LMFDB - Embedded newform 4925.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4925,2,Mod(1,4925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4925, 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("4925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.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:	$39.3263229955$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.51668 q^{2} -3.09870 q^{3} +4.33369 q^{4} +7.79844 q^{6} -2.79664 q^{7} -5.87315 q^{8} +6.60194 q^{9} +1.50057 q^{11} -13.4288 q^{12} +6.67560 q^{13} +7.03826 q^{14} +6.11348 q^{16} +7.00147 q^{17} -16.6150 q^{18} -2.25120 q^{19} +8.66596 q^{21} -3.77645 q^{22} -1.49137 q^{23} +18.1991 q^{24} -16.8004 q^{26} -11.1613 q^{27} -12.1198 q^{28} -9.20215 q^{29} +6.53831 q^{31} -3.63937 q^{32} -4.64980 q^{33} -17.6205 q^{34} +28.6107 q^{36} -0.641758 q^{37} +5.66556 q^{38} -20.6857 q^{39} +8.67754 q^{41} -21.8095 q^{42} -12.3791 q^{43} +6.50299 q^{44} +3.75330 q^{46} -5.81406 q^{47} -18.9438 q^{48} +0.821218 q^{49} -21.6955 q^{51} +28.9300 q^{52} +1.72766 q^{53} +28.0895 q^{54} +16.4251 q^{56} +6.97580 q^{57} +23.1589 q^{58} +5.59101 q^{59} -2.73144 q^{61} -16.4549 q^{62} -18.4633 q^{63} -3.06781 q^{64} +11.7021 q^{66} -4.94365 q^{67} +30.3422 q^{68} +4.62130 q^{69} -5.79185 q^{71} -38.7742 q^{72} +9.22321 q^{73} +1.61510 q^{74} -9.75601 q^{76} -4.19655 q^{77} +52.0593 q^{78} -12.8954 q^{79} +14.7798 q^{81} -21.8386 q^{82} +5.00012 q^{83} +37.5556 q^{84} +31.1542 q^{86} +28.5147 q^{87} -8.81305 q^{88} -4.45129 q^{89} -18.6693 q^{91} -6.46312 q^{92} -20.2603 q^{93} +14.6321 q^{94} +11.2773 q^{96} -12.7344 q^{97} -2.06674 q^{98} +9.90665 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) $ 49 * q - 5 * q^2 - 22 * q^3 + 49 * q^4 + 2 * q^6 - 32 * q^7 - 15 * q^8 + 51 * q^9 - 2 * q^11 - 44 * q^12 - 32 * q^13 - 8 * q^14 + 49 * q^16 - 14 * q^17 - 25 * q^18 + 4 * q^19 + 10 * q^21 - 38 * q^22 - 24 * q^23 - 2 * q^26 - 88 * q^27 - 80 * q^28 + 6 * q^29 - 2 * q^31 - 35 * q^32 - 48 * q^33 - 12 * q^34 + 43 * q^36 - 60 * q^37 - 34 * q^38 - 2 * q^41 - 36 * q^42 - 80 * q^43 - 8 * q^44 - 22 * q^46 - 48 * q^47 - 88 * q^48 + 65 * q^49 + 16 * q^51 - 96 * q^52 - 18 * q^53 + 42 * q^54 - 22 * q^56 - 40 * q^57 - 52 * q^58 + 8 * q^59 + 16 * q^61 - 38 * q^62 - 82 * q^63 + 55 * q^64 + 42 * q^66 - 146 * q^67 + 68 * q^68 - 6 * q^69 - 75 * q^72 - 78 * q^73 - 24 * q^74 - 4 * q^76 - 32 * q^77 + 38 * q^78 + 10 * q^79 + 41 * q^81 - 96 * q^82 - 64 * q^83 + 42 * q^84 - 28 * q^86 - 56 * q^87 - 72 * q^88 + 4 * q^89 - 60 * q^92 + 28 * q^93 - 8 * q^94 - 34 * q^96 - 124 * q^97 + 19 * q^98 + 22 * q^99

Coefficient data

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

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

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

$2$	−2.51668	−1.77956	−0.889781	−	0.456387i	$-0.849143\pi$
$2$	−2.51668	−1.77956	−0.889781	+	0.456387i	$0.849143\pi$
$3$	−3.09870	−1.78903	−0.894517	−	0.447033i	$-0.852481\pi$
$3$	−3.09870	−1.78903	−0.894517	+	0.447033i	$0.852481\pi$
$4$	4.33369	2.16684
$5$	0	0
$5$	0	0
$6$	7.79844	3.18370
$7$	−2.79664	−1.05703	−0.528516	−	0.848923i	$-0.677251\pi$
$7$	−2.79664	−1.05703	−0.528516	+	0.848923i	$0.677251\pi$
$8$	−5.87315	−2.07647
$9$	6.60194	2.20065
$10$	0	0
$11$	1.50057	0.452438	0.226219	−	0.974076i	$-0.427364\pi$
$11$	1.50057	0.452438	0.226219	+	0.974076i	$0.427364\pi$
$12$	−13.4288	−3.87656
$13$	6.67560	1.85148	0.925739	−	0.378163i	$-0.123444\pi$
$13$	6.67560	1.85148	0.925739	+	0.378163i	$0.123444\pi$
$14$	7.03826	1.88106
$15$	0	0
$16$	6.11348	1.52837
$17$	7.00147	1.69811	0.849053	−	0.528308i	$-0.177173\pi$
$17$	7.00147	1.69811	0.849053	+	0.528308i	$0.177173\pi$
$18$	−16.6150	−3.91619
$19$	−2.25120	−0.516462	−0.258231	−	0.966083i	$-0.583139\pi$
$19$	−2.25120	−0.516462	−0.258231	+	0.966083i	$0.583139\pi$
$20$	0	0
$21$	8.66596	1.89107
$22$	−3.77645	−0.805142
$23$	−1.49137	−0.310972	−0.155486	−	0.987838i	$-0.549694\pi$
$23$	−1.49137	−0.310972	−0.155486	+	0.987838i	$0.549694\pi$
$24$	18.1991	3.71488
$25$	0	0
$26$	−16.8004	−3.29482
$27$	−11.1613	−2.14800
$28$	−12.1198	−2.29042
$29$	−9.20215	−1.70880	−0.854399	−	0.519618i	$-0.826074\pi$
$29$	−9.20215	−1.70880	−0.854399	+	0.519618i	$0.826074\pi$
$30$	0	0
$31$	6.53831	1.17432	0.587158	−	0.809472i	$-0.300247\pi$
$31$	6.53831	1.17432	0.587158	+	0.809472i	$0.300247\pi$
$32$	−3.63937	−0.643356
$33$	−4.64980	−0.809427
$34$	−17.6205	−3.02189
$35$	0	0
$36$	28.6107	4.76846
$37$	−0.641758	−0.105504	−0.0527522	−	0.998608i	$-0.516799\pi$
$37$	−0.641758	−0.105504	−0.0527522	+	0.998608i	$0.516799\pi$
$38$	5.66556	0.919076
$39$	−20.6857	−3.31236
$40$	0	0
$41$	8.67754	1.35520	0.677602	−	0.735429i	$-0.263019\pi$
$41$	8.67754	1.35520	0.677602	+	0.735429i	$0.263019\pi$
$42$	−21.8095	−3.36527
$43$	−12.3791	−1.88779	−0.943897	−	0.330240i	$-0.892870\pi$
$43$	−12.3791	−1.88779	−0.943897	+	0.330240i	$0.892870\pi$
$44$	6.50299	0.980362
$45$	0	0
$46$	3.75330	0.553393
$47$	−5.81406	−0.848068	−0.424034	−	0.905646i	$-0.639386\pi$
$47$	−5.81406	−0.848068	−0.424034	+	0.905646i	$0.639386\pi$
$48$	−18.9438	−2.73431
$49$	0.821218	0.117317
$50$	0	0
$51$	−21.6955	−3.03797
$52$	28.9300	4.01186
$53$	1.72766	0.237313	0.118656	−	0.992935i	$-0.462141\pi$
$53$	1.72766	0.237313	0.118656	+	0.992935i	$0.462141\pi$
$54$	28.0895	3.82250
$55$	0	0
$56$	16.4251	2.19490
$57$	6.97580	0.923968
$58$	23.1589	3.04091
$59$	5.59101	0.727888	0.363944	−	0.931421i	$-0.381430\pi$
$59$	5.59101	0.727888	0.363944	+	0.931421i	$0.381430\pi$
$60$	0	0
$61$	−2.73144	−0.349725	−0.174862	−	0.984593i	$-0.555948\pi$
$61$	−2.73144	−0.349725	−0.174862	+	0.984593i	$0.555948\pi$
$62$	−16.4549	−2.08977
$63$	−18.4633	−2.32615
$64$	−3.06781	−0.383476
$65$	0	0
$66$	11.7021	1.44043
$67$	−4.94365	−0.603963	−0.301981	−	0.953314i	$-0.597648\pi$
$67$	−4.94365	−0.603963	−0.301981	+	0.953314i	$0.597648\pi$
$68$	30.3422	3.67953
$69$	4.62130	0.556339
$70$	0	0
$71$	−5.79185	−0.687366	−0.343683	−	0.939086i	$-0.611675\pi$
$71$	−5.79185	−0.687366	−0.343683	+	0.939086i	$0.611675\pi$
$72$	−38.7742	−4.56958
$73$	9.22321	1.07950	0.539748	−	0.841827i	$-0.318520\pi$
$73$	9.22321	1.07950	0.539748	+	0.841827i	$0.318520\pi$
$74$	1.61510	0.187752
$75$	0	0
$76$	−9.75601	−1.11909
$77$	−4.19655	−0.478241
$78$	52.0593	5.89455
$79$	−12.8954	−1.45085	−0.725426	−	0.688300i	$-0.758357\pi$
$79$	−12.8954	−1.45085	−0.725426	+	0.688300i	$0.758357\pi$
$80$	0	0
$81$	14.7798	1.64220
$82$	−21.8386	−2.41167
$83$	5.00012	0.548834	0.274417	−	0.961611i	$-0.411515\pi$
$83$	5.00012	0.548834	0.274417	+	0.961611i	$0.411515\pi$
$84$	37.5556	4.09765
$85$	0	0
$86$	31.1542	3.35945
$87$	28.5147	3.05710
$88$	−8.81305	−0.939475
$89$	−4.45129	−0.471836	−0.235918	−	0.971773i	$-0.575810\pi$
$89$	−4.45129	−0.471836	−0.235918	+	0.971773i	$0.575810\pi$
$90$	0	0
$91$	−18.6693	−1.95707
$92$	−6.46312	−0.673827
$93$	−20.2603	−2.10089
$94$	14.6321	1.50919
$95$	0	0
$96$	11.2773	1.15099
$97$	−12.7344	−1.29298	−0.646489	−	0.762923i	$-0.723763\pi$
$97$	−12.7344	−1.29298	−0.646489	+	0.762923i	$0.723763\pi$
$98$	−2.06674	−0.208773
$99$	9.90665	0.995655
$100$	0	0
$101$	−2.38567	−0.237383	−0.118691	−	0.992931i	$-0.537870\pi$
$101$	−2.38567	−0.237383	−0.118691	+	0.992931i	$0.537870\pi$
$102$	54.6006	5.40626
$103$	5.12451	0.504933	0.252467	−	0.967606i	$-0.418758\pi$
$103$	5.12451	0.504933	0.252467	+	0.967606i	$0.418758\pi$
$104$	−39.2068	−3.84454
$105$	0	0
$106$	−4.34798	−0.422313
$107$	−2.00560	−0.193888	−0.0969442	−	0.995290i	$-0.530907\pi$
$107$	−2.00560	−0.193888	−0.0969442	+	0.995290i	$0.530907\pi$
$108$	−48.3697	−4.65438
$109$	−11.2834	−1.08075	−0.540376	−	0.841424i	$-0.681718\pi$
$109$	−11.2834	−1.08075	−0.540376	+	0.841424i	$0.681718\pi$
$110$	0	0
$111$	1.98861	0.188751
$112$	−17.0972	−1.61553
$113$	5.35887	0.504121	0.252060	−	0.967712i	$-0.418892\pi$
$113$	5.35887	0.504121	0.252060	+	0.967712i	$0.418892\pi$
$114$	−17.5559	−1.64426
$115$	0	0
$116$	−39.8793	−3.70270
$117$	44.0719	4.07445
$118$	−14.0708	−1.29532
$119$	−19.5806	−1.79495
$120$	0	0
$121$	−8.74830	−0.795300
$122$	6.87416	0.622357
$123$	−26.8891	−2.42451
$124$	28.3350	2.54456
$125$	0	0
$126$	46.4662	4.13954
$127$	6.20549	0.550648	0.275324	−	0.961351i	$-0.411215\pi$
$127$	6.20549	0.550648	0.275324	+	0.961351i	$0.411215\pi$
$128$	14.9994	1.32578
$129$	38.3591	3.37733
$130$	0	0
$131$	12.3229	1.07666	0.538328	−	0.842736i	$-0.319056\pi$
$131$	12.3229	1.07666	0.538328	+	0.842736i	$0.319056\pi$
$132$	−20.1508	−1.75390
$133$	6.29582	0.545916
$134$	12.4416	1.07479
$135$	0	0
$136$	−41.1207	−3.52607
$137$	−1.25500	−0.107222	−0.0536109	−	0.998562i	$-0.517073\pi$
$137$	−1.25500	−0.107222	−0.0536109	+	0.998562i	$0.517073\pi$
$138$	−11.6303	−0.990040
$139$	3.33206	0.282622	0.141311	−	0.989965i	$-0.454868\pi$
$139$	3.33206	0.282622	0.141311	+	0.989965i	$0.454868\pi$
$140$	0	0
$141$	18.0160	1.51722
$142$	14.5762	1.22321
$143$	10.0172	0.837679
$144$	40.3608	3.36340
$145$	0	0
$146$	−23.2119	−1.92103
$147$	−2.54471	−0.209884
$148$	−2.78118	−0.228611
$149$	−14.1881	−1.16233	−0.581167	−	0.813784i	$-0.697404\pi$
$149$	−14.1881	−1.16233	−0.581167	+	0.813784i	$0.697404\pi$
$150$	0	0
$151$	1.14050	0.0928122	0.0464061	−	0.998923i	$-0.485223\pi$
$151$	1.14050	0.0928122	0.0464061	+	0.998923i	$0.485223\pi$
$152$	13.2217	1.07242
$153$	46.2233	3.73693
$154$	10.5614	0.851060
$155$	0	0
$156$	−89.6453	−7.17737
$157$	0.995495	0.0794492	0.0397246	−	0.999211i	$-0.487352\pi$
$157$	0.995495	0.0794492	0.0397246	+	0.999211i	$0.487352\pi$
$158$	32.4537	2.58188
$159$	−5.35350	−0.424561
$160$	0	0
$161$	4.17082	0.328707
$162$	−37.1960	−2.92239
$163$	−11.8235	−0.926087	−0.463044	−	0.886336i	$-0.653243\pi$
$163$	−11.8235	−0.926087	−0.463044	+	0.886336i	$0.653243\pi$
$164$	37.6058	2.93652
$165$	0	0
$166$	−12.5837	−0.976685
$167$	8.97926	0.694836	0.347418	−	0.937710i	$-0.387058\pi$
$167$	8.97926	0.694836	0.347418	+	0.937710i	$0.387058\pi$
$168$	−50.8965	−3.92675
$169$	31.5636	2.42797
$170$	0	0
$171$	−14.8623	−1.13655
$172$	−53.6471	−4.09055
$173$	−7.00469	−0.532556	−0.266278	−	0.963896i	$-0.585794\pi$
$173$	−7.00469	−0.532556	−0.266278	+	0.963896i	$0.585794\pi$
$174$	−71.7625	−5.44030
$175$	0	0
$176$	9.17368	0.691492
$177$	−17.3249	−1.30222
$178$	11.2025	0.839662
$179$	−18.7865	−1.40417	−0.702084	−	0.712094i	$-0.747747\pi$
$179$	−18.7865	−1.40417	−0.702084	+	0.712094i	$0.747747\pi$
$180$	0	0
$181$	7.49173	0.556856	0.278428	−	0.960457i	$-0.410187\pi$
$181$	7.49173	0.556856	0.278428	+	0.960457i	$0.410187\pi$
$182$	46.9846	3.48273
$183$	8.46390	0.625670
$184$	8.75902	0.645724
$185$	0	0
$186$	50.9887	3.73867
$187$	10.5062	0.768287
$188$	−25.1963	−1.83763
$189$	31.2142	2.27050
$190$	0	0
$191$	12.5241	0.906211	0.453106	−	0.891457i	$-0.350316\pi$
$191$	12.5241	0.906211	0.453106	+	0.891457i	$0.350316\pi$
$192$	9.50621	0.686052
$193$	−5.44211	−0.391732	−0.195866	−	0.980631i	$-0.562752\pi$
$193$	−5.44211	−0.391732	−0.195866	+	0.980631i	$0.562752\pi$
$194$	32.0483	2.30094
$195$	0	0
$196$	3.55890	0.254207
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	−24.9319	−1.77183
$199$	12.9094	0.915120	0.457560	−	0.889179i	$-0.348723\pi$
$199$	12.9094	0.915120	0.457560	+	0.889179i	$0.348723\pi$
$200$	0	0
$201$	15.3189	1.08051
$202$	6.00396	0.422437
$203$	25.7351	1.80625
$204$	−94.0213	−6.58281
$205$	0	0
$206$	−12.8968	−0.898561
$207$	−9.84591	−0.684338
$208$	40.8111	2.82974
$209$	−3.37808	−0.233667
$210$	0	0
$211$	−12.2066	−0.840339	−0.420169	−	0.907446i	$-0.638029\pi$
$211$	−12.2066	−0.840339	−0.420169	+	0.907446i	$0.638029\pi$
$212$	7.48715	0.514219
$213$	17.9472	1.22972
$214$	5.04745	0.345036
$215$	0	0
$216$	65.5521	4.46026
$217$	−18.2853	−1.24129
$218$	28.3967	1.92327
$219$	−28.5800	−1.93125
$220$	0	0
$221$	46.7390	3.14401
$222$	−5.00471	−0.335894
$223$	7.53212	0.504388	0.252194	−	0.967677i	$-0.418848\pi$
$223$	7.53212	0.504388	0.252194	+	0.967677i	$0.418848\pi$
$224$	10.1780	0.680048
$225$	0	0
$226$	−13.4866	−0.897114
$227$	13.1744	0.874417	0.437209	−	0.899360i	$-0.355967\pi$
$227$	13.1744	0.874417	0.437209	+	0.899360i	$0.355967\pi$
$228$	30.2310	2.00209
$229$	2.45091	0.161960	0.0809802	−	0.996716i	$-0.474195\pi$
$229$	2.45091	0.161960	0.0809802	+	0.996716i	$0.474195\pi$
$230$	0	0
$231$	13.0038	0.855590
$232$	54.0456	3.54827
$233$	18.4140	1.20634	0.603172	−	0.797611i	$-0.293903\pi$
$233$	18.4140	1.20634	0.603172	+	0.797611i	$0.293903\pi$
$234$	−110.915	−7.25074
$235$	0	0
$236$	24.2297	1.57722
$237$	39.9591	2.59562
$238$	49.2782	3.19423
$239$	−14.5327	−0.940043	−0.470021	−	0.882655i	$-0.655754\pi$
$239$	−14.5327	−0.940043	−0.470021	+	0.882655i	$0.655754\pi$
$240$	0	0
$241$	7.99412	0.514946	0.257473	−	0.966285i	$-0.417110\pi$
$241$	7.99412	0.514946	0.257473	+	0.966285i	$0.417110\pi$
$242$	22.0167	1.41529
$243$	−12.3141	−0.789950
$244$	−11.8372	−0.757799
$245$	0	0
$246$	67.6713	4.31456
$247$	−15.0281	−0.956217
$248$	−38.4005	−2.43843
$249$	−15.4939	−0.981884
$250$	0	0
$251$	−2.72917	−0.172264	−0.0861319	−	0.996284i	$-0.527451\pi$
$251$	−2.72917	−0.172264	−0.0861319	+	0.996284i	$0.527451\pi$
$252$	−80.0141	−5.04041
$253$	−2.23790	−0.140695
$254$	−15.6172	−0.979913
$255$	0	0
$256$	−31.6132	−1.97583
$257$	1.43751	0.0896693	0.0448347	−	0.998994i	$-0.485724\pi$
$257$	1.43751	0.0896693	0.0448347	+	0.998994i	$0.485724\pi$
$258$	−96.5376	−6.01017
$259$	1.79477	0.111521
$260$	0	0
$261$	−60.7521	−3.76046
$262$	−31.0128	−1.91598
$263$	3.08666	0.190332	0.0951659	−	0.995461i	$-0.469662\pi$
$263$	3.08666	0.190332	0.0951659	+	0.995461i	$0.469662\pi$
$264$	27.3090	1.68075
$265$	0	0
$266$	−15.8446	−0.971493
$267$	13.7932	0.844131
$268$	−21.4242	−1.30869
$269$	27.6805	1.68771	0.843855	−	0.536572i	$-0.180281\pi$
$269$	27.6805	1.68771	0.843855	+	0.536572i	$0.180281\pi$
$270$	0	0
$271$	2.06718	0.125572	0.0627861	−	0.998027i	$-0.480001\pi$
$271$	2.06718	0.125572	0.0627861	+	0.998027i	$0.480001\pi$
$272$	42.8033	2.59533
$273$	57.8505	3.50127
$274$	3.15843	0.190808
$275$	0	0
$276$	20.0273	1.20550
$277$	−16.8915	−1.01491	−0.507457	−	0.861677i	$-0.669414\pi$
$277$	−16.8915	−1.01491	−0.507457	+	0.861677i	$0.669414\pi$
$278$	−8.38574	−0.502943
$279$	43.1655	2.58425
$280$	0	0
$281$	9.29490	0.554487	0.277244	−	0.960800i	$-0.410579\pi$
$281$	9.29490	0.554487	0.277244	+	0.960800i	$0.410579\pi$
$282$	−45.3406	−2.70000
$283$	−22.2508	−1.32268	−0.661338	−	0.750088i	$-0.730011\pi$
$283$	−22.2508	−1.32268	−0.661338	+	0.750088i	$0.730011\pi$
$284$	−25.1001	−1.48941
$285$	0	0
$286$	−25.2101	−1.49070
$287$	−24.2680	−1.43249
$288$	−24.0269	−1.41580
$289$	32.0206	1.88356
$290$	0	0
$291$	39.4600	2.31318
$292$	39.9705	2.33910
$293$	−25.6859	−1.50059	−0.750295	−	0.661104i	$-0.770088\pi$
$293$	−25.6859	−1.50059	−0.750295	+	0.661104i	$0.770088\pi$
$294$	6.40422	0.373502
$295$	0	0
$296$	3.76914	0.219077
$297$	−16.7483	−0.971835
$298$	35.7070	2.06845
$299$	−9.95577	−0.575757
$300$	0	0
$301$	34.6199	1.99546
$302$	−2.87027	−0.165165
$303$	7.39246	0.424686
$304$	−13.7627	−0.789344
$305$	0	0
$306$	−116.329	−6.65010
$307$	−15.0185	−0.857149	−0.428574	−	0.903507i	$-0.640984\pi$
$307$	−15.0185	−0.857149	−0.428574	+	0.903507i	$0.640984\pi$
$308$	−18.1865	−1.03627
$309$	−15.8793	−0.903343
$310$	0	0
$311$	28.4554	1.61356	0.806779	−	0.590853i	$-0.201209\pi$
$311$	28.4554	1.61356	0.806779	+	0.590853i	$0.201209\pi$
$312$	121.490	6.87802
$313$	−18.8702	−1.06660	−0.533302	−	0.845925i	$-0.679049\pi$
$313$	−18.8702	−1.06660	−0.533302	+	0.845925i	$0.679049\pi$
$314$	−2.50535	−0.141385
$315$	0	0
$316$	−55.8849	−3.14377
$317$	5.87594	0.330025	0.165013	−	0.986291i	$-0.447233\pi$
$317$	5.87594	0.330025	0.165013	+	0.986291i	$0.447233\pi$
$318$	13.4731	0.755532
$319$	−13.8084	−0.773124
$320$	0	0
$321$	6.21474	0.346873
$322$	−10.4966	−0.584955
$323$	−15.7617	−0.877007
$324$	64.0509	3.55838
$325$	0	0
$326$	29.7560	1.64803
$327$	34.9638	1.93350
$328$	−50.9645	−2.81404
$329$	16.2599	0.896435
$330$	0	0
$331$	13.0218	0.715744	0.357872	−	0.933771i	$-0.383502\pi$
$331$	13.0218	0.715744	0.357872	+	0.933771i	$0.383502\pi$
$332$	21.6690	1.18924
$333$	−4.23685	−0.232178
$334$	−22.5979	−1.23650
$335$	0	0
$336$	52.9791	2.89025
$337$	−8.22154	−0.447856	−0.223928	−	0.974606i	$-0.571888\pi$
$337$	−8.22154	−0.447856	−0.223928	+	0.974606i	$0.571888\pi$
$338$	−79.4356	−4.32073
$339$	−16.6055	−0.901889
$340$	0	0
$341$	9.81118	0.531305
$342$	37.4037	2.02256
$343$	17.2799	0.933024
$344$	72.7043	3.91995
$345$	0	0
$346$	17.6286	0.947718
$347$	−14.9607	−0.803130	−0.401565	−	0.915831i	$-0.631534\pi$
$347$	−14.9607	−0.803130	−0.401565	+	0.915831i	$0.631534\pi$
$348$	123.574	6.62425
$349$	−17.8589	−0.955968	−0.477984	−	0.878369i	$-0.658632\pi$
$349$	−17.8589	−0.955968	−0.477984	+	0.878369i	$0.658632\pi$
$350$	0	0
$351$	−74.5085	−3.97697
$352$	−5.46112	−0.291079
$353$	−23.4918	−1.25034	−0.625171	−	0.780488i	$-0.714971\pi$
$353$	−23.4918	−1.25034	−0.625171	+	0.780488i	$0.714971\pi$
$354$	43.6012	2.31738
$355$	0	0
$356$	−19.2905	−1.02239
$357$	60.6745	3.21123
$358$	47.2796	2.49881
$359$	12.6629	0.668323	0.334162	−	0.942516i	$-0.391547\pi$
$359$	12.6629	0.668323	0.334162	+	0.942516i	$0.391547\pi$
$360$	0	0
$361$	−13.9321	−0.733267
$362$	−18.8543	−0.990961
$363$	27.1084	1.42282
$364$	−80.9068	−4.24067
$365$	0	0
$366$	−21.3009	−1.11342
$367$	−17.6279	−0.920171	−0.460085	−	0.887875i	$-0.652181\pi$
$367$	−17.6279	−0.920171	−0.460085	+	0.887875i	$0.652181\pi$
$368$	−9.11744	−0.475279
$369$	57.2886	2.98232
$370$	0	0
$371$	−4.83166	−0.250847
$372$	−87.8017	−4.55231
$373$	−4.63168	−0.239819	−0.119910	−	0.992785i	$-0.538260\pi$
$373$	−4.63168	−0.239819	−0.119910	+	0.992785i	$0.538260\pi$
$374$	−26.4407	−1.36722
$375$	0	0
$376$	34.1469	1.76099
$377$	−61.4299	−3.16380
$378$	−78.5563	−4.04050
$379$	−20.0728	−1.03107	−0.515535	−	0.856869i	$-0.672407\pi$
$379$	−20.0728	−1.03107	−0.515535	+	0.856869i	$0.672407\pi$
$380$	0	0
$381$	−19.2289	−0.985129
$382$	−31.5191	−1.61266
$383$	−13.6174	−0.695814	−0.347907	−	0.937529i	$-0.613108\pi$
$383$	−13.6174	−0.695814	−0.347907	+	0.937529i	$0.613108\pi$
$384$	−46.4788	−2.37186
$385$	0	0
$386$	13.6961	0.697111
$387$	−81.7260	−4.15437
$388$	−55.1867	−2.80168
$389$	16.3340	0.828166	0.414083	−	0.910239i	$-0.364102\pi$
$389$	16.3340	0.828166	0.414083	+	0.910239i	$0.364102\pi$
$390$	0	0
$391$	−10.4418	−0.528063
$392$	−4.82314	−0.243605
$393$	−38.1849	−1.92617
$394$	−2.51668	−0.126789
$395$	0	0
$396$	42.9323	2.15743
$397$	−5.84689	−0.293447	−0.146723	−	0.989178i	$-0.546873\pi$
$397$	−5.84689	−0.293447	−0.146723	+	0.989178i	$0.546873\pi$
$398$	−32.4888	−1.62851
$399$	−19.5088	−0.976664
$400$	0	0
$401$	−36.7175	−1.83359	−0.916793	−	0.399363i	$-0.869231\pi$
$401$	−36.7175	−1.83359	−0.916793	+	0.399363i	$0.869231\pi$
$402$	−38.5527	−1.92284
$403$	43.6472	2.17422
$404$	−10.3387	−0.514371
$405$	0	0
$406$	−64.7672	−3.21434
$407$	−0.963000	−0.0477342
$408$	127.421	6.30826
$409$	21.2972	1.05308	0.526540	−	0.850150i	$-0.323489\pi$
$409$	21.2972	1.05308	0.526540	+	0.850150i	$0.323489\pi$
$410$	0	0
$411$	3.88886	0.191823
$412$	22.2080	1.09411
$413$	−15.6361	−0.769401
$414$	24.7790	1.21782
$415$	0	0
$416$	−24.2950	−1.19116
$417$	−10.3251	−0.505621
$418$	8.50155	0.415825
$419$	32.6019	1.59271	0.796354	−	0.604831i	$-0.206759\pi$
$419$	32.6019	1.59271	0.796354	+	0.604831i	$0.206759\pi$
$420$	0	0
$421$	−18.9139	−0.921809	−0.460904	−	0.887450i	$-0.652475\pi$
$421$	−18.9139	−0.921809	−0.460904	+	0.887450i	$0.652475\pi$
$422$	30.7202	1.49544
$423$	−38.3841	−1.86630
$424$	−10.1468	−0.492773
$425$	0	0
$426$	−45.1674	−2.18837
$427$	7.63886	0.369670
$428$	−8.69163	−0.420126
$429$	−31.0402	−1.49864
$430$	0	0
$431$	−8.91965	−0.429644	−0.214822	−	0.976653i	$-0.568917\pi$
$431$	−8.91965	−0.429644	−0.214822	+	0.976653i	$0.568917\pi$
$432$	−68.2345	−3.28293
$433$	−38.5494	−1.85257	−0.926283	−	0.376828i	$-0.877015\pi$
$433$	−38.5494	−1.85257	−0.926283	+	0.376828i	$0.877015\pi$
$434$	46.0184	2.20895
$435$	0	0
$436$	−48.8987	−2.34182
$437$	3.35737	0.160605
$438$	71.9267	3.43679
$439$	0.871605	0.0415994	0.0207997	−	0.999784i	$-0.493379\pi$
$439$	0.871605	0.0415994	0.0207997	+	0.999784i	$0.493379\pi$
$440$	0	0
$441$	5.42163	0.258173
$442$	−117.627	−5.59496
$443$	21.9551	1.04312	0.521558	−	0.853216i	$-0.325351\pi$
$443$	21.9551	1.04312	0.521558	+	0.853216i	$0.325351\pi$
$444$	8.61804	0.408994
$445$	0	0
$446$	−18.9559	−0.897590
$447$	43.9647	2.07946
$448$	8.57957	0.405346
$449$	−34.0961	−1.60909	−0.804547	−	0.593889i	$-0.797592\pi$
$449$	−34.0961	−1.60909	−0.804547	+	0.593889i	$0.797592\pi$
$450$	0	0
$451$	13.0212	0.613146
$452$	23.2237	1.09235
$453$	−3.53405	−0.166044
$454$	−33.1558	−1.55608
$455$	0	0
$456$	−40.9699	−1.91859
$457$	17.7967	0.832494	0.416247	−	0.909252i	$-0.363345\pi$
$457$	17.7967	0.832494	0.416247	+	0.909252i	$0.363345\pi$
$458$	−6.16815	−0.288219
$459$	−78.1457	−3.64753
$460$	0	0
$461$	−10.2487	−0.477328	−0.238664	−	0.971102i	$-0.576709\pi$
$461$	−10.2487	−0.477328	−0.238664	+	0.971102i	$0.576709\pi$
$462$	−32.7265	−1.52258
$463$	18.6577	0.867096	0.433548	−	0.901131i	$-0.357262\pi$
$463$	18.6577	0.867096	0.433548	+	0.901131i	$0.357262\pi$
$464$	−56.2571	−2.61167
$465$	0	0
$466$	−46.3423	−2.14676
$467$	−5.79159	−0.268003	−0.134001	−	0.990981i	$-0.542783\pi$
$467$	−5.79159	−0.268003	−0.134001	+	0.990981i	$0.542783\pi$
$468$	190.994	8.82869
$469$	13.8256	0.638408
$470$	0	0
$471$	−3.08474	−0.142137
$472$	−32.8369	−1.51144
$473$	−18.5757	−0.854109
$474$	−100.564	−4.61908
$475$	0	0
$476$	−84.8563	−3.88938
$477$	11.4059	0.522241
$478$	36.5742	1.67287
$479$	−26.4049	−1.20647	−0.603236	−	0.797562i	$-0.706122\pi$
$479$	−26.4049	−1.20647	−0.603236	+	0.797562i	$0.706122\pi$
$480$	0	0
$481$	−4.28412	−0.195339
$482$	−20.1187	−0.916379
$483$	−12.9241	−0.588068
$484$	−37.9124	−1.72329
$485$	0	0
$486$	30.9907	1.40577
$487$	−13.7452	−0.622856	−0.311428	−	0.950270i	$-0.600807\pi$
$487$	−13.7452	−0.622856	−0.311428	+	0.950270i	$0.600807\pi$
$488$	16.0421	0.726193
$489$	36.6374	1.65680
$490$	0	0
$491$	35.9494	1.62238	0.811188	−	0.584786i	$-0.198822\pi$
$491$	35.9494	1.62238	0.811188	+	0.584786i	$0.198822\pi$
$492$	−116.529	−5.25353
$493$	−64.4286	−2.90172
$494$	37.8210	1.70165
$495$	0	0
$496$	39.9718	1.79479
$497$	16.1977	0.726568
$498$	38.9931	1.74732
$499$	−13.8967	−0.622102	−0.311051	−	0.950393i	$-0.600681\pi$
$499$	−13.8967	−0.622102	−0.311051	+	0.950393i	$0.600681\pi$
$500$	0	0
$501$	−27.8240	−1.24309
$502$	6.86846	0.306554
$503$	2.34843	0.104711	0.0523557	−	0.998629i	$-0.483327\pi$
$503$	2.34843	0.104711	0.0523557	+	0.998629i	$0.483327\pi$
$504$	108.438	4.83019
$505$	0	0
$506$	5.63207	0.250376
$507$	−97.8062	−4.34373
$508$	26.8927	1.19317
$509$	32.3485	1.43382	0.716911	−	0.697165i	$-0.245556\pi$
$509$	32.3485	1.43382	0.716911	+	0.697165i	$0.245556\pi$
$510$	0	0
$511$	−25.7940	−1.14106
$512$	49.5615	2.19033
$513$	25.1264	1.10936
$514$	−3.61775	−0.159572
$515$	0	0
$516$	166.236	7.31815
$517$	−8.72439	−0.383698
$518$	−4.51686	−0.198459
$519$	21.7054	0.952762
$520$	0	0
$521$	−38.6437	−1.69301	−0.846505	−	0.532381i	$-0.821297\pi$
$521$	−38.6437	−1.69301	−0.846505	+	0.532381i	$0.821297\pi$
$522$	152.894	6.69197
$523$	−35.5348	−1.55383	−0.776914	−	0.629607i	$-0.783216\pi$
$523$	−35.5348	−1.55383	−0.776914	+	0.629607i	$0.783216\pi$
$524$	53.4035	2.33294
$525$	0	0
$526$	−7.76815	−0.338708
$527$	45.7778	1.99411
$528$	−28.4265	−1.23710
$529$	−20.7758	−0.903297
$530$	0	0
$531$	36.9115	1.60182
$532$	27.2841	1.18292
$533$	57.9278	2.50913
$534$	−34.7131	−1.50218
$535$	0	0
$536$	29.0348	1.25411
$537$	58.2137	2.51211
$538$	−69.6630	−3.00339
$539$	1.23229	0.0530786
$540$	0	0
$541$	17.7932	0.764990	0.382495	−	0.923958i	$-0.375065\pi$
$541$	17.7932	0.764990	0.382495	+	0.923958i	$0.375065\pi$
$542$	−5.20244	−0.223464
$543$	−23.2146	−0.996235
$544$	−25.4810	−1.09249
$545$	0	0
$546$	−145.591	−6.23073
$547$	−17.9853	−0.768994	−0.384497	−	0.923126i	$-0.625625\pi$
$547$	−17.9853	−0.768994	−0.384497	+	0.923126i	$0.625625\pi$
$548$	−5.43877	−0.232333
$549$	−18.0328	−0.769620
$550$	0	0
$551$	20.7159	0.882528
$552$	−27.1416	−1.15522
$553$	36.0640	1.53360
$554$	42.5106	1.80610
$555$	0	0
$556$	14.4401	0.612398
$557$	13.1986	0.559241	0.279620	−	0.960111i	$-0.409791\pi$
$557$	13.1986	0.559241	0.279620	+	0.960111i	$0.409791\pi$
$558$	−108.634	−4.59884
$559$	−82.6379	−3.49521
$560$	0	0
$561$	−32.5555	−1.37449
$562$	−23.3923	−0.986745
$563$	7.69263	0.324206	0.162103	−	0.986774i	$-0.448172\pi$
$563$	7.69263	0.324206	0.162103	+	0.986774i	$0.448172\pi$
$564$	78.0759	3.28759
$565$	0	0
$566$	55.9983	2.35378
$567$	−41.3338	−1.73585
$568$	34.0164	1.42730
$569$	−6.89152	−0.288907	−0.144454	−	0.989512i	$-0.546142\pi$
$569$	−6.89152	−0.288907	−0.144454	+	0.989512i	$0.546142\pi$
$570$	0	0
$571$	1.56325	0.0654198	0.0327099	−	0.999465i	$-0.489586\pi$
$571$	1.56325	0.0654198	0.0327099	+	0.999465i	$0.489586\pi$
$572$	43.4113	1.81512
$573$	−38.8084	−1.62124
$574$	61.0748	2.54921
$575$	0	0
$576$	−20.2535	−0.843895
$577$	21.1132	0.878955	0.439477	−	0.898254i	$-0.355164\pi$
$577$	21.1132	0.878955	0.439477	+	0.898254i	$0.355164\pi$
$578$	−80.5857	−3.35192
$579$	16.8635	0.700822
$580$	0	0
$581$	−13.9836	−0.580135
$582$	−99.3081	−4.11646
$583$	2.59247	0.107369
$584$	−54.1693	−2.24154
$585$	0	0
$586$	64.6434	2.67039
$587$	39.6950	1.63839	0.819194	−	0.573517i	$-0.194421\pi$
$587$	39.6950	1.63839	0.819194	+	0.573517i	$0.194421\pi$
$588$	−11.0280	−0.454786
$589$	−14.7191	−0.606489
$590$	0	0
$591$	−3.09870	−0.127463
$592$	−3.92337	−0.161250
$593$	25.9310	1.06486	0.532429	−	0.846475i	$-0.321279\pi$
$593$	25.9310	1.06486	0.532429	+	0.846475i	$0.321279\pi$
$594$	42.1502	1.72944
$595$	0	0
$596$	−61.4868	−2.51860
$597$	−40.0022	−1.63718
$598$	25.0555	1.02460
$599$	12.6863	0.518346	0.259173	−	0.965831i	$-0.416550\pi$
$599$	12.6863	0.518346	0.259173	+	0.965831i	$0.416550\pi$
$600$	0	0
$601$	−43.2012	−1.76221	−0.881107	−	0.472918i	$-0.843201\pi$
$601$	−43.2012	−1.76221	−0.881107	+	0.472918i	$0.843201\pi$
$602$	−87.1273	−3.55104
$603$	−32.6376	−1.32911
$604$	4.94255	0.201110
$605$	0	0
$606$	−18.6045	−0.755755
$607$	36.2528	1.47146	0.735728	−	0.677278i	$-0.236840\pi$
$607$	36.2528	1.47146	0.735728	+	0.677278i	$0.236840\pi$
$608$	8.19297	0.332269
$609$	−79.7455	−3.23145
$610$	0	0
$611$	−38.8124	−1.57018
$612$	200.317	8.09735
$613$	−21.3720	−0.863206	−0.431603	−	0.902064i	$-0.642052\pi$
$613$	−21.3720	−0.863206	−0.431603	+	0.902064i	$0.642052\pi$
$614$	37.7967	1.52535
$615$	0	0
$616$	24.6470	0.993055
$617$	−38.6984	−1.55794	−0.778969	−	0.627063i	$-0.784257\pi$
$617$	−38.6984	−1.55794	−0.778969	+	0.627063i	$0.784257\pi$
$618$	39.9632	1.60756
$619$	−32.5065	−1.30655	−0.653274	−	0.757122i	$-0.726605\pi$
$619$	−32.5065	−1.30655	−0.653274	+	0.757122i	$0.726605\pi$
$620$	0	0
$621$	16.6456	0.667966
$622$	−71.6132	−2.87143
$623$	12.4487	0.498746
$624$	−126.461	−5.06251
$625$	0	0
$626$	47.4902	1.89809
$627$	10.4677	0.418038
$628$	4.31417	0.172154
$629$	−4.49325	−0.179158
$630$	0	0
$631$	−10.6698	−0.424758	−0.212379	−	0.977187i	$-0.568121\pi$
$631$	−10.6698	−0.424758	−0.212379	+	0.977187i	$0.568121\pi$
$632$	75.7369	3.01265
$633$	37.8247	1.50340
$634$	−14.7879	−0.587301
$635$	0	0
$636$	−23.2004	−0.919956
$637$	5.48212	0.217210
$638$	34.7515	1.37582
$639$	−38.2374	−1.51265
$640$	0	0
$641$	19.5872	0.773646	0.386823	−	0.922154i	$-0.373572\pi$
$641$	19.5872	0.773646	0.386823	+	0.922154i	$0.373572\pi$
$642$	−15.6405	−0.617282
$643$	−24.6413	−0.971760	−0.485880	−	0.874026i	$-0.661501\pi$
$643$	−24.6413	−0.971760	−0.485880	+	0.874026i	$0.661501\pi$
$644$	18.0750	0.712257
$645$	0	0
$646$	39.6673	1.56069
$647$	−5.53370	−0.217552	−0.108776	−	0.994066i	$-0.534693\pi$
$647$	−5.53370	−0.217552	−0.108776	+	0.994066i	$0.534693\pi$
$648$	−86.8038	−3.40998
$649$	8.38969	0.329324
$650$	0	0
$651$	56.6608	2.22071
$652$	−51.2393	−2.00669
$653$	18.3451	0.717898	0.358949	−	0.933357i	$-0.383135\pi$
$653$	18.3451	0.717898	0.358949	+	0.933357i	$0.383135\pi$
$654$	−87.9928	−3.44079
$655$	0	0
$656$	53.0499	2.07125
$657$	60.8911	2.37559
$658$	−40.9209	−1.59526
$659$	−25.6105	−0.997642	−0.498821	−	0.866705i	$-0.666233\pi$
$659$	−25.6105	−0.997642	−0.498821	+	0.866705i	$0.666233\pi$
$660$	0	0
$661$	−2.11336	−0.0822003	−0.0411002	−	0.999155i	$-0.513086\pi$
$661$	−2.11336	−0.0822003	−0.0411002	+	0.999155i	$0.513086\pi$
$662$	−32.7718	−1.27371
$663$	−144.830	−5.62474
$664$	−29.3664	−1.13964
$665$	0	0
$666$	10.6628	0.413175
$667$	13.7238	0.531387
$668$	38.9133	1.50560
$669$	−23.3398	−0.902367
$670$	0	0
$671$	−4.09870	−0.158229
$672$	−31.5387	−1.21663
$673$	−6.51416	−0.251102	−0.125551	−	0.992087i	$-0.540070\pi$
$673$	−6.51416	−0.251102	−0.125551	+	0.992087i	$0.540070\pi$
$674$	20.6910	0.796987
$675$	0	0
$676$	136.787	5.26104
$677$	−30.9997	−1.19141	−0.595707	−	0.803202i	$-0.703128\pi$
$677$	−30.9997	−1.19141	−0.595707	+	0.803202i	$0.703128\pi$
$678$	41.7909	1.60497
$679$	35.6135	1.36672
$680$	0	0
$681$	−40.8236	−1.56436
$682$	−24.6916	−0.945491
$683$	8.62247	0.329930	0.164965	−	0.986299i	$-0.447249\pi$
$683$	8.62247	0.329930	0.164965	+	0.986299i	$0.447249\pi$
$684$	−64.4086	−2.46272
$685$	0	0
$686$	−43.4879	−1.66038
$687$	−7.59462	−0.289753
$688$	−75.6793	−2.88525
$689$	11.5332	0.439379
$690$	0	0
$691$	−12.1350	−0.461639	−0.230819	−	0.972997i	$-0.574141\pi$
$691$	−12.1350	−0.461639	−0.230819	+	0.972997i	$0.574141\pi$
$692$	−30.3561	−1.15397
$693$	−27.7054	−1.05244
$694$	37.6512	1.42922
$695$	0	0
$696$	−167.471	−6.34798
$697$	60.7555	2.30128
$698$	44.9453	1.70120
$699$	−57.0596	−2.15819
$700$	0	0
$701$	32.6236	1.23218	0.616089	−	0.787677i	$-0.288716\pi$
$701$	32.6236	1.23218	0.616089	+	0.787677i	$0.288716\pi$
$702$	187.514	7.07727
$703$	1.44473	0.0544889
$704$	−4.60345	−0.173499
$705$	0	0
$706$	59.1213	2.22506
$707$	6.67186	0.250921
$708$	−75.0806	−2.82170
$709$	−7.57943	−0.284651	−0.142326	−	0.989820i	$-0.545458\pi$
$709$	−7.57943	−0.284651	−0.142326	+	0.989820i	$0.545458\pi$
$710$	0	0
$711$	−85.1350	−3.19281
$712$	26.1431	0.979754
$713$	−9.75103	−0.365179
$714$	−152.698	−5.71459
$715$	0	0
$716$	−81.4148	−3.04261
$717$	45.0325	1.68177
$718$	−31.8685	−1.18932
$719$	32.1975	1.20076	0.600382	−	0.799714i	$-0.295015\pi$
$719$	32.1975	1.20076	0.600382	+	0.799714i	$0.295015\pi$
$720$	0	0
$721$	−14.3314	−0.533731
$722$	35.0626	1.30490
$723$	−24.7714	−0.921257
$724$	32.4668	1.20662
$725$	0	0
$726$	−68.2231	−2.53200
$727$	−50.8989	−1.88773	−0.943867	−	0.330324i	$-0.892842\pi$
$727$	−50.8989	−1.88773	−0.943867	+	0.330324i	$0.892842\pi$
$728$	109.647	4.06381
$729$	−6.18162	−0.228949
$730$	0	0
$731$	−86.6719	−3.20567
$732$	36.6799	1.35573
$733$	25.8335	0.954180	0.477090	−	0.878854i	$-0.341691\pi$
$733$	25.8335	0.954180	0.477090	+	0.878854i	$0.341691\pi$
$734$	44.3639	1.63750
$735$	0	0
$736$	5.42764	0.200066
$737$	−7.41827	−0.273255
$738$	−144.177	−5.30723
$739$	−18.0128	−0.662613	−0.331307	−	0.943523i	$-0.607489\pi$
$739$	−18.0128	−0.662613	−0.331307	+	0.943523i	$0.607489\pi$
$740$	0	0
$741$	46.5677	1.71071
$742$	12.1597	0.446398
$743$	−37.3102	−1.36878	−0.684388	−	0.729118i	$-0.739931\pi$
$743$	−37.3102	−1.36878	−0.684388	+	0.729118i	$0.739931\pi$
$744$	118.992	4.36244
$745$	0	0
$746$	11.6565	0.426773
$747$	33.0105	1.20779
$748$	45.5305	1.66476
$749$	5.60894	0.204946
$750$	0	0
$751$	21.0651	0.768676	0.384338	−	0.923192i	$-0.374430\pi$
$751$	21.0651	0.768676	0.384338	+	0.923192i	$0.374430\pi$
$752$	−35.5441	−1.29616
$753$	8.45689	0.308186
$754$	154.600	5.63018
$755$	0	0
$756$	135.273	4.91982
$757$	−10.8971	−0.396063	−0.198032	−	0.980196i	$-0.563455\pi$
$757$	−10.8971	−0.396063	−0.198032	+	0.980196i	$0.563455\pi$
$758$	50.5168	1.83485
$759$	6.93457	0.251709
$760$	0	0
$761$	24.0495	0.871793	0.435897	−	0.899997i	$-0.356431\pi$
$761$	24.0495	0.871793	0.435897	+	0.899997i	$0.356431\pi$
$762$	48.3931	1.75310
$763$	31.5556	1.14239
$764$	54.2755	1.96362
$765$	0	0
$766$	34.2705	1.23825
$767$	37.3234	1.34767
$768$	97.9598	3.53482
$769$	37.7607	1.36169	0.680843	−	0.732429i	$-0.261613\pi$
$769$	37.7607	1.36169	0.680843	+	0.732429i	$0.261613\pi$
$770$	0	0
$771$	−4.45441	−0.160422
$772$	−23.5844	−0.848821
$773$	4.04749	0.145578	0.0727891	−	0.997347i	$-0.476810\pi$
$773$	4.04749	0.145578	0.0727891	+	0.997347i	$0.476810\pi$
$774$	205.678	7.39296
$775$	0	0
$776$	74.7908	2.68483
$777$	−5.56145	−0.199516
$778$	−41.1074	−1.47377
$779$	−19.5349	−0.699911
$780$	0	0
$781$	−8.69105	−0.310990
$782$	26.2786	0.939721
$783$	102.708	3.67049
$784$	5.02049	0.179303
$785$	0	0
$786$	96.0993	3.42775
$787$	−26.4661	−0.943416	−0.471708	−	0.881755i	$-0.656362\pi$
$787$	−26.4661	−0.943416	−0.471708	+	0.881755i	$0.656362\pi$
$788$	4.33369	0.154381
$789$	−9.56464	−0.340510
$790$	0	0
$791$	−14.9869	−0.532872
$792$	−58.1832	−2.06745
$793$	−18.2340	−0.647508
$794$	14.7148	0.522207
$795$	0	0
$796$	55.9451	1.98292
$797$	38.7017	1.37088	0.685442	−	0.728127i	$-0.259609\pi$
$797$	38.7017	1.37088	0.685442	+	0.728127i	$0.259609\pi$
$798$	49.0975	1.73803
$799$	−40.7070	−1.44011
$800$	0	0
$801$	−29.3871	−1.03834
$802$	92.4063	3.26298
$803$	13.8400	0.488404
$804$	66.3872	2.34130
$805$	0	0
$806$	−109.846	−3.86916
$807$	−85.7735	−3.01937
$808$	14.0114	0.492919
$809$	29.1778	1.02584	0.512919	−	0.858437i	$-0.328564\pi$
$809$	29.1778	1.02584	0.512919	+	0.858437i	$0.328564\pi$
$810$	0	0
$811$	55.7601	1.95800	0.979002	−	0.203852i	$-0.0653461\pi$
$811$	55.7601	1.95800	0.979002	+	0.203852i	$0.0653461\pi$
$812$	111.528	3.91387
$813$	−6.40557	−0.224653
$814$	2.42357	0.0849459
$815$	0	0
$816$	−132.635	−4.64314
$817$	27.8679	0.974973
$818$	−53.5984	−1.87402
$819$	−123.253	−4.30682
$820$	0	0
$821$	12.7088	0.443541	0.221771	−	0.975099i	$-0.428816\pi$
$821$	12.7088	0.443541	0.221771	+	0.975099i	$0.428816\pi$
$822$	−9.78703	−0.341362
$823$	18.3680	0.640268	0.320134	−	0.947372i	$-0.396272\pi$
$823$	18.3680	0.640268	0.320134	+	0.947372i	$0.396272\pi$
$824$	−30.0970	−1.04848
$825$	0	0
$826$	39.3510	1.36920
$827$	54.8658	1.90787	0.953936	−	0.300009i	$-0.0969898\pi$
$827$	54.8658	1.90787	0.953936	+	0.300009i	$0.0969898\pi$
$828$	−42.6691	−1.48285
$829$	9.41303	0.326928	0.163464	−	0.986549i	$-0.447733\pi$
$829$	9.41303	0.326928	0.163464	+	0.986549i	$0.447733\pi$
$830$	0	0
$831$	52.3418	1.81572
$832$	−20.4795	−0.709998
$833$	5.74973	0.199216
$834$	25.9849	0.899783
$835$	0	0
$836$	−14.6395	−0.506319
$837$	−72.9762	−2.52243
$838$	−82.0487	−2.83432
$839$	−15.8786	−0.548191	−0.274095	−	0.961702i	$-0.588378\pi$
$839$	−15.8786	−0.548191	−0.274095	+	0.961702i	$0.588378\pi$
$840$	0	0
$841$	55.6796	1.91999
$842$	47.6004	1.64042
$843$	−28.8021	−0.991997
$844$	−52.8997	−1.82088
$845$	0	0
$846$	96.6005	3.32120
$847$	24.4659	0.840658
$848$	10.5620	0.362701
$849$	68.9487	2.36631
$850$	0	0
$851$	0.957097	0.0328089
$852$	77.7776	2.66462
$853$	30.0839	1.03005	0.515027	−	0.857174i	$-0.327782\pi$
$853$	30.0839	1.03005	0.515027	+	0.857174i	$0.327782\pi$
$854$	−19.2246	−0.657851
$855$	0	0
$856$	11.7792	0.402604
$857$	−42.8903	−1.46511	−0.732553	−	0.680710i	$-0.761671\pi$
$857$	−42.8903	−1.46511	−0.732553	+	0.680710i	$0.761671\pi$
$858$	78.1184	2.66692
$859$	18.2769	0.623601	0.311801	−	0.950148i	$-0.399068\pi$
$859$	18.2769	0.623601	0.311801	+	0.950148i	$0.399068\pi$
$860$	0	0
$861$	75.1992	2.56278
$862$	22.4479	0.764579
$863$	16.2652	0.553673	0.276836	−	0.960917i	$-0.410714\pi$
$863$	16.2652	0.553673	0.276836	+	0.960917i	$0.410714\pi$
$864$	40.6202	1.38193
$865$	0	0
$866$	97.0166	3.29676
$867$	−99.2222	−3.36976
$868$	−79.2430	−2.68968
$869$	−19.3505	−0.656420
$870$	0	0
$871$	−33.0018	−1.11822
$872$	66.2690	2.24415
$873$	−84.0714	−2.84539
$874$	−8.44944	−0.285806
$875$	0	0
$876$	−123.857	−4.18473
$877$	−25.2468	−0.852524	−0.426262	−	0.904600i	$-0.640170\pi$
$877$	−25.2468	−0.852524	−0.426262	+	0.904600i	$0.640170\pi$
$878$	−2.19355	−0.0740288
$879$	79.5930	2.68461
$880$	0	0
$881$	−7.19132	−0.242282	−0.121141	−	0.992635i	$-0.538655\pi$
$881$	−7.19132	−0.242282	−0.121141	+	0.992635i	$0.538655\pi$
$882$	−13.6445	−0.459435
$883$	28.5295	0.960095	0.480048	−	0.877242i	$-0.340619\pi$
$883$	28.5295	0.960095	0.480048	+	0.877242i	$0.340619\pi$
$884$	202.552	6.81257
$885$	0	0
$886$	−55.2539	−1.85629
$887$	−34.0551	−1.14346	−0.571730	−	0.820442i	$-0.693727\pi$
$887$	−34.0551	−1.14346	−0.571730	+	0.820442i	$0.693727\pi$
$888$	−11.6794	−0.391936
$889$	−17.3545	−0.582053
$890$	0	0
$891$	22.1780	0.742992
$892$	32.6418	1.09293
$893$	13.0886	0.437995
$894$	−110.645	−3.70053
$895$	0	0
$896$	−41.9481	−1.40139
$897$	30.8499	1.03005
$898$	85.8090	2.86348
$899$	−60.1666	−2.00667
$900$	0	0
$901$	12.0962	0.402982
$902$	−32.7703	−1.09113
$903$	−107.277	−3.56995
$904$	−31.4735	−1.04679
$905$	0	0
$906$	8.89409	0.295486
$907$	15.7373	0.522547	0.261274	−	0.965265i	$-0.415858\pi$
$907$	15.7373	0.522547	0.261274	+	0.965265i	$0.415858\pi$
$908$	57.0938	1.89473
$909$	−15.7500	−0.522395
$910$	0	0
$911$	−29.2026	−0.967525	−0.483763	−	0.875199i	$-0.660730\pi$
$911$	−29.2026	−0.967525	−0.483763	+	0.875199i	$0.660730\pi$
$912$	42.6464	1.41216
$913$	7.50301	0.248313
$914$	−44.7886	−1.48148
$915$	0	0
$916$	10.6215	0.350943
$917$	−34.4627	−1.13806
$918$	196.668	6.49101
$919$	−33.6084	−1.10864	−0.554320	−	0.832304i	$-0.687022\pi$
$919$	−33.6084	−1.10864	−0.554320	+	0.832304i	$0.687022\pi$
$920$	0	0
$921$	46.5377	1.53347
$922$	25.7926	0.849435
$923$	−38.6641	−1.27264
$924$	56.3546	1.85393
$925$	0	0
$926$	−46.9554	−1.54305
$927$	33.8317	1.11118
$928$	33.4901	1.09937
$929$	−1.72906	−0.0567286	−0.0283643	−	0.999598i	$-0.509030\pi$
$929$	−1.72906	−0.0567286	−0.0283643	+	0.999598i	$0.509030\pi$
$930$	0	0
$931$	−1.84873	−0.0605896
$932$	79.8007	2.61396
$933$	−88.1747	−2.88671
$934$	14.5756	0.476928
$935$	0	0
$936$	−258.841	−8.46048
$937$	−46.5505	−1.52074	−0.760369	−	0.649491i	$-0.774982\pi$
$937$	−46.5505	−1.52074	−0.760369	+	0.649491i	$0.774982\pi$
$938$	−34.7947	−1.13609
$939$	58.4729	1.90819
$940$	0	0
$941$	−26.9576	−0.878794	−0.439397	−	0.898293i	$-0.644808\pi$
$941$	−26.9576	−0.878794	−0.439397	+	0.898293i	$0.644808\pi$
$942$	7.76331	0.252942
$943$	−12.9414	−0.421430
$944$	34.1805	1.11248
$945$	0	0
$946$	46.7490	1.51994
$947$	11.6919	0.379936	0.189968	−	0.981790i	$-0.439162\pi$
$947$	11.6919	0.379936	0.189968	+	0.981790i	$0.439162\pi$
$948$	173.170	5.62431
$949$	61.5705	1.99866
$950$	0	0
$951$	−18.2078	−0.590427
$952$	115.000	3.72717
$953$	−29.7229	−0.962821	−0.481410	−	0.876495i	$-0.659875\pi$
$953$	−29.7229	−0.962821	−0.481410	+	0.876495i	$0.659875\pi$
$954$	−28.7051	−0.929361
$955$	0	0
$956$	−62.9802	−2.03693
$957$	42.7882	1.38315
$958$	66.4529	2.14699
$959$	3.50978	0.113337
$960$	0	0
$961$	11.7496	0.379018
$962$	10.7818	0.347618
$963$	−13.2408	−0.426680
$964$	34.6440	1.11581
$965$	0	0
$966$	32.5259	1.04650
$967$	5.42656	0.174506	0.0872532	−	0.996186i	$-0.472191\pi$
$967$	5.42656	0.174506	0.0872532	+	0.996186i	$0.472191\pi$
$968$	51.3801	1.65142
$969$	48.8409	1.56900
$970$	0	0
$971$	2.85780	0.0917112	0.0458556	−	0.998948i	$-0.485399\pi$
$971$	2.85780	0.0917112	0.0458556	+	0.998948i	$0.485399\pi$
$972$	−53.3655	−1.71170
$973$	−9.31860	−0.298740
$974$	34.5924	1.10841
$975$	0	0
$976$	−16.6986	−0.534508
$977$	−15.5772	−0.498358	−0.249179	−	0.968457i	$-0.580161\pi$
$977$	−15.5772	−0.498358	−0.249179	+	0.968457i	$0.580161\pi$
$978$	−92.2048	−2.94838
$979$	−6.67946	−0.213476
$980$	0	0
$981$	−74.4922	−2.37835
$982$	−90.4733	−2.88712
$983$	40.2118	1.28256	0.641278	−	0.767308i	$-0.278404\pi$
$983$	40.2118	1.28256	0.641278	+	0.767308i	$0.278404\pi$
$984$	157.924	5.03442
$985$	0	0
$986$	162.146	5.16379
$987$	−50.3844	−1.60375
$988$	−65.1273	−2.07197
$989$	18.4618	0.587050
$990$	0	0
$991$	−38.0438	−1.20850	−0.604250	−	0.796795i	$-0.706527\pi$
$991$	−38.0438	−1.20850	−0.604250	+	0.796795i	$0.706527\pi$
$992$	−23.7954	−0.755504
$993$	−40.3507	−1.28049
$994$	−40.7646	−1.29297
$995$	0	0
$996$	−67.1456	−2.12759
$997$	−6.39660	−0.202582	−0.101291	−	0.994857i	$-0.532297\pi$
$997$	−6.39660	−0.202582	−0.101291	+	0.994857i	$0.532297\pi$
$998$	34.9736	1.10707
$999$	7.16287	0.226623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.r.1.5		49
5.2	odd	4		985.2.b.a.789.8	✓	98
5.3	odd	4		985.2.b.a.789.91	yes	98
5.4	even	2		4925.2.a.s.1.45		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.8	✓	98	5.2	odd	4
985.2.b.a.789.91	yes	98	5.3	odd	4
4925.2.a.r.1.5		49	1.1	even	1	trivial
4925.2.a.s.1.45		49	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:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51668 q^{2} -3.09870 q^{3} +4.33369 q^{4} +7.79844 q^{6} -2.79664 q^{7} -5.87315 q^{8} +6.60194 q^{9} +1.50057 q^{11} -13.4288 q^{12} +6.67560 q^{13} +7.03826 q^{14} +6.11348 q^{16} +7.00147 q^{17} -16.6150 q^{18} -2.25120 q^{19} +8.66596 q^{21} -3.77645 q^{22} -1.49137 q^{23} +18.1991 q^{24} -16.8004 q^{26} -11.1613 q^{27} -12.1198 q^{28} -9.20215 q^{29} +6.53831 q^{31} -3.63937 q^{32} -4.64980 q^{33} -17.6205 q^{34} +28.6107 q^{36} -0.641758 q^{37} +5.66556 q^{38} -20.6857 q^{39} +8.67754 q^{41} -21.8095 q^{42} -12.3791 q^{43} +6.50299 q^{44} +3.75330 q^{46} -5.81406 q^{47} -18.9438 q^{48} +0.821218 q^{49} -21.6955 q^{51} +28.9300 q^{52} +1.72766 q^{53} +28.0895 q^{54} +16.4251 q^{56} +6.97580 q^{57} +23.1589 q^{58} +5.59101 q^{59} -2.73144 q^{61} -16.4549 q^{62} -18.4633 q^{63} -3.06781 q^{64} +11.7021 q^{66} -4.94365 q^{67} +30.3422 q^{68} +4.62130 q^{69} -5.79185 q^{71} -38.7742 q^{72} +9.22321 q^{73} +1.61510 q^{74} -9.75601 q^{76} -4.19655 q^{77} +52.0593 q^{78} -12.8954 q^{79} +14.7798 q^{81} -21.8386 q^{82} +5.00012 q^{83} +37.5556 q^{84} +31.1542 q^{86} +28.5147 q^{87} -8.81305 q^{88} -4.45129 q^{89} -18.6693 q^{91} -6.46312 q^{92} -20.2603 q^{93} +14.6321 q^{94} +11.2773 q^{96} -12.7344 q^{97} -2.06674 q^{98} +9.90665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 5 q^{2} - 22 q^{3} + 49 q^{4} + 2 q^{6} - 32 q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{11} - 44 q^{12} - 32 q^{13} - 8 q^{14} + 49 q^{16} - 14 q^{17} - 25 q^{18} + 4 q^{19} + 10 q^{21} - 38 q^{22} - 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 49 * q - 5 * q^2 - 22 * q^3 + 49 * q^4 + 2 * q^6 - 32 * q^7 - 15 * q^8 + 51 * q^9 - 2 * q^11 - 44 * q^12 - 32 * q^13 - 8 * q^14 + 49 * q^16 - 14 * q^17 - 25 * q^18 + 4 * q^19 + 10 * q^21 - 38 * q^22 - 24 * q^23 - 2 * q^26 - 88 * q^27 - 80 * q^28 + 6 * q^29 - 2 * q^31 - 35 * q^32 - 48 * q^33 - 12 * q^34 + 43 * q^36 - 60 * q^37 - 34 * q^38 - 2 * q^41 - 36 * q^42 - 80 * q^43 - 8 * q^44 - 22 * q^46 - 48 * q^47 - 88 * q^48 + 65 * q^49 + 16 * q^51 - 96 * q^52 - 18 * q^53 + 42 * q^54 - 22 * q^56 - 40 * q^57 - 52 * q^58 + 8 * q^59 + 16 * q^61 - 38 * q^62 - 82 * q^63 + 55 * q^64 + 42 * q^66 - 146 * q^67 + 68 * q^68 - 6 * q^69 - 75 * q^72 - 78 * q^73 - 24 * q^74 - 4 * q^76 - 32 * q^77 + 38 * q^78 + 10 * q^79 + 41 * q^81 - 96 * q^82 - 64 * q^83 + 42 * q^84 - 28 * q^86 - 56 * q^87 - 72 * q^88 + 4 * q^89 - 60 * q^92 + 28 * q^93 - 8 * q^94 - 34 * q^96 - 124 * q^97 + 19 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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