LMFDB - Embedded newform 6025.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80490 q^{2} +0.485560 q^{3} +5.86748 q^{4} -1.36195 q^{6} +4.89210 q^{7} -10.8479 q^{8} -2.76423 q^{9} +O(q^{10})$	\(q-2.80490 q^{2} +0.485560 q^{3} +5.86748 q^{4} -1.36195 q^{6} +4.89210 q^{7} -10.8479 q^{8} -2.76423 q^{9} +0.293364 q^{11} +2.84901 q^{12} -5.26146 q^{13} -13.7219 q^{14} +18.6923 q^{16} +1.23077 q^{17} +7.75340 q^{18} +0.414242 q^{19} +2.37541 q^{21} -0.822857 q^{22} -8.19596 q^{23} -5.26730 q^{24} +14.7579 q^{26} -2.79888 q^{27} +28.7043 q^{28} +4.60427 q^{29} -3.52673 q^{31} -30.7344 q^{32} +0.142446 q^{33} -3.45218 q^{34} -16.2191 q^{36} -2.98025 q^{37} -1.16191 q^{38} -2.55475 q^{39} +3.04640 q^{41} -6.66279 q^{42} -1.50980 q^{43} +1.72131 q^{44} +22.9889 q^{46} -3.34936 q^{47} +9.07625 q^{48} +16.9327 q^{49} +0.597611 q^{51} -30.8715 q^{52} +13.0652 q^{53} +7.85058 q^{54} -53.0690 q^{56} +0.201140 q^{57} -12.9145 q^{58} -5.31389 q^{59} +8.04366 q^{61} +9.89214 q^{62} -13.5229 q^{63} +48.8223 q^{64} -0.399546 q^{66} +12.6395 q^{67} +7.22150 q^{68} -3.97963 q^{69} +8.20746 q^{71} +29.9861 q^{72} -1.83449 q^{73} +8.35930 q^{74} +2.43056 q^{76} +1.43517 q^{77} +7.16583 q^{78} +7.62903 q^{79} +6.93367 q^{81} -8.54484 q^{82} -11.6196 q^{83} +13.9377 q^{84} +4.23483 q^{86} +2.23565 q^{87} -3.18238 q^{88} -0.911955 q^{89} -25.7396 q^{91} -48.0896 q^{92} -1.71244 q^{93} +9.39462 q^{94} -14.9234 q^{96} +11.3602 q^{97} -47.4945 q^{98} -0.810925 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9	$ 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9} + q^{11} - 26 q^{12} - 11 q^{13} - q^{14} + 43 q^{16} - 20 q^{17} - 18 q^{18} + 2 q^{21} - 23 q^{22} - 79 q^{23} - 2 q^{24} + 2 q^{26} - 26 q^{27} - 30 q^{28} + 2 q^{29} + q^{31} - 68 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 16 q^{37} - 45 q^{38} - 2 q^{39} - 2 q^{41} - 19 q^{42} - 25 q^{43} + 3 q^{44} + 14 q^{46} - 88 q^{47} - 75 q^{48} + 40 q^{49} - 10 q^{51} - 18 q^{52} - 34 q^{53} + 4 q^{54} - 15 q^{56} - 51 q^{57} - 53 q^{58} + q^{59} + 9 q^{61} - 39 q^{62} - 110 q^{63} + 17 q^{64} + 26 q^{66} - 30 q^{67} - 44 q^{68} - 7 q^{69} + 5 q^{71} - 18 q^{72} - 23 q^{73} - 18 q^{74} + 43 q^{76} - 30 q^{77} - 46 q^{78} + 5 q^{79} + 44 q^{81} - 5 q^{82} - 65 q^{83} - 65 q^{84} + 40 q^{86} - 33 q^{87} - 71 q^{88} - 9 q^{89} + q^{91} - 117 q^{92} - 68 q^{93} - 72 q^{94} + 83 q^{96} + 8 q^{97} - 76 q^{98} - 40 q^{99}+O(q^{100}) $ 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9 + q^11 - 26 * q^12 - 11 * q^13 - q^14 + 43 * q^16 - 20 * q^17 - 18 * q^18 + 2 * q^21 - 23 * q^22 - 79 * q^23 - 2 * q^24 + 2 * q^26 - 26 * q^27 - 30 * q^28 + 2 * q^29 + q^31 - 68 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 16 * q^37 - 45 * q^38 - 2 * q^39 - 2 * q^41 - 19 * q^42 - 25 * q^43 + 3 * q^44 + 14 * q^46 - 88 * q^47 - 75 * q^48 + 40 * q^49 - 10 * q^51 - 18 * q^52 - 34 * q^53 + 4 * q^54 - 15 * q^56 - 51 * q^57 - 53 * q^58 + q^59 + 9 * q^61 - 39 * q^62 - 110 * q^63 + 17 * q^64 + 26 * q^66 - 30 * q^67 - 44 * q^68 - 7 * q^69 + 5 * q^71 - 18 * q^72 - 23 * q^73 - 18 * q^74 + 43 * q^76 - 30 * q^77 - 46 * q^78 + 5 * q^79 + 44 * q^81 - 5 * q^82 - 65 * q^83 - 65 * q^84 + 40 * q^86 - 33 * q^87 - 71 * q^88 - 9 * q^89 + q^91 - 117 * q^92 - 68 * q^93 - 72 * q^94 + 83 * q^96 + 8 * q^97 - 76 * q^98 - 40 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.80490	−1.98337	−0.991683	−	0.128706i	$-0.958918\pi$
$2$	−2.80490	−1.98337	−0.991683	+	0.128706i	$0.958918\pi$
$3$	0.485560	0.280338	0.140169	−	0.990128i	$-0.455235\pi$
$3$	0.485560	0.280338	0.140169	+	0.990128i	$0.455235\pi$
$4$	5.86748	2.93374
$5$	0	0
$5$	0	0
$6$	−1.36195	−0.556013
$7$	4.89210	1.84904	0.924520	−	0.381133i	$-0.124466\pi$
$7$	4.89210	1.84904	0.924520	+	0.381133i	$0.124466\pi$
$8$	−10.8479	−3.83531
$9$	−2.76423	−0.921411
$10$	0	0
$11$	0.293364	0.0884525	0.0442262	−	0.999022i	$-0.485918\pi$
$11$	0.293364	0.0884525	0.0442262	+	0.999022i	$0.485918\pi$
$12$	2.84901	0.822439
$13$	−5.26146	−1.45927	−0.729633	−	0.683839i	$-0.760309\pi$
$13$	−5.26146	−1.45927	−0.729633	+	0.683839i	$0.760309\pi$
$14$	−13.7219	−3.66732
$15$	0	0
$16$	18.6923	4.67309
$17$	1.23077	0.298505	0.149252	−	0.988799i	$-0.452313\pi$
$17$	1.23077	0.298505	0.149252	+	0.988799i	$0.452313\pi$
$18$	7.75340	1.82749
$19$	0.414242	0.0950337	0.0475169	−	0.998870i	$-0.484869\pi$
$19$	0.414242	0.0950337	0.0475169	+	0.998870i	$0.484869\pi$
$20$	0	0
$21$	2.37541	0.518357
$22$	−0.822857	−0.175434
$23$	−8.19596	−1.70898	−0.854488	−	0.519471i	$-0.826129\pi$
$23$	−8.19596	−1.70898	−0.854488	+	0.519471i	$0.826129\pi$
$24$	−5.26730	−1.07518
$25$	0	0
$26$	14.7579	2.89426
$27$	−2.79888	−0.538645
$28$	28.7043	5.42460
$29$	4.60427	0.854992	0.427496	−	0.904017i	$-0.359396\pi$
$29$	4.60427	0.854992	0.427496	+	0.904017i	$0.359396\pi$
$30$	0	0
$31$	−3.52673	−0.633420	−0.316710	−	0.948522i	$-0.602578\pi$
$31$	−3.52673	−0.633420	−0.316710	+	0.948522i	$0.602578\pi$
$32$	−30.7344	−5.43313
$33$	0.142446	0.0247966
$34$	−3.45218	−0.592044
$35$	0	0
$36$	−16.2191	−2.70318
$37$	−2.98025	−0.489950	−0.244975	−	0.969529i	$-0.578780\pi$
$37$	−2.98025	−0.489950	−0.244975	+	0.969529i	$0.578780\pi$
$38$	−1.16191	−0.188487
$39$	−2.55475	−0.409088
$40$	0	0
$41$	3.04640	0.475767	0.237884	−	0.971294i	$-0.423546\pi$
$41$	3.04640	0.475767	0.237884	+	0.971294i	$0.423546\pi$
$42$	−6.66279	−1.02809
$43$	−1.50980	−0.230242	−0.115121	−	0.993351i	$-0.536726\pi$
$43$	−1.50980	−0.230242	−0.115121	+	0.993351i	$0.536726\pi$
$44$	1.72131	0.259497
$45$	0	0
$46$	22.9889	3.38952
$47$	−3.34936	−0.488554	−0.244277	−	0.969706i	$-0.578551\pi$
$47$	−3.34936	−0.488554	−0.244277	+	0.969706i	$0.578551\pi$
$48$	9.07625	1.31004
$49$	16.9327	2.41895
$50$	0	0
$51$	0.597611	0.0836823
$52$	−30.8715	−4.28111
$53$	13.0652	1.79464	0.897320	−	0.441380i	$-0.145511\pi$
$53$	13.0652	1.79464	0.897320	+	0.441380i	$0.145511\pi$
$54$	7.85058	1.06833
$55$	0	0
$56$	−53.0690	−7.09165
$57$	0.201140	0.0266416
$58$	−12.9145	−1.69576
$59$	−5.31389	−0.691810	−0.345905	−	0.938270i	$-0.612428\pi$
$59$	−5.31389	−0.691810	−0.345905	+	0.938270i	$0.612428\pi$
$60$	0	0
$61$	8.04366	1.02989	0.514943	−	0.857225i	$-0.327813\pi$
$61$	8.04366	1.02989	0.514943	+	0.857225i	$0.327813\pi$
$62$	9.89214	1.25630
$63$	−13.5229	−1.70373
$64$	48.8223	6.10279
$65$	0	0
$66$	−0.399546	−0.0491807
$67$	12.6395	1.54416	0.772079	−	0.635526i	$-0.219217\pi$
$67$	12.6395	1.54416	0.772079	+	0.635526i	$0.219217\pi$
$68$	7.22150	0.875735
$69$	−3.97963	−0.479091
$70$	0	0
$71$	8.20746	0.974047	0.487023	−	0.873389i	$-0.338083\pi$
$71$	8.20746	0.974047	0.487023	+	0.873389i	$0.338083\pi$
$72$	29.9861	3.53390
$73$	−1.83449	−0.214711	−0.107356	−	0.994221i	$-0.534238\pi$
$73$	−1.83449	−0.214711	−0.107356	+	0.994221i	$0.534238\pi$
$74$	8.35930	0.971749
$75$	0	0
$76$	2.43056	0.278804
$77$	1.43517	0.163552
$78$	7.16583	0.811371
$79$	7.62903	0.858333	0.429166	−	0.903225i	$-0.358807\pi$
$79$	7.62903	0.858333	0.429166	+	0.903225i	$0.358807\pi$
$80$	0	0
$81$	6.93367	0.770408
$82$	−8.54484	−0.943620
$83$	−11.6196	−1.27542	−0.637709	−	0.770278i	$-0.720118\pi$
$83$	−11.6196	−1.27542	−0.637709	+	0.770278i	$0.720118\pi$
$84$	13.9377	1.52072
$85$	0	0
$86$	4.23483	0.456653
$87$	2.23565	0.239687
$88$	−3.18238	−0.339243
$89$	−0.911955	−0.0966671	−0.0483335	−	0.998831i	$-0.515391\pi$
$89$	−0.911955	−0.0966671	−0.0483335	+	0.998831i	$0.515391\pi$
$90$	0	0
$91$	−25.7396	−2.69824
$92$	−48.0896	−5.01369
$93$	−1.71244	−0.177572
$94$	9.39462	0.968981
$95$	0	0
$96$	−14.9234	−1.52311
$97$	11.3602	1.15345	0.576726	−	0.816938i	$-0.304330\pi$
$97$	11.3602	1.15345	0.576726	+	0.816938i	$0.304330\pi$
$98$	−47.4945	−4.79767
$99$	−0.810925	−0.0815011
$100$	0	0
$101$	−10.3350	−1.02837	−0.514184	−	0.857680i	$-0.671905\pi$
$101$	−10.3350	−1.02837	−0.514184	+	0.857680i	$0.671905\pi$
$102$	−1.67624	−0.165973
$103$	−16.6251	−1.63812	−0.819058	−	0.573710i	$-0.805504\pi$
$103$	−16.6251	−1.63812	−0.819058	+	0.573710i	$0.805504\pi$
$104$	57.0758	5.59674
$105$	0	0
$106$	−36.6466	−3.55943
$107$	−16.5697	−1.60185	−0.800925	−	0.598765i	$-0.795658\pi$
$107$	−16.5697	−1.60185	−0.800925	+	0.598765i	$0.795658\pi$
$108$	−16.4224	−1.58024
$109$	−1.37145	−0.131361	−0.0656803	−	0.997841i	$-0.520922\pi$
$109$	−1.37145	−0.131361	−0.0656803	+	0.997841i	$0.520922\pi$
$110$	0	0
$111$	−1.44709	−0.137352
$112$	91.4448	8.64073
$113$	−1.62125	−0.152514	−0.0762570	−	0.997088i	$-0.524297\pi$
$113$	−1.62125	−0.152514	−0.0762570	+	0.997088i	$0.524297\pi$
$114$	−0.564177	−0.0528400
$115$	0	0
$116$	27.0155	2.50832
$117$	14.5439	1.34458
$118$	14.9050	1.37211
$119$	6.02104	0.551948
$120$	0	0
$121$	−10.9139	−0.992176
$122$	−22.5617	−2.04264
$123$	1.47921	0.133376
$124$	−20.6930	−1.85829
$125$	0	0
$126$	37.9304	3.37911
$127$	2.03498	0.180575	0.0902877	−	0.995916i	$-0.471221\pi$
$127$	2.03498	0.180575	0.0902877	+	0.995916i	$0.471221\pi$
$128$	−75.4730	−6.67094
$129$	−0.733096	−0.0645455
$130$	0	0
$131$	−15.6819	−1.37013	−0.685066	−	0.728481i	$-0.740227\pi$
$131$	−15.6819	−1.37013	−0.685066	+	0.728481i	$0.740227\pi$
$132$	0.835797	0.0727468
$133$	2.02652	0.175721
$134$	−35.4525	−3.06263
$135$	0	0
$136$	−13.3512	−1.14486
$137$	−8.31634	−0.710513	−0.355257	−	0.934769i	$-0.615606\pi$
$137$	−8.31634	−0.710513	−0.355257	+	0.934769i	$0.615606\pi$
$138$	11.1625	0.950213
$139$	10.7854	0.914802	0.457401	−	0.889260i	$-0.348780\pi$
$139$	10.7854	0.914802	0.457401	+	0.889260i	$0.348780\pi$
$140$	0	0
$141$	−1.62631	−0.136960
$142$	−23.0211	−1.93189
$143$	−1.54352	−0.129076
$144$	−51.6700	−4.30583
$145$	0	0
$146$	5.14557	0.425851
$147$	8.22182	0.678124
$148$	−17.4865	−1.43738
$149$	−14.4547	−1.18417	−0.592087	−	0.805874i	$-0.701696\pi$
$149$	−14.4547	−1.18417	−0.592087	+	0.805874i	$0.701696\pi$
$150$	0	0
$151$	0.715215	0.0582034	0.0291017	−	0.999576i	$-0.490735\pi$
$151$	0.715215	0.0582034	0.0291017	+	0.999576i	$0.490735\pi$
$152$	−4.49366	−0.364484
$153$	−3.40213	−0.275046
$154$	−4.02550	−0.324384
$155$	0	0
$156$	−14.9900	−1.20016
$157$	−1.56453	−0.124863	−0.0624315	−	0.998049i	$-0.519886\pi$
$157$	−1.56453	−0.124863	−0.0624315	+	0.998049i	$0.519886\pi$
$158$	−21.3987	−1.70239
$159$	6.34393	0.503106
$160$	0	0
$161$	−40.0955	−3.15997
$162$	−19.4483	−1.52800
$163$	−14.1695	−1.10984	−0.554919	−	0.831904i	$-0.687251\pi$
$163$	−14.1695	−1.10984	−0.554919	+	0.831904i	$0.687251\pi$
$164$	17.8747	1.39578
$165$	0	0
$166$	32.5919	2.52962
$167$	−22.8972	−1.77184	−0.885918	−	0.463842i	$-0.846470\pi$
$167$	−22.8972	−1.77184	−0.885918	+	0.463842i	$0.846470\pi$
$168$	−25.7682	−1.98806
$169$	14.6829	1.12946
$170$	0	0
$171$	−1.14506	−0.0875651
$172$	−8.85869	−0.675469
$173$	17.0277	1.29459	0.647295	−	0.762239i	$-0.275900\pi$
$173$	17.0277	1.29459	0.647295	+	0.762239i	$0.275900\pi$
$174$	−6.27078	−0.475387
$175$	0	0
$176$	5.48366	0.413346
$177$	−2.58021	−0.193941
$178$	2.55795	0.191726
$179$	1.17924	0.0881408	0.0440704	−	0.999028i	$-0.485967\pi$
$179$	1.17924	0.0881408	0.0440704	+	0.999028i	$0.485967\pi$
$180$	0	0
$181$	−23.7380	−1.76443	−0.882216	−	0.470845i	$-0.843949\pi$
$181$	−23.7380	−1.76443	−0.882216	+	0.470845i	$0.843949\pi$
$182$	72.1970	5.35160
$183$	3.90568	0.288716
$184$	88.9090	6.55446
$185$	0	0
$186$	4.80323	0.352190
$187$	0.361062	0.0264035
$188$	−19.6523	−1.43329
$189$	−13.6924	−0.995976
$190$	0	0
$191$	−5.67680	−0.410759	−0.205379	−	0.978682i	$-0.565843\pi$
$191$	−5.67680	−0.410759	−0.205379	+	0.978682i	$0.565843\pi$
$192$	23.7062	1.71084
$193$	3.58059	0.257736	0.128868	−	0.991662i	$-0.458866\pi$
$193$	3.58059	0.257736	0.128868	+	0.991662i	$0.458866\pi$
$194$	−31.8642	−2.28772
$195$	0	0
$196$	99.3520	7.09657
$197$	−8.96834	−0.638968	−0.319484	−	0.947592i	$-0.603510\pi$
$197$	−8.96834	−0.638968	−0.319484	+	0.947592i	$0.603510\pi$
$198$	2.27457	0.161646
$199$	−15.3249	−1.08635	−0.543176	−	0.839619i	$-0.682778\pi$
$199$	−15.3249	−1.08635	−0.543176	+	0.839619i	$0.682778\pi$
$200$	0	0
$201$	6.13722	0.432886
$202$	28.9886	2.03963
$203$	22.5246	1.58092
$204$	3.50647	0.245502
$205$	0	0
$206$	46.6317	3.24898
$207$	22.6555	1.57467
$208$	−98.3490	−6.81928
$209$	0.121524	0.00840597
$210$	0	0
$211$	20.2258	1.39240	0.696200	−	0.717848i	$-0.254873\pi$
$211$	20.2258	1.39240	0.696200	+	0.717848i	$0.254873\pi$
$212$	76.6597	5.26501
$213$	3.98521	0.273062
$214$	46.4763	3.17705
$215$	0	0
$216$	30.3620	2.06587
$217$	−17.2531	−1.17122
$218$	3.84677	0.260536
$219$	−0.890756	−0.0601917
$220$	0	0
$221$	−6.47563	−0.435598
$222$	4.05894	0.272418
$223$	2.63931	0.176741	0.0883705	−	0.996088i	$-0.471834\pi$
$223$	2.63931	0.176741	0.0883705	+	0.996088i	$0.471834\pi$
$224$	−150.356	−10.0461
$225$	0	0
$226$	4.54744	0.302491
$227$	−6.92682	−0.459749	−0.229875	−	0.973220i	$-0.573832\pi$
$227$	−6.92682	−0.459749	−0.229875	+	0.973220i	$0.573832\pi$
$228$	1.18018	0.0781594
$229$	−18.7926	−1.24185	−0.620923	−	0.783871i	$-0.713242\pi$
$229$	−18.7926	−1.24185	−0.620923	+	0.783871i	$0.713242\pi$
$230$	0	0
$231$	0.696859	0.0458499
$232$	−49.9467	−3.27916
$233$	−24.4846	−1.60404	−0.802018	−	0.597299i	$-0.796240\pi$
$233$	−24.4846	−1.60404	−0.802018	+	0.597299i	$0.796240\pi$
$234$	−40.7942	−2.66680
$235$	0	0
$236$	−31.1792	−2.02959
$237$	3.70435	0.240623
$238$	−16.8884	−1.09471
$239$	−9.79381	−0.633509	−0.316755	−	0.948508i	$-0.602593\pi$
$239$	−9.79381	−0.633509	−0.316755	+	0.948508i	$0.602593\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	30.6125	1.96785
$243$	11.7634	0.754619
$244$	47.1960	3.02142
$245$	0	0
$246$	−4.14903	−0.264533
$247$	−2.17952	−0.138680
$248$	38.2576	2.42936
$249$	−5.64201	−0.357548
$250$	0	0
$251$	−20.3241	−1.28285	−0.641424	−	0.767186i	$-0.721656\pi$
$251$	−20.3241	−1.28285	−0.641424	+	0.767186i	$0.721656\pi$
$252$	−79.3453	−4.99829
$253$	−2.40440	−0.151163
$254$	−5.70792	−0.358147
$255$	0	0
$256$	114.050	7.12812
$257$	22.5050	1.40382	0.701910	−	0.712265i	$-0.252331\pi$
$257$	22.5050	1.40382	0.701910	+	0.712265i	$0.252331\pi$
$258$	2.05626	0.128017
$259$	−14.5797	−0.905937
$260$	0	0
$261$	−12.7273	−0.787799
$262$	43.9861	2.71747
$263$	−19.0779	−1.17640	−0.588198	−	0.808717i	$-0.700163\pi$
$263$	−19.0779	−1.17640	−0.588198	+	0.808717i	$0.700163\pi$
$264$	−1.54524	−0.0951027
$265$	0	0
$266$	−5.68418	−0.348519
$267$	−0.442809	−0.0270995
$268$	74.1619	4.53016
$269$	−1.24310	−0.0757929	−0.0378965	−	0.999282i	$-0.512066\pi$
$269$	−1.24310	−0.0757929	−0.0378965	+	0.999282i	$0.512066\pi$
$270$	0	0
$271$	−19.7957	−1.20251	−0.601253	−	0.799059i	$-0.705332\pi$
$271$	−19.7957	−1.20251	−0.601253	+	0.799059i	$0.705332\pi$
$272$	23.0059	1.39494
$273$	−12.4981	−0.756420
$274$	23.3265	1.40921
$275$	0	0
$276$	−23.3504	−1.40553
$277$	−0.142971	−0.00859032	−0.00429516	−	0.999991i	$-0.501367\pi$
$277$	−0.142971	−0.00859032	−0.00429516	+	0.999991i	$0.501367\pi$
$278$	−30.2519	−1.81439
$279$	9.74870	0.583640
$280$	0	0
$281$	−20.4199	−1.21815	−0.609076	−	0.793112i	$-0.708460\pi$
$281$	−20.4199	−1.21815	−0.609076	+	0.793112i	$0.708460\pi$
$282$	4.56165	0.271642
$283$	15.0482	0.894521	0.447261	−	0.894404i	$-0.352400\pi$
$283$	15.0482	0.894521	0.447261	+	0.894404i	$0.352400\pi$
$284$	48.1571	2.85760
$285$	0	0
$286$	4.32943	0.256004
$287$	14.9033	0.879713
$288$	84.9570	5.00614
$289$	−15.4852	−0.910895
$290$	0	0
$291$	5.51605	0.323356
$292$	−10.7638	−0.629906
$293$	23.4523	1.37010	0.685049	−	0.728497i	$-0.259781\pi$
$293$	23.4523	1.37010	0.685049	+	0.728497i	$0.259781\pi$
$294$	−23.0614	−1.34497
$295$	0	0
$296$	32.3294	1.87911
$297$	−0.821090	−0.0476445
$298$	40.5440	2.34865
$299$	43.1227	2.49385
$300$	0	0
$301$	−7.38607	−0.425726
$302$	−2.00611	−0.115439
$303$	−5.01824	−0.288291
$304$	7.74316	0.444101
$305$	0	0
$306$	9.54263	0.545516
$307$	−20.3269	−1.16012	−0.580058	−	0.814575i	$-0.696970\pi$
$307$	−20.3269	−1.16012	−0.580058	+	0.814575i	$0.696970\pi$
$308$	8.42080	0.479820
$309$	−8.07247	−0.459227
$310$	0	0
$311$	−13.5008	−0.765559	−0.382779	−	0.923840i	$-0.625033\pi$
$311$	−13.5008	−0.765559	−0.382779	+	0.923840i	$0.625033\pi$
$312$	27.7137	1.56898
$313$	−10.2379	−0.578683	−0.289342	−	0.957226i	$-0.593436\pi$
$313$	−10.2379	−0.578683	−0.289342	+	0.957226i	$0.593436\pi$
$314$	4.38835	0.247649
$315$	0	0
$316$	44.7632	2.51813
$317$	10.0193	0.562742	0.281371	−	0.959599i	$-0.409211\pi$
$317$	10.0193	0.562742	0.281371	+	0.959599i	$0.409211\pi$
$318$	−17.7941	−0.997844
$319$	1.35073	0.0756262
$320$	0	0
$321$	−8.04556	−0.449060
$322$	112.464	6.26737
$323$	0.509836	0.0283680
$324$	40.6832	2.26018
$325$	0	0
$326$	39.7440	2.20122
$327$	−0.665919	−0.0368254
$328$	−33.0470	−1.82472
$329$	−16.3854	−0.903356
$330$	0	0
$331$	−16.4440	−0.903846	−0.451923	−	0.892057i	$-0.649262\pi$
$331$	−16.4440	−0.903846	−0.451923	+	0.892057i	$0.649262\pi$
$332$	−68.1778	−3.74174
$333$	8.23809	0.451445
$334$	64.2243	3.51420
$335$	0	0
$336$	44.4019	2.42232
$337$	−14.5892	−0.794725	−0.397363	−	0.917662i	$-0.630075\pi$
$337$	−14.5892	−0.794725	−0.397363	+	0.917662i	$0.630075\pi$
$338$	−41.1842	−2.24013
$339$	−0.787212	−0.0427555
$340$	0	0
$341$	−1.03462	−0.0560275
$342$	3.21179	0.173674
$343$	48.5916	2.62370
$344$	16.3781	0.883048
$345$	0	0
$346$	−47.7610	−2.56765
$347$	25.6230	1.37551	0.687757	−	0.725941i	$-0.258595\pi$
$347$	25.6230	1.37551	0.687757	+	0.725941i	$0.258595\pi$
$348$	13.1176	0.703179
$349$	21.1248	1.13078	0.565391	−	0.824823i	$-0.308725\pi$
$349$	21.1248	1.13078	0.565391	+	0.824823i	$0.308725\pi$
$350$	0	0
$351$	14.7262	0.786026
$352$	−9.01636	−0.480574
$353$	19.3369	1.02920	0.514601	−	0.857430i	$-0.327940\pi$
$353$	19.3369	1.02920	0.514601	+	0.857430i	$0.327940\pi$
$354$	7.23725	0.384655
$355$	0	0
$356$	−5.35088	−0.283596
$357$	2.92357	0.154732
$358$	−3.30766	−0.174816
$359$	−20.3503	−1.07405	−0.537023	−	0.843567i	$-0.680451\pi$
$359$	−20.3503	−1.07405	−0.537023	+	0.843567i	$0.680451\pi$
$360$	0	0
$361$	−18.8284	−0.990969
$362$	66.5828	3.49951
$363$	−5.29937	−0.278145
$364$	−151.026	−7.91594
$365$	0	0
$366$	−10.9551	−0.572630
$367$	12.2120	0.637463	0.318732	−	0.947845i	$-0.396743\pi$
$367$	12.2120	0.637463	0.318732	+	0.947845i	$0.396743\pi$
$368$	−153.202	−7.98619
$369$	−8.42094	−0.438377
$370$	0	0
$371$	63.9162	3.31836
$372$	−10.0477	−0.520949
$373$	5.25928	0.272315	0.136158	−	0.990687i	$-0.456525\pi$
$373$	5.25928	0.272315	0.136158	+	0.990687i	$0.456525\pi$
$374$	−1.01274	−0.0523678
$375$	0	0
$376$	36.3335	1.87376
$377$	−24.2252	−1.24766
$378$	38.4059	1.97538
$379$	4.78330	0.245702	0.122851	−	0.992425i	$-0.460796\pi$
$379$	4.78330	0.245702	0.122851	+	0.992425i	$0.460796\pi$
$380$	0	0
$381$	0.988105	0.0506222
$382$	15.9229	0.814684
$383$	27.0830	1.38388	0.691938	−	0.721957i	$-0.256757\pi$
$383$	27.0830	1.38388	0.691938	+	0.721957i	$0.256757\pi$
$384$	−36.6467	−1.87012
$385$	0	0
$386$	−10.0432	−0.511185
$387$	4.17342	0.212147
$388$	66.6556	3.38393
$389$	−24.1422	−1.22406	−0.612029	−	0.790835i	$-0.709646\pi$
$389$	−24.1422	−1.22406	−0.612029	+	0.790835i	$0.709646\pi$
$390$	0	0
$391$	−10.0873	−0.510138
$392$	−183.684	−9.27743
$393$	−7.61449	−0.384100
$394$	25.1553	1.26731
$395$	0	0
$396$	−4.75809	−0.239103
$397$	−2.17095	−0.108957	−0.0544784	−	0.998515i	$-0.517350\pi$
$397$	−2.17095	−0.108957	−0.0544784	+	0.998515i	$0.517350\pi$
$398$	42.9848	2.15463
$399$	0.983995	0.0492614
$400$	0	0
$401$	5.66878	0.283085	0.141543	−	0.989932i	$-0.454794\pi$
$401$	5.66878	0.283085	0.141543	+	0.989932i	$0.454794\pi$
$402$	−17.2143	−0.858572
$403$	18.5558	0.924328
$404$	−60.6402	−3.01696
$405$	0	0
$406$	−63.1792	−3.13553
$407$	−0.874296	−0.0433373
$408$	−6.48282	−0.320948
$409$	26.5294	1.31179	0.655897	−	0.754851i	$-0.272291\pi$
$409$	26.5294	1.31179	0.655897	+	0.754851i	$0.272291\pi$
$410$	0	0
$411$	−4.03808	−0.199184
$412$	−97.5472	−4.80581
$413$	−25.9961	−1.27918
$414$	−63.5466	−3.12314
$415$	0	0
$416$	161.708	7.92838
$417$	5.23694	0.256454
$418$	−0.340862	−0.0166721
$419$	−13.2165	−0.645669	−0.322835	−	0.946455i	$-0.604636\pi$
$419$	−13.2165	−0.645669	−0.322835	+	0.946455i	$0.604636\pi$
$420$	0	0
$421$	−40.0192	−1.95042	−0.975208	−	0.221289i	$-0.928974\pi$
$421$	−40.0192	−1.95042	−0.975208	+	0.221289i	$0.928974\pi$
$422$	−56.7313	−2.76164
$423$	9.25840	0.450159
$424$	−141.730	−6.88301
$425$	0	0
$426$	−11.1781	−0.541583
$427$	39.3504	1.90430
$428$	−97.2222	−4.69941
$429$	−0.749472	−0.0361848
$430$	0	0
$431$	4.81780	0.232065	0.116033	−	0.993245i	$-0.462982\pi$
$431$	4.81780	0.232065	0.116033	+	0.993245i	$0.462982\pi$
$432$	−52.3176	−2.51713
$433$	−10.5739	−0.508150	−0.254075	−	0.967185i	$-0.581771\pi$
$433$	−10.5739	−0.508150	−0.254075	+	0.967185i	$0.581771\pi$
$434$	48.3934	2.32296
$435$	0	0
$436$	−8.04692	−0.385378
$437$	−3.39512	−0.162410
$438$	2.49848	0.119382
$439$	−32.9387	−1.57208	−0.786039	−	0.618177i	$-0.787872\pi$
$439$	−32.9387	−1.57208	−0.786039	+	0.618177i	$0.787872\pi$
$440$	0	0
$441$	−46.8058	−2.22885
$442$	18.1635	0.863950
$443$	−1.00657	−0.0478237	−0.0239118	−	0.999714i	$-0.507612\pi$
$443$	−1.00657	−0.0478237	−0.0239118	+	0.999714i	$0.507612\pi$
$444$	−8.49076	−0.402954
$445$	0	0
$446$	−7.40300	−0.350542
$447$	−7.01862	−0.331969
$448$	238.844	11.2843
$449$	23.1350	1.09181	0.545904	−	0.837848i	$-0.316186\pi$
$449$	23.1350	1.09181	0.545904	+	0.837848i	$0.316186\pi$
$450$	0	0
$451$	0.893702	0.0420828
$452$	−9.51263	−0.447436
$453$	0.347280	0.0163166
$454$	19.4290	0.911850
$455$	0	0
$456$	−2.18194	−0.102179
$457$	−7.25525	−0.339386	−0.169693	−	0.985497i	$-0.554278\pi$
$457$	−7.25525	−0.339386	−0.169693	+	0.985497i	$0.554278\pi$
$458$	52.7113	2.46304
$459$	−3.44477	−0.160788
$460$	0	0
$461$	5.65731	0.263487	0.131744	−	0.991284i	$-0.457942\pi$
$461$	5.65731	0.263487	0.131744	+	0.991284i	$0.457942\pi$
$462$	−1.95462	−0.0909372
$463$	−25.3579	−1.17848	−0.589241	−	0.807957i	$-0.700573\pi$
$463$	−25.3579	−1.17848	−0.589241	+	0.807957i	$0.700573\pi$
$464$	86.0647	3.99545
$465$	0	0
$466$	68.6768	3.18139
$467$	17.8922	0.827954	0.413977	−	0.910287i	$-0.364139\pi$
$467$	17.8922	0.827954	0.413977	+	0.910287i	$0.364139\pi$
$468$	85.3360	3.94466
$469$	61.8336	2.85521
$470$	0	0
$471$	−0.759673	−0.0350039
$472$	57.6446	2.65331
$473$	−0.442919	−0.0203654
$474$	−10.3903	−0.477244
$475$	0	0
$476$	35.3283	1.61927
$477$	−36.1152	−1.65360
$478$	27.4707	1.25648
$479$	43.5447	1.98961	0.994805	−	0.101802i	$-0.0324608\pi$
$479$	43.5447	1.98961	0.994805	+	0.101802i	$0.0324608\pi$
$480$	0	0
$481$	15.6804	0.714967
$482$	−2.80490	−0.127760
$483$	−19.4688	−0.885859
$484$	−64.0373	−2.91079
$485$	0	0
$486$	−32.9951	−1.49669
$487$	−23.0345	−1.04379	−0.521897	−	0.853008i	$-0.674776\pi$
$487$	−23.0345	−1.04379	−0.521897	+	0.853008i	$0.674776\pi$
$488$	−87.2569	−3.94993
$489$	−6.88013	−0.311130
$490$	0	0
$491$	34.4555	1.55495	0.777476	−	0.628912i	$-0.216499\pi$
$491$	34.4555	1.55495	0.777476	+	0.628912i	$0.216499\pi$
$492$	8.67922	0.391289
$493$	5.66679	0.255219
$494$	6.11334	0.275052
$495$	0	0
$496$	−65.9229	−2.96002
$497$	40.1517	1.80105
$498$	15.8253	0.709149
$499$	−16.9539	−0.758961	−0.379481	−	0.925200i	$-0.623897\pi$
$499$	−16.9539	−0.758961	−0.379481	+	0.925200i	$0.623897\pi$
$500$	0	0
$501$	−11.1179	−0.496713
$502$	57.0072	2.54436
$503$	1.70105	0.0758460	0.0379230	−	0.999281i	$-0.487926\pi$
$503$	1.70105	0.0758460	0.0379230	+	0.999281i	$0.487926\pi$
$504$	146.695	6.53432
$505$	0	0
$506$	6.74410	0.299812
$507$	7.12945	0.316630
$508$	11.9402	0.529761
$509$	9.92106	0.439743	0.219872	−	0.975529i	$-0.429436\pi$
$509$	9.92106	0.439743	0.219872	+	0.975529i	$0.429436\pi$
$510$	0	0
$511$	−8.97452	−0.397010
$512$	−168.953	−7.46673
$513$	−1.15941	−0.0511894
$514$	−63.1242	−2.78429
$515$	0	0
$516$	−4.30142	−0.189360
$517$	−0.982579	−0.0432138
$518$	40.8946	1.79680
$519$	8.26796	0.362923
$520$	0	0
$521$	14.5785	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$521$	14.5785	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$522$	35.6988	1.56249
$523$	−39.5152	−1.72788	−0.863940	−	0.503595i	$-0.832010\pi$
$523$	−39.5152	−1.72788	−0.863940	+	0.503595i	$0.832010\pi$
$524$	−92.0131	−4.01961
$525$	0	0
$526$	53.5118	2.33322
$527$	−4.34059	−0.189079
$528$	2.66264	0.115877
$529$	44.1738	1.92060
$530$	0	0
$531$	14.6888	0.637441
$532$	11.8905	0.515520
$533$	−16.0285	−0.694271
$534$	1.24204	0.0537481
$535$	0	0
$536$	−137.112	−5.92233
$537$	0.572594	0.0247092
$538$	3.48676	0.150325
$539$	4.96743	0.213962
$540$	0	0
$541$	−27.7865	−1.19464	−0.597318	−	0.802004i	$-0.703767\pi$
$541$	−27.7865	−1.19464	−0.597318	+	0.802004i	$0.703767\pi$
$542$	55.5251	2.38501
$543$	−11.5262	−0.494637
$544$	−37.8269	−1.62181
$545$	0	0
$546$	35.0560	1.50026
$547$	−17.5527	−0.750498	−0.375249	−	0.926924i	$-0.622443\pi$
$547$	−17.5527	−0.750498	−0.375249	+	0.926924i	$0.622443\pi$
$548$	−48.7960	−2.08446
$549$	−22.2346	−0.948947
$550$	0	0
$551$	1.90729	0.0812531
$552$	43.1706	1.83746
$553$	37.3220	1.58709
$554$	0.401021	0.0170377
$555$	0	0
$556$	63.2829	2.68379
$557$	0.607354	0.0257344	0.0128672	−	0.999917i	$-0.495904\pi$
$557$	0.607354	0.0257344	0.0128672	+	0.999917i	$0.495904\pi$
$558$	−27.3442	−1.15757
$559$	7.94372	0.335984
$560$	0	0
$561$	0.175317	0.00740191
$562$	57.2760	2.41604
$563$	14.0579	0.592469	0.296235	−	0.955115i	$-0.404269\pi$
$563$	14.0579	0.592469	0.296235	+	0.955115i	$0.404269\pi$
$564$	−9.54235	−0.401806
$565$	0	0
$566$	−42.2087	−1.77416
$567$	33.9202	1.42452
$568$	−89.0337	−3.73577
$569$	−28.1667	−1.18081	−0.590404	−	0.807108i	$-0.701032\pi$
$569$	−28.1667	−1.18081	−0.590404	+	0.807108i	$0.701032\pi$
$570$	0	0
$571$	2.25969	0.0945652	0.0472826	−	0.998882i	$-0.484944\pi$
$571$	2.25969	0.0945652	0.0472826	+	0.998882i	$0.484944\pi$
$572$	−9.05658	−0.378674
$573$	−2.75642	−0.115151
$574$	−41.8022	−1.74479
$575$	0	0
$576$	−134.956	−5.62317
$577$	27.6667	1.15178	0.575889	−	0.817528i	$-0.304656\pi$
$577$	27.6667	1.15178	0.575889	+	0.817528i	$0.304656\pi$
$578$	43.4345	1.80664
$579$	1.73859	0.0722533
$580$	0	0
$581$	−56.8443	−2.35830
$582$	−15.4720	−0.641334
$583$	3.83285	0.158740
$584$	19.9004	0.823484
$585$	0	0
$586$	−65.7814	−2.71741
$587$	−2.92834	−0.120865	−0.0604327	−	0.998172i	$-0.519248\pi$
$587$	−2.92834	−0.120865	−0.0604327	+	0.998172i	$0.519248\pi$
$588$	48.2414	1.98944
$589$	−1.46092	−0.0601962
$590$	0	0
$591$	−4.35467	−0.179127
$592$	−55.7078	−2.28958
$593$	−30.1960	−1.24000	−0.620000	−	0.784602i	$-0.712868\pi$
$593$	−30.1960	−1.24000	−0.620000	+	0.784602i	$0.712868\pi$
$594$	2.30308	0.0944964
$595$	0	0
$596$	−84.8126	−3.47406
$597$	−7.44115	−0.304546
$598$	−120.955	−4.94622
$599$	−33.9350	−1.38654	−0.693272	−	0.720676i	$-0.743832\pi$
$599$	−33.9350	−1.38654	−0.693272	+	0.720676i	$0.743832\pi$
$600$	0	0
$601$	2.34145	0.0955096	0.0477548	−	0.998859i	$-0.484793\pi$
$601$	2.34145	0.0955096	0.0477548	+	0.998859i	$0.484793\pi$
$602$	20.7172	0.844371
$603$	−34.9385	−1.42280
$604$	4.19651	0.170754
$605$	0	0
$606$	14.0757	0.571786
$607$	22.5745	0.916269	0.458135	−	0.888883i	$-0.348518\pi$
$607$	22.5745	0.916269	0.458135	+	0.888883i	$0.348518\pi$
$608$	−12.7315	−0.516330
$609$	10.9370	0.443191
$610$	0	0
$611$	17.6225	0.712930
$612$	−19.9619	−0.806912
$613$	−17.1684	−0.693426	−0.346713	−	0.937971i	$-0.612702\pi$
$613$	−17.1684	−0.693426	−0.346713	+	0.937971i	$0.612702\pi$
$614$	57.0149	2.30094
$615$	0	0
$616$	−15.5685	−0.627274
$617$	10.4893	0.422282	0.211141	−	0.977456i	$-0.432282\pi$
$617$	10.4893	0.422282	0.211141	+	0.977456i	$0.432282\pi$
$618$	22.6425	0.910814
$619$	19.3404	0.777355	0.388678	−	0.921374i	$-0.372932\pi$
$619$	19.3404	0.777355	0.388678	+	0.921374i	$0.372932\pi$
$620$	0	0
$621$	22.9395	0.920531
$622$	37.8684	1.51838
$623$	−4.46138	−0.178741
$624$	−47.7543	−1.91170
$625$	0	0
$626$	28.7164	1.14774
$627$	0.0590070	0.00235651
$628$	−9.17985	−0.366316
$629$	−3.66799	−0.146252
$630$	0	0
$631$	41.3811	1.64736	0.823678	−	0.567058i	$-0.191918\pi$
$631$	41.3811	1.64736	0.823678	+	0.567058i	$0.191918\pi$
$632$	−82.7590	−3.29197
$633$	9.82082	0.390343
$634$	−28.1033	−1.11612
$635$	0	0
$636$	37.2229	1.47598
$637$	−89.0905	−3.52989
$638$	−3.78866	−0.149994
$639$	−22.6873	−0.897497
$640$	0	0
$641$	1.42372	0.0562335	0.0281167	−	0.999605i	$-0.491049\pi$
$641$	1.42372	0.0562335	0.0281167	+	0.999605i	$0.491049\pi$
$642$	22.5670	0.890649
$643$	−15.7532	−0.621246	−0.310623	−	0.950533i	$-0.600538\pi$
$643$	−15.7532	−0.621246	−0.310623	+	0.950533i	$0.600538\pi$
$644$	−235.259	−9.27052
$645$	0	0
$646$	−1.43004	−0.0562642
$647$	−2.54574	−0.100083	−0.0500417	−	0.998747i	$-0.515935\pi$
$647$	−2.54574	−0.100083	−0.0500417	+	0.998747i	$0.515935\pi$
$648$	−75.2158	−2.95475
$649$	−1.55890	−0.0611923
$650$	0	0
$651$	−8.37743	−0.328337
$652$	−83.1391	−3.25598
$653$	−16.8077	−0.657736	−0.328868	−	0.944376i	$-0.606667\pi$
$653$	−16.8077	−0.657736	−0.328868	+	0.944376i	$0.606667\pi$
$654$	1.86784	0.0730382
$655$	0	0
$656$	56.9443	2.22330
$657$	5.07096	0.197837
$658$	45.9594	1.79168
$659$	−0.968386	−0.0377230	−0.0188615	−	0.999822i	$-0.506004\pi$
$659$	−0.968386	−0.0377230	−0.0188615	+	0.999822i	$0.506004\pi$
$660$	0	0
$661$	17.6448	0.686305	0.343152	−	0.939280i	$-0.388505\pi$
$661$	17.6448	0.686305	0.343152	+	0.939280i	$0.388505\pi$
$662$	46.1239	1.79266
$663$	−3.14431	−0.122115
$664$	126.048	4.89162
$665$	0	0
$666$	−23.1070	−0.895380
$667$	−37.7365	−1.46116
$668$	−134.349	−5.19810
$669$	1.28154	0.0495472
$670$	0	0
$671$	2.35972	0.0910959
$672$	−73.0068	−2.81630
$673$	−18.9836	−0.731764	−0.365882	−	0.930661i	$-0.619233\pi$
$673$	−18.9836	−0.731764	−0.365882	+	0.930661i	$0.619233\pi$
$674$	40.9213	1.57623
$675$	0	0
$676$	86.1519	3.31353
$677$	16.2408	0.624186	0.312093	−	0.950052i	$-0.398970\pi$
$677$	16.2408	0.624186	0.312093	+	0.950052i	$0.398970\pi$
$678$	2.20805	0.0847998
$679$	55.5752	2.13278
$680$	0	0
$681$	−3.36338	−0.128885
$682$	2.90199	0.111123
$683$	−7.58433	−0.290206	−0.145103	−	0.989417i	$-0.546351\pi$
$683$	−7.58433	−0.290206	−0.145103	+	0.989417i	$0.546351\pi$
$684$	−6.71863	−0.256893
$685$	0	0
$686$	−136.295	−5.20375
$687$	−9.12491	−0.348137
$688$	−28.2216	−1.07594
$689$	−68.7419	−2.61886
$690$	0	0
$691$	−36.2043	−1.37728	−0.688639	−	0.725104i	$-0.741791\pi$
$691$	−36.2043	−1.37728	−0.688639	+	0.725104i	$0.741791\pi$
$692$	99.9095	3.79799
$693$	−3.96713	−0.150699
$694$	−71.8700	−2.72815
$695$	0	0
$696$	−24.2521	−0.919274
$697$	3.74940	0.142019
$698$	−59.2529	−2.24276
$699$	−11.8887	−0.449673
$700$	0	0
$701$	−1.45134	−0.0548164	−0.0274082	−	0.999624i	$-0.508725\pi$
$701$	−1.45134	−0.0548164	−0.0274082	+	0.999624i	$0.508725\pi$
$702$	−41.3055	−1.55898
$703$	−1.23454	−0.0465617
$704$	14.3227	0.539807
$705$	0	0
$706$	−54.2382	−2.04128
$707$	−50.5597	−1.90149
$708$	−15.1393	−0.568972
$709$	−2.61113	−0.0980631	−0.0490315	−	0.998797i	$-0.515613\pi$
$709$	−2.61113	−0.0980631	−0.0490315	+	0.998797i	$0.515613\pi$
$710$	0	0
$711$	−21.0884	−0.790877
$712$	9.89280	0.370748
$713$	28.9050	1.08250
$714$	−8.20034	−0.306890
$715$	0	0
$716$	6.91919	0.258582
$717$	−4.75548	−0.177597
$718$	57.0805	2.13023
$719$	45.4376	1.69454	0.847269	−	0.531164i	$-0.178245\pi$
$719$	45.4376	1.69454	0.847269	+	0.531164i	$0.178245\pi$
$720$	0	0
$721$	−81.3315	−3.02894
$722$	52.8118	1.96545
$723$	0.485560	0.0180582
$724$	−139.282	−5.17638
$725$	0	0
$726$	14.8642	0.551663
$727$	18.0545	0.669603	0.334801	−	0.942289i	$-0.391331\pi$
$727$	18.0545	0.669603	0.334801	+	0.942289i	$0.391331\pi$
$728$	279.220	10.3486
$729$	−15.0892	−0.558859
$730$	0	0
$731$	−1.85821	−0.0687282
$732$	22.9165	0.847018
$733$	43.4237	1.60389	0.801946	−	0.597396i	$-0.203798\pi$
$733$	43.4237	1.60389	0.801946	+	0.597396i	$0.203798\pi$
$734$	−34.2536	−1.26432
$735$	0	0
$736$	251.898	9.28508
$737$	3.70796	0.136585
$738$	23.6199	0.869462
$739$	−5.78820	−0.212922	−0.106461	−	0.994317i	$-0.533952\pi$
$739$	−5.78820	−0.212922	−0.106461	+	0.994317i	$0.533952\pi$
$740$	0	0
$741$	−1.05829	−0.0388772
$742$	−179.279	−6.58153
$743$	−14.7525	−0.541217	−0.270608	−	0.962690i	$-0.587225\pi$
$743$	−14.7525	−0.541217	−0.270608	+	0.962690i	$0.587225\pi$
$744$	18.5764	0.681043
$745$	0	0
$746$	−14.7518	−0.540101
$747$	32.1193	1.17518
$748$	2.11853	0.0774610
$749$	−81.0605	−2.96189
$750$	0	0
$751$	24.3152	0.887275	0.443637	−	0.896206i	$-0.353688\pi$
$751$	24.3152	0.887275	0.443637	+	0.896206i	$0.353688\pi$
$752$	−62.6073	−2.28305
$753$	−9.86859	−0.359631
$754$	67.9493	2.47457
$755$	0	0
$756$	−80.3399	−2.92193
$757$	−30.2691	−1.10015	−0.550074	−	0.835116i	$-0.685401\pi$
$757$	−30.2691	−1.10015	−0.550074	+	0.835116i	$0.685401\pi$
$758$	−13.4167	−0.487317
$759$	−1.16748	−0.0423768
$760$	0	0
$761$	−21.7886	−0.789835	−0.394917	−	0.918717i	$-0.629227\pi$
$761$	−21.7886	−0.789835	−0.394917	+	0.918717i	$0.629227\pi$
$762$	−2.77154	−0.100402
$763$	−6.70925	−0.242891
$764$	−33.3085	−1.20506
$765$	0	0
$766$	−75.9651	−2.74473
$767$	27.9588	1.00953
$768$	55.3780	1.99828
$769$	−40.3622	−1.45550	−0.727749	−	0.685844i	$-0.759433\pi$
$769$	−40.3622	−1.45550	−0.727749	+	0.685844i	$0.759433\pi$
$770$	0	0
$771$	10.9275	0.393544
$772$	21.0090	0.756131
$773$	41.4770	1.49182	0.745912	−	0.666044i	$-0.232014\pi$
$773$	41.4770	1.49182	0.745912	+	0.666044i	$0.232014\pi$
$774$	−11.7060	−0.420765
$775$	0	0
$776$	−123.234	−4.42385
$777$	−7.07930	−0.253969
$778$	67.7165	2.42775
$779$	1.26195	0.0452139
$780$	0	0
$781$	2.40777	0.0861569
$782$	28.2939	1.01179
$783$	−12.8868	−0.460537
$784$	316.511	11.3040
$785$	0	0
$786$	21.3579	0.761811
$787$	3.16267	0.112737	0.0563685	−	0.998410i	$-0.482048\pi$
$787$	3.16267	0.112737	0.0563685	+	0.998410i	$0.482048\pi$
$788$	−52.6215	−1.87456
$789$	−9.26348	−0.329789
$790$	0	0
$791$	−7.93130	−0.282005
$792$	8.79684	0.312582
$793$	−42.3214	−1.50288
$794$	6.08929	0.216101
$795$	0	0
$796$	−89.9184	−3.18707
$797$	−44.2274	−1.56661	−0.783307	−	0.621635i	$-0.786469\pi$
$797$	−44.2274	−1.56661	−0.783307	+	0.621635i	$0.786469\pi$
$798$	−2.76001	−0.0977033
$799$	−4.12228	−0.145836
$800$	0	0
$801$	2.52086	0.0890701
$802$	−15.9004	−0.561462
$803$	−0.538173	−0.0189917
$804$	36.0100	1.26998
$805$	0	0
$806$	−52.0471	−1.83328
$807$	−0.603598	−0.0212476
$808$	112.113	3.94411
$809$	39.8111	1.39968	0.699841	−	0.714298i	$-0.253254\pi$
$809$	39.8111	1.39968	0.699841	+	0.714298i	$0.253254\pi$
$810$	0	0
$811$	−7.47104	−0.262344	−0.131172	−	0.991360i	$-0.541874\pi$
$811$	−7.47104	−0.262344	−0.131172	+	0.991360i	$0.541874\pi$
$812$	132.162	4.63799
$813$	−9.61202	−0.337108
$814$	2.45232	0.0859536
$815$	0	0
$816$	11.1708	0.391055
$817$	−0.625421	−0.0218807
$818$	−74.4123	−2.60177
$819$	71.1502	2.48619
$820$	0	0
$821$	23.4576	0.818674	0.409337	−	0.912383i	$-0.365760\pi$
$821$	23.4576	0.818674	0.409337	+	0.912383i	$0.365760\pi$
$822$	11.3264	0.395055
$823$	26.4031	0.920356	0.460178	−	0.887827i	$-0.347786\pi$
$823$	26.4031	0.920356	0.460178	+	0.887827i	$0.347786\pi$
$824$	180.347	6.28269
$825$	0	0
$826$	72.9166	2.53709
$827$	32.5331	1.13129	0.565644	−	0.824650i	$-0.308628\pi$
$827$	32.5331	1.13129	0.565644	+	0.824650i	$0.308628\pi$
$828$	132.931	4.61967
$829$	−10.2130	−0.354711	−0.177356	−	0.984147i	$-0.556754\pi$
$829$	−10.2130	−0.354711	−0.177356	+	0.984147i	$0.556754\pi$
$830$	0	0
$831$	−0.0694212	−0.00240819
$832$	−256.877	−8.90559
$833$	20.8402	0.722069
$834$	−14.6891	−0.508642
$835$	0	0
$836$	0.713038	0.0246609
$837$	9.87090	0.341188
$838$	37.0711	1.28060
$839$	−18.8360	−0.650291	−0.325146	−	0.945664i	$-0.605413\pi$
$839$	−18.8360	−0.650291	−0.325146	+	0.945664i	$0.605413\pi$
$840$	0	0
$841$	−7.80066	−0.268988
$842$	112.250	3.86839
$843$	−9.91511	−0.341494
$844$	118.674	4.08494
$845$	0	0
$846$	−25.9689	−0.892829
$847$	−53.3921	−1.83457
$848$	244.219	8.38651
$849$	7.30679	0.250768
$850$	0	0
$851$	24.4260	0.837312
$852$	23.3832	0.801094
$853$	−47.4205	−1.62365	−0.811824	−	0.583902i	$-0.801525\pi$
$853$	−47.4205	−1.62365	−0.811824	+	0.583902i	$0.801525\pi$
$854$	−110.374	−3.77692
$855$	0	0
$856$	179.746	6.14359
$857$	46.9658	1.60432	0.802161	−	0.597108i	$-0.203684\pi$
$857$	46.9658	1.60432	0.802161	+	0.597108i	$0.203684\pi$
$858$	2.10220	0.0717678
$859$	−2.42136	−0.0826158	−0.0413079	−	0.999146i	$-0.513152\pi$
$859$	−2.42136	−0.0826158	−0.0413079	+	0.999146i	$0.513152\pi$
$860$	0	0
$861$	7.23643	0.246617
$862$	−13.5134	−0.460270
$863$	−24.1408	−0.821762	−0.410881	−	0.911689i	$-0.634779\pi$
$863$	−24.1408	−0.821762	−0.410881	+	0.911689i	$0.634779\pi$
$864$	86.0219	2.92652
$865$	0	0
$866$	29.6588	1.00785
$867$	−7.51900	−0.255359
$868$	−101.232	−3.43605
$869$	2.23808	0.0759217
$870$	0	0
$871$	−66.5021	−2.25334
$872$	14.8773	0.503809
$873$	−31.4022	−1.06280
$874$	9.52297	0.322119
$875$	0	0
$876$	−5.22649	−0.176587
$877$	−4.71231	−0.159123	−0.0795617	−	0.996830i	$-0.525352\pi$
$877$	−4.71231	−0.159123	−0.0795617	+	0.996830i	$0.525352\pi$
$878$	92.3898	3.11801
$879$	11.3875	0.384091
$880$	0	0
$881$	30.4040	1.02434	0.512169	−	0.858885i	$-0.328842\pi$
$881$	30.4040	1.02434	0.512169	+	0.858885i	$0.328842\pi$
$882$	131.286	4.42062
$883$	−41.5977	−1.39987	−0.699937	−	0.714205i	$-0.746788\pi$
$883$	−41.5977	−1.39987	−0.699937	+	0.714205i	$0.746788\pi$
$884$	−37.9956	−1.27793
$885$	0	0
$886$	2.82334	0.0948518
$887$	6.61151	0.221993	0.110996	−	0.993821i	$-0.464596\pi$
$887$	6.61151	0.221993	0.110996	+	0.993821i	$0.464596\pi$
$888$	15.6979	0.526786
$889$	9.95533	0.333891
$890$	0	0
$891$	2.03409	0.0681445
$892$	15.4861	0.518512
$893$	−1.38745	−0.0464291
$894$	19.6865	0.658416
$895$	0	0
$896$	−369.222	−12.3348
$897$	20.9387	0.699121
$898$	−64.8914	−2.16545
$899$	−16.2380	−0.541569
$900$	0	0
$901$	16.0802	0.535709
$902$	−2.50675	−0.0834656
$903$	−3.58638	−0.119347
$904$	17.5871	0.584939
$905$	0	0
$906$	−0.974086	−0.0323618
$907$	52.9504	1.75819	0.879095	−	0.476647i	$-0.158148\pi$
$907$	52.9504	1.75819	0.879095	+	0.476647i	$0.158148\pi$
$908$	−40.6430	−1.34878
$909$	28.5682	0.947548
$910$	0	0
$911$	25.7946	0.854613	0.427306	−	0.904107i	$-0.359463\pi$
$911$	25.7946	0.854613	0.427306	+	0.904107i	$0.359463\pi$
$912$	3.75977	0.124498
$913$	−3.40877	−0.112814
$914$	20.3503	0.673127
$915$	0	0
$916$	−110.265	−3.64325
$917$	−76.7173	−2.53343
$918$	9.66224	0.318901
$919$	34.2783	1.13074	0.565368	−	0.824839i	$-0.308734\pi$
$919$	34.2783	1.13074	0.565368	+	0.824839i	$0.308734\pi$
$920$	0	0
$921$	−9.86992	−0.325225
$922$	−15.8682	−0.522592
$923$	−43.1832	−1.42139
$924$	4.08880	0.134512
$925$	0	0
$926$	71.1264	2.33736
$927$	45.9555	1.50938
$928$	−141.510	−4.64528
$929$	−18.2685	−0.599371	−0.299686	−	0.954038i	$-0.596882\pi$
$929$	−18.2685	−0.599371	−0.299686	+	0.954038i	$0.596882\pi$
$930$	0	0
$931$	7.01423	0.229882
$932$	−143.663	−4.70583
$933$	−6.55543	−0.214615
$934$	−50.1860	−1.64214
$935$	0	0
$936$	−157.771	−5.15690
$937$	−40.1480	−1.31158	−0.655789	−	0.754944i	$-0.727664\pi$
$937$	−40.1480	−1.31158	−0.655789	+	0.754944i	$0.727664\pi$
$938$	−173.437	−5.66293
$939$	−4.97114	−0.162227
$940$	0	0
$941$	38.5010	1.25510	0.627549	−	0.778577i	$-0.284058\pi$
$941$	38.5010	1.25510	0.627549	+	0.778577i	$0.284058\pi$
$942$	2.13081	0.0694255
$943$	−24.9681	−0.813075
$944$	−99.3291	−3.23289
$945$	0	0
$946$	1.24234	0.0403921
$947$	−30.0400	−0.976168	−0.488084	−	0.872797i	$-0.662304\pi$
$947$	−30.0400	−0.976168	−0.488084	+	0.872797i	$0.662304\pi$
$948$	21.7352	0.705926
$949$	9.65211	0.313321
$950$	0	0
$951$	4.86499	0.157758
$952$	−65.3156	−2.11689
$953$	−36.2218	−1.17334	−0.586670	−	0.809826i	$-0.699561\pi$
$953$	−36.2218	−1.17334	−0.586670	+	0.809826i	$0.699561\pi$
$954$	101.300	3.27970
$955$	0	0
$956$	−57.4650	−1.85855
$957$	0.655859	0.0212009
$958$	−122.139	−3.94612
$959$	−40.6844	−1.31377
$960$	0	0
$961$	−18.5622	−0.598779
$962$	−43.9821	−1.41804
$963$	45.8024	1.47596
$964$	5.86748	0.188979
$965$	0	0
$966$	54.6080	1.75698
$967$	−31.7730	−1.02175	−0.510876	−	0.859654i	$-0.670679\pi$
$967$	−31.7730	−1.02175	−0.510876	+	0.859654i	$0.670679\pi$
$968$	118.393	3.80530
$969$	0.247556	0.00795264
$970$	0	0
$971$	44.4074	1.42510	0.712551	−	0.701620i	$-0.247540\pi$
$971$	44.4074	1.42510	0.712551	+	0.701620i	$0.247540\pi$
$972$	69.0212	2.21386
$973$	52.7631	1.69151
$974$	64.6097	2.07023
$975$	0	0
$976$	150.355	4.81274
$977$	−2.66944	−0.0854028	−0.0427014	−	0.999088i	$-0.513596\pi$
$977$	−2.66944	−0.0854028	−0.0427014	+	0.999088i	$0.513596\pi$
$978$	19.2981	0.617085
$979$	−0.267535	−0.00855044
$980$	0	0
$981$	3.79099	0.121037
$982$	−96.6442	−3.08404
$983$	−36.0653	−1.15030	−0.575152	−	0.818047i	$-0.695057\pi$
$983$	−36.0653	−1.15030	−0.575152	+	0.818047i	$0.695057\pi$
$984$	−16.0463	−0.511537
$985$	0	0
$986$	−15.8948	−0.506193
$987$	−7.95609	−0.253245
$988$	−12.7883	−0.406849
$989$	12.3742	0.393477
$990$	0	0
$991$	−45.8246	−1.45567	−0.727833	−	0.685754i	$-0.759472\pi$
$991$	−45.8246	−1.45567	−0.727833	+	0.685754i	$0.759472\pi$
$992$	108.392	3.44145
$993$	−7.98456	−0.253382
$994$	−112.622	−3.57214
$995$	0	0
$996$	−33.1044	−1.04895
$997$	−45.1187	−1.42892	−0.714462	−	0.699674i	$-0.753328\pi$
$997$	−45.1187	−1.42892	−0.714462	+	0.699674i	$0.753328\pi$
$998$	47.5541	1.50530
$999$	8.34135	0.263909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.m.1.1	✓	40
5.4	even	2		6025.2.a.n.1.40	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.1	✓	40	1.1	even	1	trivial
6025.2.a.n.1.40	yes	40	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80490 q^{2} +0.485560 q^{3} +5.86748 q^{4} -1.36195 q^{6} +4.89210 q^{7} -10.8479 q^{8} -2.76423 q^{9} +O(q^{10})\)	\(q-2.80490 q^{2} +0.485560 q^{3} +5.86748 q^{4} -1.36195 q^{6} +4.89210 q^{7} -10.8479 q^{8} -2.76423 q^{9} +0.293364 q^{11} +2.84901 q^{12} -5.26146 q^{13} -13.7219 q^{14} +18.6923 q^{16} +1.23077 q^{17} +7.75340 q^{18} +0.414242 q^{19} +2.37541 q^{21} -0.822857 q^{22} -8.19596 q^{23} -5.26730 q^{24} +14.7579 q^{26} -2.79888 q^{27} +28.7043 q^{28} +4.60427 q^{29} -3.52673 q^{31} -30.7344 q^{32} +0.142446 q^{33} -3.45218 q^{34} -16.2191 q^{36} -2.98025 q^{37} -1.16191 q^{38} -2.55475 q^{39} +3.04640 q^{41} -6.66279 q^{42} -1.50980 q^{43} +1.72131 q^{44} +22.9889 q^{46} -3.34936 q^{47} +9.07625 q^{48} +16.9327 q^{49} +0.597611 q^{51} -30.8715 q^{52} +13.0652 q^{53} +7.85058 q^{54} -53.0690 q^{56} +0.201140 q^{57} -12.9145 q^{58} -5.31389 q^{59} +8.04366 q^{61} +9.89214 q^{62} -13.5229 q^{63} +48.8223 q^{64} -0.399546 q^{66} +12.6395 q^{67} +7.22150 q^{68} -3.97963 q^{69} +8.20746 q^{71} +29.9861 q^{72} -1.83449 q^{73} +8.35930 q^{74} +2.43056 q^{76} +1.43517 q^{77} +7.16583 q^{78} +7.62903 q^{79} +6.93367 q^{81} -8.54484 q^{82} -11.6196 q^{83} +13.9377 q^{84} +4.23483 q^{86} +2.23565 q^{87} -3.18238 q^{88} -0.911955 q^{89} -25.7396 q^{91} -48.0896 q^{92} -1.71244 q^{93} +9.39462 q^{94} -14.9234 q^{96} +11.3602 q^{97} -47.4945 q^{98} -0.810925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9} + q^{11} - 26 q^{12} - 11 q^{13} - q^{14} + 43 q^{16} - 20 q^{17} - 18 q^{18} + 2 q^{21} - 23 q^{22} - 79 q^{23} - 2 q^{24} + 2 q^{26} - 26 q^{27} - 30 q^{28} + 2 q^{29} + q^{31} - 68 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 16 q^{37} - 45 q^{38} - 2 q^{39} - 2 q^{41} - 19 q^{42} - 25 q^{43} + 3 q^{44} + 14 q^{46} - 88 q^{47} - 75 q^{48} + 40 q^{49} - 10 q^{51} - 18 q^{52} - 34 q^{53} + 4 q^{54} - 15 q^{56} - 51 q^{57} - 53 q^{58} + q^{59} + 9 q^{61} - 39 q^{62} - 110 q^{63} + 17 q^{64} + 26 q^{66} - 30 q^{67} - 44 q^{68} - 7 q^{69} + 5 q^{71} - 18 q^{72} - 23 q^{73} - 18 q^{74} + 43 q^{76} - 30 q^{77} - 46 q^{78} + 5 q^{79} + 44 q^{81} - 5 q^{82} - 65 q^{83} - 65 q^{84} + 40 q^{86} - 33 q^{87} - 71 q^{88} - 9 q^{89} + q^{91} - 117 q^{92} - 68 q^{93} - 72 q^{94} + 83 q^{96} + 8 q^{97} - 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9 + q^11 - 26 * q^12 - 11 * q^13 - q^14 + 43 * q^16 - 20 * q^17 - 18 * q^18 + 2 * q^21 - 23 * q^22 - 79 * q^23 - 2 * q^24 + 2 * q^26 - 26 * q^27 - 30 * q^28 + 2 * q^29 + q^31 - 68 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 16 * q^37 - 45 * q^38 - 2 * q^39 - 2 * q^41 - 19 * q^42 - 25 * q^43 + 3 * q^44 + 14 * q^46 - 88 * q^47 - 75 * q^48 + 40 * q^49 - 10 * q^51 - 18 * q^52 - 34 * q^53 + 4 * q^54 - 15 * q^56 - 51 * q^57 - 53 * q^58 + q^59 + 9 * q^61 - 39 * q^62 - 110 * q^63 + 17 * q^64 + 26 * q^66 - 30 * q^67 - 44 * q^68 - 7 * q^69 + 5 * q^71 - 18 * q^72 - 23 * q^73 - 18 * q^74 + 43 * q^76 - 30 * q^77 - 46 * q^78 + 5 * q^79 + 44 * q^81 - 5 * q^82 - 65 * q^83 - 65 * q^84 + 40 * q^86 - 33 * q^87 - 71 * q^88 - 9 * q^89 + q^91 - 117 * q^92 - 68 * q^93 - 72 * q^94 + 83 * q^96 + 8 * q^97 - 76 * q^98 - 40 * q^99

Introduction

L-functions

Modular forms

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Database

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