$	−40.2420	−1.35733
$880$	0	0
$881$	32.1627	1.08359	0.541795	−	0.840511i	$-0.317745\pi$
$881$	32.1627	1.08359	0.541795	+	0.840511i	$0.317745\pi$
$882$	−115.899	−3.90251
$883$	−3.99847	−0.134559	−0.0672795	−	0.997734i	$-0.521432\pi$
$883$	−3.99847	−0.134559	−0.0672795	+	0.997734i	$0.521432\pi$
$884$	1.86884	0.0628561
$885$	−21.4932	−0.722484
$886$	−1.85033	−0.0621629
$887$	−14.5709	−0.489244	−0.244622	−	0.969618i	$-0.578664\pi$
$887$	−14.5709	−0.489244	−0.244622	+	0.969618i	$0.578664\pi$
$888$	66.7716	2.24071
$889$	−26.9013	−0.902242
$890$	4.30203	0.144204
$891$	0	0
$892$	0.252059	0.00843956
$893$	−49.0358	−1.64092
$894$	−48.0936	−1.60849
$895$	12.6133	0.421615
$896$	−8.15913	−0.272577
$897$	−7.30990	−0.244071
$898$	38.9278	1.29904
$899$	−1.59686	−0.0532584
$900$	−5.86962	−0.195654
$901$	−8.76568	−0.292027
$902$	0	0
$903$	114.625	3.81449
$904$	25.7243	0.855578
$905$	14.8932	0.495068
$906$	13.2788	0.441159
$907$	19.5452	0.648987	0.324494	−	0.945888i	$-0.394806\pi$
$907$	19.5452	0.648987	0.324494	+	0.945888i	$0.394806\pi$
$908$	15.4908	0.514080
$909$	43.1161	1.43007
$910$	−5.00483	−0.165909
$911$	−42.2982	−1.40140	−0.700701	−	0.713455i	$-0.747129\pi$
$911$	−42.2982	−1.40140	−0.700701	+	0.713455i	$0.747129\pi$
$912$	−40.0377	−1.32578
$913$	0	0
$914$	−36.4537	−1.20578
$915$	49.9669	1.65186
$916$	−18.4713	−0.610309
$917$	26.8832	0.887762
$918$	−50.4299	−1.66443
$919$	20.4894	0.675884	0.337942	−	0.941167i	$-0.390269\pi$
$919$	20.4894	0.675884	0.337942	+	0.941167i	$0.390269\pi$
$920$	6.73744	0.222127
$921$	67.6316	2.22854
$922$	−0.961492	−0.0316651
$923$	−15.1767	−0.499548
$924$	0	0
$925$	−6.49884	−0.213680
$926$	−21.1728	−0.695781
$927$	100.914	3.31446
$928$	14.2229	0.466890
$929$	1.78487	0.0585595	0.0292798	−	0.999571i	$-0.490679\pi$
$929$	1.78487	0.0585595	0.0292798	+	0.999571i	$0.490679\pi$
$930$	1.63119	0.0534887
$931$	−73.9188	−2.42259
$932$	−1.67959	−0.0550169
$933$	−55.7926	−1.82657
$934$	32.6385	1.06796
$935$	0	0
$936$	25.0720	0.819503
$937$	13.1804	0.430583	0.215292	−	0.976550i	$-0.430930\pi$
$937$	13.1804	0.430583	0.215292	+	0.976550i	$0.430930\pi$
$938$	16.0715	0.524752
$939$	50.7885	1.65742
$940$	−6.01005	−0.196026
$941$	22.1738	0.722847	0.361423	−	0.932402i	$-0.382291\pi$
$941$	22.1738	0.722847	0.361423	+	0.932402i	$0.382291\pi$
$942$	16.7343	0.545233
$943$	14.7634	0.480762
$944$	−13.1332	−0.427448
$945$	−76.0381	−2.47352
$946$	0	0
$947$	43.5166	1.41410	0.707050	−	0.707164i	$-0.250026\pi$
$947$	43.5166	1.41410	0.707050	+	0.707164i	$0.250026\pi$
$948$	29.1612	0.947111
$949$	−9.79960	−0.318108
$950$	6.64904	0.215723
$951$	11.1403	0.361250
$952$	−35.3190	−1.14470
$953$	−46.4467	−1.50456	−0.752278	−	0.658846i	$-0.771045\pi$
$953$	−46.4467	−1.50456	−0.752278	+	0.658846i	$0.771045\pi$
$954$	−31.1426	−1.00828
$955$	19.3365	0.625714
$956$	7.82257	0.253000
$957$	0	0
$958$	−12.7935	−0.413338
$959$	−51.8955	−1.67579
$960$	−28.1516	−0.908589
$961$	−30.8135	−0.993983
$962$	7.35136	0.237017
$963$	26.8270	0.864490
$964$	−14.6151	−0.470720
$965$	15.2407	0.490614
$966$	36.5848	1.17710
$967$	42.3549	1.36204	0.681021	−	0.732264i	$-0.261536\pi$
$967$	42.3549	1.36204	0.681021	+	0.732264i	$0.261536\pi$
$968$	0	0
$969$	−50.9091	−1.63543
$970$	−0.380748	−0.0122251
$971$	−31.9327	−1.02477	−0.512385	−	0.858756i	$-0.671238\pi$
$971$	−31.9327	−1.02477	−0.512385	+	0.858756i	$0.671238\pi$
$972$	42.0832	1.34982
$973$	8.65760	0.277550
$974$	−30.9008	−0.990125
$975$	−3.33877	−0.106926
$976$	30.5318	0.977298
$977$	19.1141	0.611514	0.305757	−	0.952110i	$-0.401090\pi$
$977$	19.1141	0.611514	0.305757	+	0.952110i	$0.401090\pi$
$978$	70.8503	2.26554
$979$	0	0
$980$	−9.05982	−0.289405
$981$	18.1617	0.579859
$982$	−2.20497	−0.0703633
$983$	−8.03743	−0.256354	−0.128177	−	0.991751i	$-0.540913\pi$
$983$	−8.03743	−0.256354	−0.128177	+	0.991751i	$0.540913\pi$
$984$	−69.2814	−2.20861
$985$	−17.5628	−0.559596
$986$	−10.8492	−0.345508
$987$	−123.234	−3.92258
$988$	4.23466	0.134722
$989$	−16.9887	−0.540211
$990$	0	0
$991$	−22.6134	−0.718338	−0.359169	−	0.933273i	$-0.616940\pi$
$991$	−22.6134	−0.718338	−0.359169	+	0.933273i	$0.616940\pi$
$992$	−1.66146	−0.0527514
$993$	79.4635	2.52170
$994$	75.9569	2.40921
$995$	26.1437	0.828811
$996$	24.6771	0.781924
$997$	7.68488	0.243383	0.121691	−	0.992568i	$-0.461168\pi$
$997$	7.68488	0.243383	0.121691	+	0.992568i	$0.461168\pi$
$998$	−29.3691	−0.929662
$999$	111.689	3.53368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7865.2.a.w.1.4		9
11.10	odd	2		715.2.a.i.1.6	✓	9
33.32	even	2		6435.2.a.bq.1.4		9
55.54	odd	2		3575.2.a.t.1.4		9
143.142	odd	2		9295.2.a.t.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
715.2.a.i.1.6	✓	9	11.10	odd	2
3575.2.a.t.1.4		9	55.54	odd	2
6435.2.a.bq.1.4		9	33.32	even	2
7865.2.a.w.1.4		9	1.1	even	1	trivial
9295.2.a.t.1.4		9	143.142	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:	\( N \)	\(=\)	\( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7865.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13118 q^{2} -3.33877 q^{3} -0.720430 q^{4} +1.00000 q^{5} +3.77675 q^{6} +4.42443 q^{7} +3.07730 q^{8} +8.14739 q^{9} -1.13118 q^{10} +2.40535 q^{12} +1.00000 q^{13} -5.00483 q^{14} -3.33877 q^{15} -2.04012 q^{16} -2.59407 q^{17} -9.21618 q^{18} -5.87796 q^{19} -0.720430 q^{20} -14.7722 q^{21} +2.18940 q^{23} -10.2744 q^{24} +1.00000 q^{25} -1.13118 q^{26} -17.1860 q^{27} -3.18749 q^{28} -3.69729 q^{29} +3.77675 q^{30} +0.431901 q^{31} -3.84685 q^{32} +2.93436 q^{34} +4.42443 q^{35} -5.86962 q^{36} -6.49884 q^{37} +6.64904 q^{38} -3.33877 q^{39} +3.07730 q^{40} +6.74312 q^{41} +16.7100 q^{42} -7.75955 q^{43} +8.14739 q^{45} -2.47661 q^{46} +8.34232 q^{47} +6.81150 q^{48} +12.5756 q^{49} -1.13118 q^{50} +8.66101 q^{51} -0.720430 q^{52} +3.37912 q^{53} +19.4404 q^{54} +13.6153 q^{56} +19.6252 q^{57} +4.18230 q^{58} +6.43744 q^{59} +2.40535 q^{60} -14.9657 q^{61} -0.488559 q^{62} +36.0476 q^{63} +8.43173 q^{64} +1.00000 q^{65} -3.21119 q^{67} +1.86884 q^{68} -7.30990 q^{69} -5.00483 q^{70} -15.1767 q^{71} +25.0720 q^{72} -9.79960 q^{73} +7.35136 q^{74} -3.33877 q^{75} +4.23466 q^{76} +3.77675 q^{78} +12.1235 q^{79} -2.04012 q^{80} +32.9378 q^{81} -7.62768 q^{82} +10.2593 q^{83} +10.6423 q^{84} -2.59407 q^{85} +8.77745 q^{86} +12.3444 q^{87} -3.80313 q^{89} -9.21618 q^{90} +4.42443 q^{91} -1.57731 q^{92} -1.44202 q^{93} -9.43667 q^{94} -5.87796 q^{95} +12.8438 q^{96} +0.336593 q^{97} -14.2253 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 2 q^{3} + 15 q^{4} + 9 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} - q^{10} + 6 q^{12} + 9 q^{13} + 16 q^{14} + 2 q^{15} + 15 q^{16} - 13 q^{17} - 3 q^{18} + 3 q^{19} + 15 q^{20} - 14 q^{21}+ \cdots - 24 q^{98}+O(q^{100}) \) 9 * q - q^2 + 2 * q^3 + 15 * q^4 + 9 * q^5 + q^6 + 4 * q^7 - 3 * q^8 + 23 * q^9 - q^10 + 6 * q^12 + 9 * q^13 + 16 * q^14 + 2 * q^15 + 15 * q^16 - 13 * q^17 - 3 * q^18 + 3 * q^19 + 15 * q^20 - 14 * q^21 - 2 * q^24 + 9 * q^25 - q^26 + 8 * q^27 + 27 * q^28 - 16 * q^29 + q^30 + 12 * q^31 + 18 * q^32 + 14 * q^34 + 4 * q^35 - 4 * q^36 + q^37 - 9 * q^38 + 2 * q^39 - 3 * q^40 - 5 * q^41 - 11 * q^42 + 3 * q^43 + 23 * q^45 - 4 * q^46 + 27 * q^47 + q^48 + 31 * q^49 - q^50 + 15 * q^52 + 34 * q^53 + 49 * q^54 + 73 * q^56 + 16 * q^57 - 38 * q^58 - 4 * q^59 + 6 * q^60 - 30 * q^61 + 8 * q^62 + 30 * q^63 + 3 * q^64 + 9 * q^65 - 11 * q^67 - 52 * q^68 + 52 * q^69 + 16 * q^70 - 4 * q^71 + 8 * q^72 + 12 * q^73 + 16 * q^74 + 2 * q^75 + 16 * q^76 + q^78 + 16 * q^79 + 15 * q^80 + 61 * q^81 - 42 * q^82 - 34 * q^83 + 79 * q^84 - 13 * q^85 + 52 * q^86 + 64 * q^87 - 10 * q^89 - 3 * q^90 + 4 * q^91 + 7 * q^92 - 22 * q^93 + 18 * q^94 + 3 * q^95 - 15 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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