LMFDB - Embedded newform 6025.2.a.j.1.7

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:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.96718 q^{2} -2.44636 q^{3} +1.86979 q^{4} +4.81242 q^{6} -2.28114 q^{7} +0.256155 q^{8} +2.98466 q^{9} +O(q^{10})$	\(q-1.96718 q^{2} -2.44636 q^{3} +1.86979 q^{4} +4.81242 q^{6} -2.28114 q^{7} +0.256155 q^{8} +2.98466 q^{9} +4.45596 q^{11} -4.57416 q^{12} +2.62871 q^{13} +4.48741 q^{14} -4.24347 q^{16} +5.62386 q^{17} -5.87136 q^{18} -4.92868 q^{19} +5.58049 q^{21} -8.76566 q^{22} +2.14068 q^{23} -0.626646 q^{24} -5.17115 q^{26} +0.0375205 q^{27} -4.26525 q^{28} -1.59959 q^{29} -3.29431 q^{31} +7.83535 q^{32} -10.9009 q^{33} -11.0631 q^{34} +5.58068 q^{36} +3.32475 q^{37} +9.69558 q^{38} -6.43077 q^{39} +4.94337 q^{41} -10.9778 q^{42} -5.48142 q^{43} +8.33169 q^{44} -4.21109 q^{46} -9.60820 q^{47} +10.3811 q^{48} -1.79639 q^{49} -13.7580 q^{51} +4.91513 q^{52} -4.43708 q^{53} -0.0738095 q^{54} -0.584326 q^{56} +12.0573 q^{57} +3.14667 q^{58} -2.35301 q^{59} +7.22425 q^{61} +6.48049 q^{62} -6.80844 q^{63} -6.92658 q^{64} +21.4439 q^{66} -10.6210 q^{67} +10.5154 q^{68} -5.23686 q^{69} +3.03152 q^{71} +0.764536 q^{72} -10.4106 q^{73} -6.54038 q^{74} -9.21557 q^{76} -10.1647 q^{77} +12.6505 q^{78} -7.36888 q^{79} -9.04578 q^{81} -9.72448 q^{82} -1.30901 q^{83} +10.4343 q^{84} +10.7829 q^{86} +3.91317 q^{87} +1.14142 q^{88} +14.3314 q^{89} -5.99647 q^{91} +4.00260 q^{92} +8.05906 q^{93} +18.9010 q^{94} -19.1681 q^{96} -8.92318 q^{97} +3.53382 q^{98} +13.2995 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * 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 $

$2$	−1.96718	−1.39100	−0.695502	−	0.718524i	$-0.744818\pi$
$2$	−1.96718	−1.39100	−0.695502	+	0.718524i	$0.744818\pi$
$3$	−2.44636	−1.41240	−0.706202	−	0.708010i	$-0.749593\pi$
$3$	−2.44636	−1.41240	−0.706202	+	0.708010i	$0.749593\pi$
$4$	1.86979	0.934893
$5$	0	0
$5$	0	0
$6$	4.81242	1.96466
$7$	−2.28114	−0.862191	−0.431095	−	0.902306i	$-0.641873\pi$
$7$	−2.28114	−0.862191	−0.431095	+	0.902306i	$0.641873\pi$
$8$	0.256155	0.0905644
$9$	2.98466	0.994888
$10$	0	0
$11$	4.45596	1.34352	0.671761	−	0.740768i	$-0.265538\pi$
$11$	4.45596	1.34352	0.671761	+	0.740768i	$0.265538\pi$
$12$	−4.57416	−1.32045
$13$	2.62871	0.729074	0.364537	−	0.931189i	$-0.381227\pi$
$13$	2.62871	0.729074	0.364537	+	0.931189i	$0.381227\pi$
$14$	4.48741	1.19931
$15$	0	0
$16$	−4.24347	−1.06087
$17$	5.62386	1.36399	0.681993	−	0.731358i	$-0.261113\pi$
$17$	5.62386	1.36399	0.681993	+	0.731358i	$0.261113\pi$
$18$	−5.87136	−1.38389
$19$	−4.92868	−1.13072	−0.565358	−	0.824845i	$-0.691262\pi$
$19$	−4.92868	−1.13072	−0.565358	+	0.824845i	$0.691262\pi$
$20$	0	0
$21$	5.58049	1.21776
$22$	−8.76566	−1.86884
$23$	2.14068	0.446362	0.223181	−	0.974777i	$-0.428356\pi$
$23$	2.14068	0.446362	0.223181	+	0.974777i	$0.428356\pi$
$24$	−0.626646	−0.127914
$25$	0	0
$26$	−5.17115	−1.01415
$27$	0.0375205	0.00722082
$28$	−4.26525	−0.806056
$29$	−1.59959	−0.297036	−0.148518	−	0.988910i	$-0.547450\pi$
$29$	−1.59959	−0.297036	−0.148518	+	0.988910i	$0.547450\pi$
$30$	0	0
$31$	−3.29431	−0.591676	−0.295838	−	0.955238i	$-0.595599\pi$
$31$	−3.29431	−0.591676	−0.295838	+	0.955238i	$0.595599\pi$
$32$	7.83535	1.38511
$33$	−10.9009	−1.89760
$34$	−11.0631	−1.89731
$35$	0	0
$36$	5.58068	0.930113
$37$	3.32475	0.546586	0.273293	−	0.961931i	$-0.411887\pi$
$37$	3.32475	0.546586	0.273293	+	0.961931i	$0.411887\pi$
$38$	9.69558	1.57283
$39$	−6.43077	−1.02975
$40$	0	0
$41$	4.94337	0.772025	0.386012	−	0.922494i	$-0.373852\pi$
$41$	4.94337	0.772025	0.386012	+	0.922494i	$0.373852\pi$
$42$	−10.9778	−1.69391
$43$	−5.48142	−0.835909	−0.417954	−	0.908468i	$-0.637253\pi$
$43$	−5.48142	−0.835909	−0.417954	+	0.908468i	$0.637253\pi$
$44$	8.33169	1.25605
$45$	0	0
$46$	−4.21109	−0.620891
$47$	−9.60820	−1.40150	−0.700750	−	0.713407i	$-0.747151\pi$
$47$	−9.60820	−1.40150	−0.700750	+	0.713407i	$0.747151\pi$
$48$	10.3811	1.49838
$49$	−1.79639	−0.256627
$50$	0	0
$51$	−13.7580	−1.92650
$52$	4.91513	0.681606
$53$	−4.43708	−0.609479	−0.304740	−	0.952436i	$-0.598569\pi$
$53$	−4.43708	−0.609479	−0.304740	+	0.952436i	$0.598569\pi$
$54$	−0.0738095	−0.0100442
$55$	0	0
$56$	−0.584326	−0.0780838
$57$	12.0573	1.59703
$58$	3.14667	0.413179
$59$	−2.35301	−0.306336	−0.153168	−	0.988200i	$-0.548948\pi$
$59$	−2.35301	−0.306336	−0.153168	+	0.988200i	$0.548948\pi$
$60$	0	0
$61$	7.22425	0.924970	0.462485	−	0.886627i	$-0.346958\pi$
$61$	7.22425	0.924970	0.462485	+	0.886627i	$0.346958\pi$
$62$	6.48049	0.823023
$63$	−6.80844	−0.857783
$64$	−6.92658	−0.865823
$65$	0	0
$66$	21.4439	2.63957
$67$	−10.6210	−1.29756	−0.648780	−	0.760976i	$-0.724721\pi$
$67$	−10.6210	−1.29756	−0.648780	+	0.760976i	$0.724721\pi$
$68$	10.5154	1.27518
$69$	−5.23686	−0.630443
$70$	0	0
$71$	3.03152	0.359776	0.179888	−	0.983687i	$-0.442427\pi$
$71$	3.03152	0.359776	0.179888	+	0.983687i	$0.442427\pi$
$72$	0.764536	0.0901014
$73$	−10.4106	−1.21847	−0.609233	−	0.792992i	$-0.708522\pi$
$73$	−10.4106	−1.21847	−0.609233	+	0.792992i	$0.708522\pi$
$74$	−6.54038	−0.760303
$75$	0	0
$76$	−9.21557	−1.05710
$77$	−10.1647	−1.15837
$78$	12.6505	1.43238
$79$	−7.36888	−0.829064	−0.414532	−	0.910035i	$-0.636055\pi$
$79$	−7.36888	−0.829064	−0.414532	+	0.910035i	$0.636055\pi$
$80$	0	0
$81$	−9.04578	−1.00509
$82$	−9.72448	−1.07389
$83$	−1.30901	−0.143682	−0.0718411	−	0.997416i	$-0.522887\pi$
$83$	−1.30901	−0.143682	−0.0718411	+	0.997416i	$0.522887\pi$
$84$	10.4343	1.13848
$85$	0	0
$86$	10.7829	1.16275
$87$	3.91317	0.419535
$88$	1.14142	0.121675
$89$	14.3314	1.51913	0.759563	−	0.650434i	$-0.225413\pi$
$89$	14.3314	1.51913	0.759563	+	0.650434i	$0.225413\pi$
$90$	0	0
$91$	−5.99647	−0.628601
$92$	4.00260	0.417300
$93$	8.05906	0.835686
$94$	18.9010	1.94949
$95$	0	0
$96$	−19.1681	−1.95633
$97$	−8.92318	−0.906012	−0.453006	−	0.891508i	$-0.649648\pi$
$97$	−8.92318	−0.906012	−0.453006	+	0.891508i	$0.649648\pi$
$98$	3.53382	0.356969
$99$	13.2995	1.33665
$100$	0	0
$101$	7.20217	0.716643	0.358321	−	0.933598i	$-0.383349\pi$
$101$	7.20217	0.716643	0.358321	+	0.933598i	$0.383349\pi$
$102$	27.0644	2.67977
$103$	18.8324	1.85561	0.927806	−	0.373062i	$-0.121692\pi$
$103$	18.8324	1.85561	0.927806	+	0.373062i	$0.121692\pi$
$104$	0.673358	0.0660282
$105$	0	0
$106$	8.72851	0.847788
$107$	12.8315	1.24047	0.620233	−	0.784417i	$-0.287038\pi$
$107$	12.8315	1.24047	0.620233	+	0.784417i	$0.287038\pi$
$108$	0.0701553	0.00675070
$109$	−16.9292	−1.62153	−0.810764	−	0.585373i	$-0.800948\pi$
$109$	−16.9292	−1.62153	−0.810764	+	0.585373i	$0.800948\pi$
$110$	0	0
$111$	−8.13353	−0.772001
$112$	9.67997	0.914671
$113$	9.40040	0.884315	0.442158	−	0.896937i	$-0.354213\pi$
$113$	9.40040	0.884315	0.442158	+	0.896937i	$0.354213\pi$
$114$	−23.7189	−2.22147
$115$	0	0
$116$	−2.99089	−0.277697
$117$	7.84583	0.725347
$118$	4.62879	0.426115
$119$	−12.8288	−1.17602
$120$	0	0
$121$	8.85557	0.805052
$122$	−14.2114	−1.28664
$123$	−12.0932	−1.09041
$124$	−6.15965	−0.553153
$125$	0	0
$126$	13.3934	1.19318
$127$	−5.46708	−0.485125	−0.242562	−	0.970136i	$-0.577988\pi$
$127$	−5.46708	−0.485125	−0.242562	+	0.970136i	$0.577988\pi$
$128$	−2.04490	−0.180745
$129$	13.4095	1.18064
$130$	0	0
$131$	−14.4311	−1.26085	−0.630427	−	0.776248i	$-0.717120\pi$
$131$	−14.4311	−1.26085	−0.630427	+	0.776248i	$0.717120\pi$
$132$	−20.3823	−1.77405
$133$	11.2430	0.974893
$134$	20.8934	1.80491
$135$	0	0
$136$	1.44058	0.123529
$137$	4.11769	0.351798	0.175899	−	0.984408i	$-0.443717\pi$
$137$	4.11769	0.351798	0.175899	+	0.984408i	$0.443717\pi$
$138$	10.3018	0.876949
$139$	9.46090	0.802463	0.401231	−	0.915977i	$-0.368582\pi$
$139$	9.46090	0.802463	0.401231	+	0.915977i	$0.368582\pi$
$140$	0	0
$141$	23.5051	1.97949
$142$	−5.96355	−0.500450
$143$	11.7134	0.979527
$144$	−12.6653	−1.05544
$145$	0	0
$146$	20.4794	1.69489
$147$	4.39461	0.362461
$148$	6.21657	0.510999
$149$	3.57361	0.292761	0.146381	−	0.989228i	$-0.453238\pi$
$149$	3.57361	0.292761	0.146381	+	0.989228i	$0.453238\pi$
$150$	0	0
$151$	−11.6041	−0.944328	−0.472164	−	0.881511i	$-0.656527\pi$
$151$	−11.6041	−0.944328	−0.472164	+	0.881511i	$0.656527\pi$
$152$	−1.26251	−0.102403
$153$	16.7853	1.35701
$154$	19.9957	1.61130
$155$	0	0
$156$	−12.0242	−0.962704
$157$	−7.77747	−0.620709	−0.310355	−	0.950621i	$-0.600448\pi$
$157$	−7.77747	−0.620709	−0.310355	+	0.950621i	$0.600448\pi$
$158$	14.4959	1.15323
$159$	10.8547	0.860832
$160$	0	0
$161$	−4.88318	−0.384849
$162$	17.7946	1.39808
$163$	−1.13709	−0.0890635	−0.0445318	−	0.999008i	$-0.514180\pi$
$163$	−1.13709	−0.0890635	−0.0445318	+	0.999008i	$0.514180\pi$
$164$	9.24304	0.721760
$165$	0	0
$166$	2.57505	0.199863
$167$	−8.55880	−0.662299	−0.331150	−	0.943578i	$-0.607437\pi$
$167$	−8.55880	−0.662299	−0.331150	+	0.943578i	$0.607437\pi$
$168$	1.42947	0.110286
$169$	−6.08986	−0.468451
$170$	0	0
$171$	−14.7104	−1.12494
$172$	−10.2491	−0.781485
$173$	−4.75621	−0.361608	−0.180804	−	0.983519i	$-0.557870\pi$
$173$	−4.75621	−0.361608	−0.180804	+	0.983519i	$0.557870\pi$
$174$	−7.69789	−0.583576
$175$	0	0
$176$	−18.9087	−1.42530
$177$	5.75631	0.432671
$178$	−28.1924	−2.11311
$179$	−6.42794	−0.480447	−0.240223	−	0.970718i	$-0.577221\pi$
$179$	−6.42794	−0.480447	−0.240223	+	0.970718i	$0.577221\pi$
$180$	0	0
$181$	18.4620	1.37227	0.686135	−	0.727474i	$-0.259306\pi$
$181$	18.4620	1.37227	0.686135	+	0.727474i	$0.259306\pi$
$182$	11.7961	0.874387
$183$	−17.6731	−1.30643
$184$	0.548345	0.0404245
$185$	0	0
$186$	−15.8536	−1.16244
$187$	25.0597	1.83255
$188$	−17.9653	−1.31025
$189$	−0.0855896	−0.00622573
$190$	0	0
$191$	14.8871	1.07720	0.538598	−	0.842563i	$-0.318954\pi$
$191$	14.8871	1.07720	0.538598	+	0.842563i	$0.318954\pi$
$192$	16.9449	1.22289
$193$	−12.5187	−0.901119	−0.450559	−	0.892746i	$-0.648775\pi$
$193$	−12.5187	−0.901119	−0.450559	+	0.892746i	$0.648775\pi$
$194$	17.5535	1.26027
$195$	0	0
$196$	−3.35886	−0.239919
$197$	14.4254	1.02777	0.513884	−	0.857860i	$-0.328206\pi$
$197$	14.4254	1.02777	0.513884	+	0.857860i	$0.328206\pi$
$198$	−26.1625	−1.85929
$199$	−17.7404	−1.25758	−0.628791	−	0.777574i	$-0.716450\pi$
$199$	−17.7404	−1.25758	−0.628791	+	0.777574i	$0.716450\pi$
$200$	0	0
$201$	25.9827	1.83268
$202$	−14.1679	−0.996853
$203$	3.64889	0.256102
$204$	−25.7245	−1.80107
$205$	0	0
$206$	−37.0467	−2.58117
$207$	6.38919	0.444080
$208$	−11.1549	−0.773452
$209$	−21.9620	−1.51914
$210$	0	0
$211$	19.2909	1.32804	0.664022	−	0.747713i	$-0.268848\pi$
$211$	19.2909	1.32804	0.664022	+	0.747713i	$0.268848\pi$
$212$	−8.29638	−0.569798
$213$	−7.41619	−0.508149
$214$	−25.2418	−1.72549
$215$	0	0
$216$	0.00961106	0.000653950 0
$217$	7.51479	0.510137
$218$	33.3028	2.25555
$219$	25.4680	1.72097
$220$	0	0
$221$	14.7835	0.994448
$222$	16.0001	1.07386
$223$	−0.542807	−0.0363490	−0.0181745	−	0.999835i	$-0.505785\pi$
$223$	−0.542807	−0.0363490	−0.0181745	+	0.999835i	$0.505785\pi$
$224$	−17.8736	−1.19423
$225$	0	0
$226$	−18.4923	−1.23009
$227$	6.56591	0.435795	0.217898	−	0.975972i	$-0.430080\pi$
$227$	6.56591	0.435795	0.217898	+	0.975972i	$0.430080\pi$
$228$	22.5446	1.49305
$229$	17.4502	1.15314	0.576570	−	0.817048i	$-0.304391\pi$
$229$	17.4502	1.15314	0.576570	+	0.817048i	$0.304391\pi$
$230$	0	0
$231$	24.8664	1.63609
$232$	−0.409743	−0.0269009
$233$	26.4097	1.73016	0.865078	−	0.501638i	$-0.167269\pi$
$233$	26.4097	1.73016	0.865078	+	0.501638i	$0.167269\pi$
$234$	−15.4341	−1.00896
$235$	0	0
$236$	−4.39963	−0.286391
$237$	18.0269	1.17097
$238$	25.2366	1.63584
$239$	−2.77451	−0.179468	−0.0897342	−	0.995966i	$-0.528602\pi$
$239$	−2.77451	−0.179468	−0.0897342	+	0.995966i	$0.528602\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−17.4205	−1.11983
$243$	22.0166	1.41237
$244$	13.5078	0.864748
$245$	0	0
$246$	23.7896	1.51677
$247$	−12.9561	−0.824376
$248$	−0.843854	−0.0535848
$249$	3.20230	0.202938
$250$	0	0
$251$	−21.9978	−1.38849	−0.694245	−	0.719739i	$-0.744262\pi$
$251$	−21.9978	−1.38849	−0.694245	+	0.719739i	$0.744262\pi$
$252$	−12.7303	−0.801935
$253$	9.53876	0.599697
$254$	10.7547	0.674811
$255$	0	0
$256$	17.8758	1.11724
$257$	−28.1110	−1.75351	−0.876757	−	0.480934i	$-0.840298\pi$
$257$	−28.1110	−1.75351	−0.876757	+	0.480934i	$0.840298\pi$
$258$	−26.3789	−1.64228
$259$	−7.58423	−0.471261
$260$	0	0
$261$	−4.77423	−0.295518
$262$	28.3886	1.75385
$263$	2.06600	0.127395	0.0636975	−	0.997969i	$-0.479711\pi$
$263$	2.06600	0.127395	0.0636975	+	0.997969i	$0.479711\pi$
$264$	−2.79231	−0.171855
$265$	0	0
$266$	−22.1170	−1.35608
$267$	−35.0597	−2.14562
$268$	−19.8590	−1.21308
$269$	−16.5247	−1.00753	−0.503765	−	0.863841i	$-0.668052\pi$
$269$	−16.5247	−1.00753	−0.503765	+	0.863841i	$0.668052\pi$
$270$	0	0
$271$	30.0857	1.82757	0.913787	−	0.406193i	$-0.133144\pi$
$271$	30.0857	1.82757	0.913787	+	0.406193i	$0.133144\pi$
$272$	−23.8647	−1.44701
$273$	14.6695	0.887839
$274$	−8.10023	−0.489353
$275$	0	0
$276$	−9.79180	−0.589397
$277$	11.6508	0.700030	0.350015	−	0.936744i	$-0.386176\pi$
$277$	11.6508	0.700030	0.350015	+	0.936744i	$0.386176\pi$
$278$	−18.6113	−1.11623
$279$	−9.83241	−0.588651
$280$	0	0
$281$	20.8519	1.24392	0.621959	−	0.783050i	$-0.286337\pi$
$281$	20.8519	1.24392	0.621959	+	0.783050i	$0.286337\pi$
$282$	−46.2387	−2.75347
$283$	−11.3460	−0.674450	−0.337225	−	0.941424i	$-0.609488\pi$
$283$	−11.3460	−0.674450	−0.337225	+	0.941424i	$0.609488\pi$
$284$	5.66830	0.336352
$285$	0	0
$286$	−23.0424	−1.36253
$287$	−11.2765	−0.665632
$288$	23.3859	1.37803
$289$	14.6278	0.860460
$290$	0	0
$291$	21.8293	1.27966
$292$	−19.4655	−1.13913
$293$	1.76745	0.103255	0.0516277	−	0.998666i	$-0.483559\pi$
$293$	1.76745	0.103255	0.0516277	+	0.998666i	$0.483559\pi$
$294$	−8.64498	−0.504185
$295$	0	0
$296$	0.851652	0.0495013
$297$	0.167190	0.00970134
$298$	−7.02992	−0.407232
$299$	5.62722	0.325431
$300$	0	0
$301$	12.5039	0.720713
$302$	22.8273	1.31356
$303$	−17.6191	−1.01219
$304$	20.9147	1.19954
$305$	0	0
$306$	−33.0197	−1.88761
$307$	−25.6469	−1.46374	−0.731872	−	0.681442i	$-0.761353\pi$
$307$	−25.6469	−1.46374	−0.731872	+	0.681442i	$0.761353\pi$
$308$	−19.0058	−1.08295
$309$	−46.0708	−2.62088
$310$	0	0
$311$	12.6811	0.719082	0.359541	−	0.933129i	$-0.382933\pi$
$311$	12.6811	0.719082	0.359541	+	0.933129i	$0.382933\pi$
$312$	−1.64727	−0.0932586
$313$	−1.96223	−0.110912	−0.0554560	−	0.998461i	$-0.517661\pi$
$313$	−1.96223	−0.110912	−0.0554560	+	0.998461i	$0.517661\pi$
$314$	15.2997	0.863409
$315$	0	0
$316$	−13.7782	−0.775086
$317$	−28.4346	−1.59705	−0.798523	−	0.601964i	$-0.794385\pi$
$317$	−28.4346	−1.59705	−0.798523	+	0.601964i	$0.794385\pi$
$318$	−21.3531	−1.19742
$319$	−7.12770	−0.399075
$320$	0	0
$321$	−31.3904	−1.75204
$322$	9.60609	0.535326
$323$	−27.7182	−1.54228
$324$	−16.9137	−0.939648
$325$	0	0
$326$	2.23685	0.123888
$327$	41.4150	2.29025
$328$	1.26627	0.0699180
$329$	21.9177	1.20836
$330$	0	0
$331$	−2.28894	−0.125812	−0.0629059	−	0.998019i	$-0.520037\pi$
$331$	−2.28894	−0.125812	−0.0629059	+	0.998019i	$0.520037\pi$
$332$	−2.44756	−0.134328
$333$	9.92327	0.543792
$334$	16.8367	0.921261
$335$	0	0
$336$	−23.6807	−1.29189
$337$	21.4281	1.16727	0.583633	−	0.812018i	$-0.301631\pi$
$337$	21.4281	1.16727	0.583633	+	0.812018i	$0.301631\pi$
$338$	11.9798	0.651617
$339$	−22.9967	−1.24901
$340$	0	0
$341$	−14.6793	−0.794929
$342$	28.9381	1.56479
$343$	20.0658	1.08345
$344$	−1.40409	−0.0757036
$345$	0	0
$346$	9.35631	0.502999
$347$	5.99732	0.321953	0.160976	−	0.986958i	$-0.448536\pi$
$347$	5.99732	0.321953	0.160976	+	0.986958i	$0.448536\pi$
$348$	7.31678	0.392221
$349$	−12.7874	−0.684496	−0.342248	−	0.939610i	$-0.611188\pi$
$349$	−12.7874	−0.684496	−0.342248	+	0.939610i	$0.611188\pi$
$350$	0	0
$351$	0.0986307	0.00526452
$352$	34.9140	1.86092
$353$	−15.9530	−0.849095	−0.424547	−	0.905406i	$-0.639567\pi$
$353$	−15.9530	−0.849095	−0.424547	+	0.905406i	$0.639567\pi$
$354$	−11.3237	−0.601847
$355$	0	0
$356$	26.7967	1.42022
$357$	31.3839	1.66101
$358$	12.6449	0.668304
$359$	−19.3657	−1.02208	−0.511041	−	0.859557i	$-0.670740\pi$
$359$	−19.3657	−1.02208	−0.511041	+	0.859557i	$0.670740\pi$
$360$	0	0
$361$	5.29188	0.278520
$362$	−36.3180	−1.90883
$363$	−21.6639	−1.13706
$364$	−11.2121	−0.587675
$365$	0	0
$366$	34.7661	1.81725
$367$	11.3546	0.592705	0.296352	−	0.955079i	$-0.404230\pi$
$367$	11.3546	0.592705	0.296352	+	0.955079i	$0.404230\pi$
$368$	−9.08390	−0.473531
$369$	14.7543	0.768078
$370$	0	0
$371$	10.1216	0.525487
$372$	15.0687	0.781276
$373$	8.06116	0.417391	0.208695	−	0.977981i	$-0.433078\pi$
$373$	8.06116	0.417391	0.208695	+	0.977981i	$0.433078\pi$
$374$	−49.2969	−2.54908
$375$	0	0
$376$	−2.46119	−0.126926
$377$	−4.20486	−0.216561
$378$	0.168370	0.00866001
$379$	11.8158	0.606936	0.303468	−	0.952842i	$-0.401855\pi$
$379$	11.8158	0.606936	0.303468	+	0.952842i	$0.401855\pi$
$380$	0	0
$381$	13.3744	0.685193
$382$	−29.2856	−1.49838
$383$	−17.0789	−0.872690	−0.436345	−	0.899779i	$-0.643727\pi$
$383$	−17.0789	−0.872690	−0.436345	+	0.899779i	$0.643727\pi$
$384$	5.00255	0.255285
$385$	0	0
$386$	24.6266	1.25346
$387$	−16.3602	−0.831635
$388$	−16.6844	−0.847024
$389$	9.53828	0.483610	0.241805	−	0.970325i	$-0.422261\pi$
$389$	9.53828	0.483610	0.241805	+	0.970325i	$0.422261\pi$
$390$	0	0
$391$	12.0389	0.608831
$392$	−0.460154	−0.0232413
$393$	35.3037	1.78084
$394$	−28.3773	−1.42963
$395$	0	0
$396$	24.8673	1.24963
$397$	31.7271	1.59234	0.796169	−	0.605074i	$-0.206857\pi$
$397$	31.7271	1.59234	0.796169	+	0.605074i	$0.206857\pi$
$398$	34.8985	1.74930
$399$	−27.5044	−1.37694
$400$	0	0
$401$	35.4522	1.77040	0.885200	−	0.465210i	$-0.154021\pi$
$401$	35.4522	1.77040	0.885200	+	0.465210i	$0.154021\pi$
$402$	−51.1126	−2.54927
$403$	−8.65980	−0.431375
$404$	13.4665	0.669984
$405$	0	0
$406$	−7.17801	−0.356239
$407$	14.8150	0.734350
$408$	−3.52417	−0.174473
$409$	−5.01367	−0.247910	−0.123955	−	0.992288i	$-0.539558\pi$
$409$	−5.01367	−0.247910	−0.123955	+	0.992288i	$0.539558\pi$
$410$	0	0
$411$	−10.0733	−0.496881
$412$	35.2126	1.73480
$413$	5.36756	0.264120
$414$	−12.5687	−0.617717
$415$	0	0
$416$	20.5969	1.00985
$417$	−23.1447	−1.13340
$418$	43.2031	2.11313
$419$	−9.85630	−0.481512	−0.240756	−	0.970586i	$-0.577395\pi$
$419$	−9.85630	−0.481512	−0.240756	+	0.970586i	$0.577395\pi$
$420$	0	0
$421$	−26.4352	−1.28837	−0.644187	−	0.764868i	$-0.722804\pi$
$421$	−26.4352	−1.28837	−0.644187	+	0.764868i	$0.722804\pi$
$422$	−37.9487	−1.84731
$423$	−28.6772	−1.39434
$424$	−1.13658	−0.0551972
$425$	0	0
$426$	14.5890	0.706838
$427$	−16.4795	−0.797501
$428$	23.9921	1.15970
$429$	−28.6553	−1.38349
$430$	0	0
$431$	−18.0005	−0.867055	−0.433527	−	0.901140i	$-0.642731\pi$
$431$	−18.0005	−0.867055	−0.433527	+	0.901140i	$0.642731\pi$
$432$	−0.159217	−0.00766034
$433$	18.0386	0.866878	0.433439	−	0.901183i	$-0.357300\pi$
$433$	18.0386	0.866878	0.433439	+	0.901183i	$0.357300\pi$
$434$	−14.7829	−0.709603
$435$	0	0
$436$	−31.6541	−1.51595
$437$	−10.5507	−0.504708
$438$	−50.1000	−2.39387
$439$	−26.8755	−1.28270	−0.641349	−	0.767249i	$-0.721625\pi$
$439$	−26.8755	−1.28270	−0.641349	+	0.767249i	$0.721625\pi$
$440$	0	0
$441$	−5.36162	−0.255315
$442$	−29.0818	−1.38328
$443$	10.3748	0.492920	0.246460	−	0.969153i	$-0.420733\pi$
$443$	10.3748	0.492920	0.246460	+	0.969153i	$0.420733\pi$
$444$	−15.2080	−0.721738
$445$	0	0
$446$	1.06780	0.0505617
$447$	−8.74232	−0.413498
$448$	15.8005	0.746504
$449$	−18.4245	−0.869507	−0.434753	−	0.900550i	$-0.643164\pi$
$449$	−18.4245	−0.869507	−0.434753	+	0.900550i	$0.643164\pi$
$450$	0	0
$451$	22.0274	1.03723
$452$	17.5767	0.826740
$453$	28.3878	1.33377
$454$	−12.9163	−0.606193
$455$	0	0
$456$	3.08854	0.144634
$457$	−40.6233	−1.90028	−0.950139	−	0.311826i	$-0.899059\pi$
$457$	−40.6233	−1.90028	−0.950139	+	0.311826i	$0.899059\pi$
$458$	−34.3276	−1.60402
$459$	0.211010	0.00984911
$460$	0	0
$461$	−16.0196	−0.746109	−0.373055	−	0.927809i	$-0.621690\pi$
$461$	−16.0196	−0.746109	−0.373055	+	0.927809i	$0.621690\pi$
$462$	−48.9167	−2.27581
$463$	11.7038	0.543921	0.271960	−	0.962308i	$-0.412328\pi$
$463$	11.7038	0.543921	0.271960	+	0.962308i	$0.412328\pi$
$464$	6.78781	0.315116
$465$	0	0
$466$	−51.9525	−2.40665
$467$	5.54477	0.256581	0.128291	−	0.991737i	$-0.459051\pi$
$467$	5.54477	0.256581	0.128291	+	0.991737i	$0.459051\pi$
$468$	14.6700	0.678122
$469$	24.2280	1.11874
$470$	0	0
$471$	19.0265	0.876693
$472$	−0.602736	−0.0277432
$473$	−24.4250	−1.12306
$474$	−35.4621	−1.62883
$475$	0	0
$476$	−23.9872	−1.09945
$477$	−13.2432	−0.606363
$478$	5.45796	0.249641
$479$	−39.4308	−1.80164	−0.900819	−	0.434195i	$-0.857033\pi$
$479$	−39.4308	−1.80164	−0.900819	+	0.434195i	$0.857033\pi$
$480$	0	0
$481$	8.73983	0.398502
$482$	1.96718	0.0896025
$483$	11.9460	0.543562
$484$	16.5580	0.752637
$485$	0	0
$486$	−43.3106	−1.96461
$487$	−20.9425	−0.948996	−0.474498	−	0.880257i	$-0.657370\pi$
$487$	−20.9425	−0.948996	−0.474498	+	0.880257i	$0.657370\pi$
$488$	1.85053	0.0837694
$489$	2.78172	0.125794
$490$	0	0
$491$	−23.6349	−1.06663	−0.533315	−	0.845917i	$-0.679054\pi$
$491$	−23.6349	−1.06663	−0.533315	+	0.845917i	$0.679054\pi$
$492$	−22.6118	−1.01942
$493$	−8.99587	−0.405153
$494$	25.4869	1.14671
$495$	0	0
$496$	13.9793	0.627690
$497$	−6.91534	−0.310195
$498$	−6.29949	−0.282287
$499$	42.3144	1.89425	0.947126	−	0.320861i	$-0.103972\pi$
$499$	42.3144	1.89425	0.947126	+	0.320861i	$0.103972\pi$
$500$	0	0
$501$	20.9379	0.935435
$502$	43.2736	1.93140
$503$	16.3945	0.730994	0.365497	−	0.930813i	$-0.380899\pi$
$503$	16.3945	0.730994	0.365497	+	0.930813i	$0.380899\pi$
$504$	−1.74402	−0.0776846
$505$	0	0
$506$	−18.7644	−0.834181
$507$	14.8980	0.661642
$508$	−10.2223	−0.453540
$509$	−15.6277	−0.692687	−0.346343	−	0.938108i	$-0.612577\pi$
$509$	−15.6277	−0.692687	−0.346343	+	0.938108i	$0.612577\pi$
$510$	0	0
$511$	23.7480	1.05055
$512$	−31.0751	−1.37334
$513$	−0.184927	−0.00816471
$514$	55.2992	2.43914
$515$	0	0
$516$	25.0729	1.10377
$517$	−42.8137	−1.88295
$518$	14.9195	0.655527
$519$	11.6354	0.510737
$520$	0	0
$521$	−32.2030	−1.41084	−0.705419	−	0.708790i	$-0.749241\pi$
$521$	−32.2030	−1.41084	−0.705419	+	0.708790i	$0.749241\pi$
$522$	9.39176	0.411066
$523$	4.11741	0.180042	0.0900209	−	0.995940i	$-0.471307\pi$
$523$	4.11741	0.180042	0.0900209	+	0.995940i	$0.471307\pi$
$524$	−26.9831	−1.17876
$525$	0	0
$526$	−4.06419	−0.177207
$527$	−18.5267	−0.807038
$528$	46.2575	2.01310
$529$	−18.4175	−0.800761
$530$	0	0
$531$	−7.02295	−0.304770
$532$	21.0220	0.911421
$533$	12.9947	0.562863
$534$	68.9687	2.98457
$535$	0	0
$536$	−2.72062	−0.117513
$537$	15.7250	0.678586
$538$	32.5071	1.40148
$539$	−8.00464	−0.344784
$540$	0	0
$541$	−31.7237	−1.36391	−0.681954	−	0.731395i	$-0.738870\pi$
$541$	−31.7237	−1.36391	−0.681954	+	0.731395i	$0.738870\pi$
$542$	−59.1838	−2.54216
$543$	−45.1647	−1.93820
$544$	44.0649	1.88927
$545$	0	0
$546$	−28.8575	−1.23499
$547$	10.5709	0.451979	0.225990	−	0.974130i	$-0.427438\pi$
$547$	10.5709	0.451979	0.225990	+	0.974130i	$0.427438\pi$
$548$	7.69920	0.328893
$549$	21.5619	0.920241
$550$	0	0
$551$	7.88386	0.335864
$552$	−1.34145	−0.0570958
$553$	16.8095	0.714811
$554$	−22.9192	−0.973745
$555$	0	0
$556$	17.6899	0.750217
$557$	−24.7793	−1.04993	−0.524966	−	0.851123i	$-0.675922\pi$
$557$	−24.7793	−1.04993	−0.524966	+	0.851123i	$0.675922\pi$
$558$	19.3421	0.818816
$559$	−14.4091	−0.609440
$560$	0	0
$561$	−61.3050	−2.58830
$562$	−41.0193	−1.73030
$563$	−45.1674	−1.90358	−0.951790	−	0.306751i	$-0.900758\pi$
$563$	−45.1674	−1.90358	−0.951790	+	0.306751i	$0.900758\pi$
$564$	43.9495	1.85061
$565$	0	0
$566$	22.3196	0.938163
$567$	20.6347	0.866576
$568$	0.776540	0.0325829
$569$	−22.1747	−0.929611	−0.464805	−	0.885413i	$-0.653876\pi$
$569$	−22.1747	−0.929611	−0.464805	+	0.885413i	$0.653876\pi$
$570$	0	0
$571$	1.78827	0.0748366	0.0374183	−	0.999300i	$-0.488087\pi$
$571$	1.78827	0.0748366	0.0374183	+	0.999300i	$0.488087\pi$
$572$	21.9016	0.915753
$573$	−36.4193	−1.52144
$574$	22.1829	0.925897
$575$	0	0
$576$	−20.6735	−0.861396
$577$	−34.1895	−1.42333	−0.711664	−	0.702521i	$-0.752058\pi$
$577$	−34.1895	−1.42333	−0.711664	+	0.702521i	$0.752058\pi$
$578$	−28.7755	−1.19690
$579$	30.6253	1.27274
$580$	0	0
$581$	2.98603	0.123882
$582$	−42.9421	−1.78001
$583$	−19.7714	−0.818849
$584$	−2.66672	−0.110350
$585$	0	0
$586$	−3.47689	−0.143629
$587$	−8.50133	−0.350887	−0.175444	−	0.984489i	$-0.556136\pi$
$587$	−8.50133	−0.350887	−0.175444	+	0.984489i	$0.556136\pi$
$588$	8.21698	0.338863
$589$	16.2366	0.669017
$590$	0	0
$591$	−35.2897	−1.45162
$592$	−14.1085	−0.579856
$593$	−35.4195	−1.45451	−0.727253	−	0.686370i	$-0.759203\pi$
$593$	−35.4195	−1.45451	−0.727253	+	0.686370i	$0.759203\pi$
$594$	−0.328892	−0.0134946
$595$	0	0
$596$	6.68188	0.273701
$597$	43.3993	1.77622
$598$	−11.0697	−0.452676
$599$	−8.37422	−0.342161	−0.171081	−	0.985257i	$-0.554726\pi$
$599$	−8.37422	−0.342161	−0.171081	+	0.985257i	$0.554726\pi$
$600$	0	0
$601$	40.7418	1.66189	0.830946	−	0.556353i	$-0.187800\pi$
$601$	40.7418	1.66189	0.830946	+	0.556353i	$0.187800\pi$
$602$	−24.5974	−1.00251
$603$	−31.7001	−1.29093
$604$	−21.6972	−0.882845
$605$	0	0
$606$	34.6599	1.40796
$607$	−13.3854	−0.543295	−0.271647	−	0.962397i	$-0.587568\pi$
$607$	−13.3854	−0.543295	−0.271647	+	0.962397i	$0.587568\pi$
$608$	−38.6179	−1.56616
$609$	−8.92649	−0.361720
$610$	0	0
$611$	−25.2572	−1.02180
$612$	31.3850	1.26866
$613$	19.1539	0.773618	0.386809	−	0.922160i	$-0.373577\pi$
$613$	19.1539	0.773618	0.386809	+	0.922160i	$0.373577\pi$
$614$	50.4519	2.03607
$615$	0	0
$616$	−2.60373	−0.104907
$617$	12.7690	0.514060	0.257030	−	0.966403i	$-0.417256\pi$
$617$	12.7690	0.514060	0.257030	+	0.966403i	$0.417256\pi$
$618$	90.6294	3.64565
$619$	34.3688	1.38140	0.690700	−	0.723141i	$-0.257302\pi$
$619$	34.3688	1.38140	0.690700	+	0.723141i	$0.257302\pi$
$620$	0	0
$621$	0.0803192	0.00322310
$622$	−24.9461	−1.00025
$623$	−32.6920	−1.30978
$624$	27.2888	1.09243
$625$	0	0
$626$	3.86006	0.154279
$627$	53.7269	2.14564
$628$	−14.5422	−0.580297
$629$	18.6979	0.745536
$630$	0	0
$631$	−26.4763	−1.05401	−0.527003	−	0.849863i	$-0.676684\pi$
$631$	−26.4763	−1.05401	−0.527003	+	0.849863i	$0.676684\pi$
$632$	−1.88758	−0.0750837
$633$	−47.1925	−1.87574
$634$	55.9359	2.22150
$635$	0	0
$636$	20.2959	0.804785
$637$	−4.72220	−0.187100
$638$	14.0215	0.555115
$639$	9.04808	0.357936
$640$	0	0
$641$	24.8295	0.980707	0.490353	−	0.871524i	$-0.336868\pi$
$641$	24.8295	0.980707	0.490353	+	0.871524i	$0.336868\pi$
$642$	61.7505	2.43710
$643$	−12.6532	−0.498992	−0.249496	−	0.968376i	$-0.580265\pi$
$643$	−12.6532	−0.498992	−0.249496	+	0.968376i	$0.580265\pi$
$644$	−9.13051	−0.359792
$645$	0	0
$646$	54.5266	2.14532
$647$	33.6084	1.32128	0.660642	−	0.750701i	$-0.270284\pi$
$647$	33.6084	1.32128	0.660642	+	0.750701i	$0.270284\pi$
$648$	−2.31712	−0.0910251
$649$	−10.4849	−0.411569
$650$	0	0
$651$	−18.3839	−0.720520
$652$	−2.12611	−0.0832648
$653$	−19.4343	−0.760525	−0.380262	−	0.924879i	$-0.624166\pi$
$653$	−19.4343	−0.760525	−0.380262	+	0.924879i	$0.624166\pi$
$654$	−81.4706	−3.18575
$655$	0	0
$656$	−20.9771	−0.819016
$657$	−31.0720	−1.21224
$658$	−43.1159	−1.68083
$659$	−41.7837	−1.62766	−0.813831	−	0.581102i	$-0.802622\pi$
$659$	−41.7837	−1.62766	−0.813831	+	0.581102i	$0.802622\pi$
$660$	0	0
$661$	−50.5702	−1.96695	−0.983477	−	0.181033i	$-0.942056\pi$
$661$	−50.5702	−1.96695	−0.983477	+	0.181033i	$0.942056\pi$
$662$	4.50276	0.175005
$663$	−36.1658	−1.40456
$664$	−0.335309	−0.0130125
$665$	0	0
$666$	−19.5208	−0.756416
$667$	−3.42420	−0.132586
$668$	−16.0031	−0.619179
$669$	1.32790	0.0513396
$670$	0	0
$671$	32.1909	1.24272
$672$	43.7251	1.68673
$673$	−7.28621	−0.280863	−0.140431	−	0.990090i	$-0.544849\pi$
$673$	−7.28621	−0.280863	−0.140431	+	0.990090i	$0.544849\pi$
$674$	−42.1530	−1.62367
$675$	0	0
$676$	−11.3867	−0.437951
$677$	31.9411	1.22759	0.613797	−	0.789464i	$-0.289641\pi$
$677$	31.9411	1.22759	0.613797	+	0.789464i	$0.289641\pi$
$678$	45.2387	1.73738
$679$	20.3550	0.781155
$680$	0	0
$681$	−16.0626	−0.615519
$682$	28.8768	1.10575
$683$	−22.5646	−0.863409	−0.431705	−	0.902015i	$-0.642088\pi$
$683$	−22.5646	−0.863409	−0.431705	+	0.902015i	$0.642088\pi$
$684$	−27.5054	−1.05169
$685$	0	0
$686$	−39.4730	−1.50709
$687$	−42.6894	−1.62870
$688$	23.2603	0.886789
$689$	−11.6638	−0.444356
$690$	0	0
$691$	−7.23482	−0.275226	−0.137613	−	0.990486i	$-0.543943\pi$
$691$	−7.23482	−0.275226	−0.137613	+	0.990486i	$0.543943\pi$
$692$	−8.89310	−0.338065
$693$	−30.3381	−1.15245
$694$	−11.7978	−0.447838
$695$	0	0
$696$	1.00238	0.0379950
$697$	27.8008	1.05303
$698$	25.1552	0.952137
$699$	−64.6075	−2.44368
$700$	0	0
$701$	0.510517	0.0192820	0.00964098	−	0.999954i	$-0.496931\pi$
$701$	0.510517	0.0192820	0.00964098	+	0.999954i	$0.496931\pi$
$702$	−0.194024	−0.00732297
$703$	−16.3866	−0.618034
$704$	−30.8646	−1.16325
$705$	0	0
$706$	31.3824	1.18109
$707$	−16.4292	−0.617883
$708$	10.7631	0.404501
$709$	44.3436	1.66536	0.832679	−	0.553756i	$-0.186806\pi$
$709$	44.3436	1.66536	0.832679	+	0.553756i	$0.186806\pi$
$710$	0	0
$711$	−21.9936	−0.824825
$712$	3.67106	0.137579
$713$	−7.05205	−0.264101
$714$	−61.7377	−2.31047
$715$	0	0
$716$	−12.0189	−0.449166
$717$	6.78745	0.253482
$718$	38.0957	1.42172
$719$	50.3859	1.87908	0.939539	−	0.342441i	$-0.111254\pi$
$719$	50.3859	1.87908	0.939539	+	0.342441i	$0.111254\pi$
$720$	0	0
$721$	−42.9594	−1.59989
$722$	−10.4101	−0.387422
$723$	2.44636	0.0909810
$724$	34.5200	1.28293
$725$	0	0
$726$	42.6167	1.58165
$727$	−8.99743	−0.333696	−0.166848	−	0.985983i	$-0.553359\pi$
$727$	−8.99743	−0.333696	−0.166848	+	0.985983i	$0.553359\pi$
$728$	−1.53603	−0.0569289
$729$	−26.7232	−0.989749
$730$	0	0
$731$	−30.8268	−1.14017
$732$	−33.0449	−1.22137
$733$	−31.0077	−1.14530	−0.572648	−	0.819801i	$-0.694084\pi$
$733$	−31.0077	−1.14530	−0.572648	+	0.819801i	$0.694084\pi$
$734$	−22.3365	−0.824455
$735$	0	0
$736$	16.7729	0.618259
$737$	−47.3267	−1.74330
$738$	−29.0243	−1.06840
$739$	26.8680	0.988356	0.494178	−	0.869361i	$-0.335469\pi$
$739$	26.8680	0.988356	0.494178	+	0.869361i	$0.335469\pi$
$740$	0	0
$741$	31.6952	1.16435
$742$	−19.9110	−0.730955
$743$	−30.6022	−1.12269	−0.561343	−	0.827583i	$-0.689715\pi$
$743$	−30.6022	−1.12269	−0.561343	+	0.827583i	$0.689715\pi$
$744$	2.06437	0.0756834
$745$	0	0
$746$	−15.8577	−0.580592
$747$	−3.90695	−0.142948
$748$	46.8563	1.71323
$749$	−29.2704	−1.06952
$750$	0	0
$751$	15.8698	0.579099	0.289549	−	0.957163i	$-0.406495\pi$
$751$	15.8698	0.579099	0.289549	+	0.957163i	$0.406495\pi$
$752$	40.7721	1.48681
$753$	53.8145	1.96111
$754$	8.27171	0.301238
$755$	0	0
$756$	−0.160034	−0.00582039
$757$	−9.85154	−0.358060	−0.179030	−	0.983844i	$-0.557296\pi$
$757$	−9.85154	−0.358060	−0.179030	+	0.983844i	$0.557296\pi$
$758$	−23.2437	−0.844251
$759$	−23.3352	−0.847015
$760$	0	0
$761$	−5.90987	−0.214233	−0.107116	−	0.994246i	$-0.534162\pi$
$761$	−5.90987	−0.214233	−0.107116	+	0.994246i	$0.534162\pi$
$762$	−26.3099	−0.953106
$763$	38.6180	1.39807
$764$	27.8358	1.00706
$765$	0	0
$766$	33.5972	1.21392
$767$	−6.18540	−0.223342
$768$	−43.7307	−1.57799
$769$	45.7837	1.65100	0.825501	−	0.564401i	$-0.190893\pi$
$769$	45.7837	1.65100	0.825501	+	0.564401i	$0.190893\pi$
$770$	0	0
$771$	68.7695	2.47667
$772$	−23.4074	−0.842450
$773$	9.57855	0.344517	0.172258	−	0.985052i	$-0.444894\pi$
$773$	9.57855	0.344517	0.172258	+	0.985052i	$0.444894\pi$
$774$	32.1834	1.15681
$775$	0	0
$776$	−2.28572	−0.0820524
$777$	18.5537	0.665612
$778$	−18.7635	−0.672704
$779$	−24.3643	−0.872941
$780$	0	0
$781$	13.5083	0.483367
$782$	−23.6826	−0.846887
$783$	−0.0600174	−0.00214485
$784$	7.62293	0.272248
$785$	0	0
$786$	−69.4487	−2.47715
$787$	5.51780	0.196689	0.0983443	−	0.995152i	$-0.468645\pi$
$787$	5.51780	0.196689	0.0983443	+	0.995152i	$0.468645\pi$
$788$	26.9724	0.960853
$789$	−5.05418	−0.179933
$790$	0	0
$791$	−21.4437	−0.762449
$792$	3.40674	0.121053
$793$	18.9905	0.674372
$794$	−62.4129	−2.21495
$795$	0	0
$796$	−33.1707	−1.17570
$797$	36.9269	1.30802	0.654008	−	0.756487i	$-0.273086\pi$
$797$	36.9269	1.30802	0.654008	+	0.756487i	$0.273086\pi$
$798$	54.1061	1.91534
$799$	−54.0352	−1.91163
$800$	0	0
$801$	42.7744	1.51136
$802$	−69.7408	−2.46263
$803$	−46.3891	−1.63704
$804$	48.5821	1.71336
$805$	0	0
$806$	17.0354	0.600045
$807$	40.4254	1.42304
$808$	1.84487	0.0649024
$809$	−36.9774	−1.30005	−0.650027	−	0.759911i	$-0.725243\pi$
$809$	−36.9774	−1.30005	−0.650027	+	0.759911i	$0.725243\pi$
$810$	0	0
$811$	−4.57407	−0.160617	−0.0803087	−	0.996770i	$-0.525591\pi$
$811$	−4.57407	−0.160617	−0.0803087	+	0.996770i	$0.525591\pi$
$812$	6.82264	0.239428
$813$	−73.6003	−2.58128
$814$	−29.1437	−1.02148
$815$	0	0
$816$	58.3816	2.04376
$817$	27.0162	0.945176
$818$	9.86278	0.344844
$819$	−17.8974	−0.625387
$820$	0	0
$821$	−40.2763	−1.40565	−0.702827	−	0.711361i	$-0.748079\pi$
$821$	−40.2763	−1.40565	−0.702827	+	0.711361i	$0.748079\pi$
$822$	19.8160	0.691164
$823$	−33.9371	−1.18297	−0.591487	−	0.806315i	$-0.701459\pi$
$823$	−33.9371	−1.18297	−0.591487	+	0.806315i	$0.701459\pi$
$824$	4.82402	0.168053
$825$	0	0
$826$	−10.5589	−0.367392
$827$	17.9372	0.623737	0.311869	−	0.950125i	$-0.399045\pi$
$827$	17.9372	0.623737	0.311869	+	0.950125i	$0.399045\pi$
$828$	11.9464	0.415167
$829$	55.1694	1.91611	0.958056	−	0.286582i	$-0.0925191\pi$
$829$	55.1694	1.91611	0.958056	+	0.286582i	$0.0925191\pi$
$830$	0	0
$831$	−28.5021	−0.988726
$832$	−18.2080	−0.631249
$833$	−10.1026	−0.350036
$834$	45.5298	1.57657
$835$	0	0
$836$	−41.0642	−1.42024
$837$	−0.123604	−0.00427239
$838$	19.3891	0.669785
$839$	22.2855	0.769381	0.384690	−	0.923046i	$-0.374308\pi$
$839$	22.2855	0.769381	0.384690	+	0.923046i	$0.374308\pi$
$840$	0	0
$841$	−26.4413	−0.911769
$842$	52.0028	1.79214
$843$	−51.0111	−1.75692
$844$	36.0699	1.24158
$845$	0	0
$846$	56.4132	1.93953
$847$	−20.2008	−0.694108
$848$	18.8286	0.646577
$849$	27.7564	0.952597
$850$	0	0
$851$	7.11722	0.243975
$852$	−13.8667	−0.475065
$853$	−26.5876	−0.910341	−0.455170	−	0.890404i	$-0.650422\pi$
$853$	−26.5876	−0.910341	−0.455170	+	0.890404i	$0.650422\pi$
$854$	32.4182	1.10933
$855$	0	0
$856$	3.28685	0.112342
$857$	16.0685	0.548891	0.274445	−	0.961603i	$-0.411506\pi$
$857$	16.0685	0.548891	0.274445	+	0.961603i	$0.411506\pi$
$858$	56.3700	1.92444
$859$	−10.7279	−0.366031	−0.183016	−	0.983110i	$-0.558586\pi$
$859$	−10.7279	−0.366031	−0.183016	+	0.983110i	$0.558586\pi$
$860$	0	0
$861$	27.5864	0.940142
$862$	35.4102	1.20608
$863$	25.9168	0.882217	0.441109	−	0.897454i	$-0.354585\pi$
$863$	25.9168	0.882217	0.441109	+	0.897454i	$0.354585\pi$
$864$	0.293986	0.0100016
$865$	0	0
$866$	−35.4850	−1.20583
$867$	−35.7849	−1.21532
$868$	14.0510	0.476924
$869$	−32.8354	−1.11387
$870$	0	0
$871$	−27.9195	−0.946018
$872$	−4.33651	−0.146853
$873$	−26.6327	−0.901380
$874$	20.7551	0.702052
$875$	0	0
$876$	47.6197	1.60892
$877$	19.4278	0.656029	0.328015	−	0.944673i	$-0.393621\pi$
$877$	19.4278	0.656029	0.328015	+	0.944673i	$0.393621\pi$
$878$	52.8689	1.78424
$879$	−4.32381	−0.145839
$880$	0	0
$881$	−6.32510	−0.213098	−0.106549	−	0.994307i	$-0.533980\pi$
$881$	−6.32510	−0.213098	−0.106549	+	0.994307i	$0.533980\pi$
$882$	10.5473	0.355145
$883$	−29.1204	−0.979979	−0.489989	−	0.871728i	$-0.662999\pi$
$883$	−29.1204	−0.979979	−0.489989	+	0.871728i	$0.662999\pi$
$884$	27.6420	0.929702
$885$	0	0
$886$	−20.4090	−0.685654
$887$	14.3156	0.480670	0.240335	−	0.970690i	$-0.422743\pi$
$887$	14.3156	0.480670	0.240335	+	0.970690i	$0.422743\pi$
$888$	−2.08344	−0.0699158
$889$	12.4712	0.418270
$890$	0	0
$891$	−40.3076	−1.35036
$892$	−1.01493	−0.0339825
$893$	47.3557	1.58470
$894$	17.1977	0.575177
$895$	0	0
$896$	4.66470	0.155837
$897$	−13.7662	−0.459640
$898$	36.2443	1.20949
$899$	5.26954	0.175749
$900$	0	0
$901$	−24.9535	−0.831322
$902$	−43.3319	−1.44279
$903$	−30.5890	−1.01794
$904$	2.40796	0.0800875
$905$	0	0
$906$	−55.8438	−1.85528
$907$	−47.0417	−1.56200	−0.780998	−	0.624534i	$-0.785289\pi$
$907$	−47.0417	−1.56200	−0.780998	+	0.624534i	$0.785289\pi$
$908$	12.2769	0.407422
$909$	21.4961	0.712979
$910$	0	0
$911$	−17.6314	−0.584154	−0.292077	−	0.956395i	$-0.594346\pi$
$911$	−17.6314	−0.584154	−0.292077	+	0.956395i	$0.594346\pi$
$912$	−51.1649	−1.69424
$913$	−5.83289	−0.193040
$914$	79.9132	2.64330
$915$	0	0
$916$	32.6281	1.07806
$917$	32.9195	1.08710
$918$	−0.415094	−0.0137002
$919$	−16.9262	−0.558344	−0.279172	−	0.960241i	$-0.590060\pi$
$919$	−16.9262	−0.558344	−0.279172	+	0.960241i	$0.590060\pi$
$920$	0	0
$921$	62.7414	2.06740
$922$	31.5135	1.03784
$923$	7.96901	0.262303
$924$	46.4949	1.52957
$925$	0	0
$926$	−23.0234	−0.756596
$927$	56.2084	1.84613
$928$	−12.5333	−0.411427
$929$	20.8615	0.684442	0.342221	−	0.939619i	$-0.388821\pi$
$929$	20.8615	0.684442	0.342221	+	0.939619i	$0.388821\pi$
$930$	0	0
$931$	8.85383	0.290173
$932$	49.3804	1.61751
$933$	−31.0226	−1.01563
$934$	−10.9075	−0.356906
$935$	0	0
$936$	2.00975	0.0656906
$937$	−51.3276	−1.67680	−0.838400	−	0.545055i	$-0.816509\pi$
$937$	−51.3276	−1.67680	−0.838400	+	0.545055i	$0.816509\pi$
$938$	−47.6607	−1.55618
$939$	4.80033	0.156653
$940$	0	0
$941$	−0.914038	−0.0297968	−0.0148984	−	0.999889i	$-0.504742\pi$
$941$	−0.914038	−0.0297968	−0.0148984	+	0.999889i	$0.504742\pi$
$942$	−37.4284	−1.21948
$943$	10.5821	0.344602
$944$	9.98495	0.324982
$945$	0	0
$946$	48.0483	1.56218
$947$	−22.5317	−0.732182	−0.366091	−	0.930579i	$-0.619304\pi$
$947$	−22.5317	−0.732182	−0.366091	+	0.930579i	$0.619304\pi$
$948$	33.7065	1.09474
$949$	−27.3664	−0.888352
$950$	0	0
$951$	69.5612	2.25568
$952$	−3.28617	−0.106505
$953$	51.0113	1.65242	0.826209	−	0.563364i	$-0.190493\pi$
$953$	51.0113	1.65242	0.826209	+	0.563364i	$0.190493\pi$
$954$	26.0517	0.843454
$955$	0	0
$956$	−5.18774	−0.167784
$957$	17.4369	0.563655
$958$	77.5673	2.50609
$959$	−9.39304	−0.303317
$960$	0	0
$961$	−20.1475	−0.649920
$962$	−17.1928	−0.554318
$963$	38.2977	1.23412
$964$	−1.86979	−0.0602217
$965$	0	0
$966$	−23.4999	−0.756098
$967$	14.3183	0.460445	0.230223	−	0.973138i	$-0.426055\pi$
$967$	14.3183	0.460445	0.230223	+	0.973138i	$0.426055\pi$
$968$	2.26840	0.0729090
$969$	67.8086	2.17833
$970$	0	0
$971$	37.5871	1.20623	0.603113	−	0.797656i	$-0.293927\pi$
$971$	37.5871	1.20623	0.603113	+	0.797656i	$0.293927\pi$
$972$	41.1664	1.32041
$973$	−21.5817	−0.691876
$974$	41.1976	1.32006
$975$	0	0
$976$	−30.6559	−0.981271
$977$	38.5849	1.23444	0.617220	−	0.786790i	$-0.288259\pi$
$977$	38.5849	1.23444	0.617220	+	0.786790i	$0.288259\pi$
$978$	−5.47214	−0.174980
$979$	63.8602	2.04098
$980$	0	0
$981$	−50.5281	−1.61324
$982$	46.4941	1.48369
$983$	−5.62474	−0.179401	−0.0897006	−	0.995969i	$-0.528591\pi$
$983$	−5.62474	−0.179401	−0.0897006	+	0.995969i	$0.528591\pi$
$984$	−3.09774	−0.0987525
$985$	0	0
$986$	17.6965	0.563570
$987$	−53.6185	−1.70669
$988$	−24.2251	−0.770703
$989$	−11.7339	−0.373118
$990$	0	0
$991$	−12.7247	−0.404212	−0.202106	−	0.979364i	$-0.564779\pi$
$991$	−12.7247	−0.404212	−0.202106	+	0.979364i	$0.564779\pi$
$992$	−25.8121	−0.819535
$993$	5.59958	0.177697
$994$	13.6037	0.431483
$995$	0	0
$996$	5.98762	0.189725
$997$	39.5405	1.25226	0.626130	−	0.779719i	$-0.284638\pi$
$997$	39.5405	1.25226	0.626130	+	0.779719i	$0.284638\pi$
$998$	−83.2399	−2.63491
$999$	0.124746	0.00394680

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.j.1.7		25
5.4	even	2		1205.2.a.e.1.19	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.19	✓	25	5.4	even	2
6025.2.a.j.1.7		25	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.96718 q^{2} -2.44636 q^{3} +1.86979 q^{4} +4.81242 q^{6} -2.28114 q^{7} +0.256155 q^{8} +2.98466 q^{9} +O(q^{10})\)	\(q-1.96718 q^{2} -2.44636 q^{3} +1.86979 q^{4} +4.81242 q^{6} -2.28114 q^{7} +0.256155 q^{8} +2.98466 q^{9} +4.45596 q^{11} -4.57416 q^{12} +2.62871 q^{13} +4.48741 q^{14} -4.24347 q^{16} +5.62386 q^{17} -5.87136 q^{18} -4.92868 q^{19} +5.58049 q^{21} -8.76566 q^{22} +2.14068 q^{23} -0.626646 q^{24} -5.17115 q^{26} +0.0375205 q^{27} -4.26525 q^{28} -1.59959 q^{29} -3.29431 q^{31} +7.83535 q^{32} -10.9009 q^{33} -11.0631 q^{34} +5.58068 q^{36} +3.32475 q^{37} +9.69558 q^{38} -6.43077 q^{39} +4.94337 q^{41} -10.9778 q^{42} -5.48142 q^{43} +8.33169 q^{44} -4.21109 q^{46} -9.60820 q^{47} +10.3811 q^{48} -1.79639 q^{49} -13.7580 q^{51} +4.91513 q^{52} -4.43708 q^{53} -0.0738095 q^{54} -0.584326 q^{56} +12.0573 q^{57} +3.14667 q^{58} -2.35301 q^{59} +7.22425 q^{61} +6.48049 q^{62} -6.80844 q^{63} -6.92658 q^{64} +21.4439 q^{66} -10.6210 q^{67} +10.5154 q^{68} -5.23686 q^{69} +3.03152 q^{71} +0.764536 q^{72} -10.4106 q^{73} -6.54038 q^{74} -9.21557 q^{76} -10.1647 q^{77} +12.6505 q^{78} -7.36888 q^{79} -9.04578 q^{81} -9.72448 q^{82} -1.30901 q^{83} +10.4343 q^{84} +10.7829 q^{86} +3.91317 q^{87} +1.14142 q^{88} +14.3314 q^{89} -5.99647 q^{91} +4.00260 q^{92} +8.05906 q^{93} +18.9010 q^{94} -19.1681 q^{96} -8.92318 q^{97} +3.53382 q^{98} +13.2995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Introduction

L-functions

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