LMFDB - Embedded newform 6025.2.a.m.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.15543 q^{2} +0.792453 q^{3} +2.64590 q^{4} -1.70808 q^{6} -4.02126 q^{7} -1.39219 q^{8} -2.37202 q^{9} +O(q^{10})$	\(q-2.15543 q^{2} +0.792453 q^{3} +2.64590 q^{4} -1.70808 q^{6} -4.02126 q^{7} -1.39219 q^{8} -2.37202 q^{9} +1.26468 q^{11} +2.09675 q^{12} -4.88804 q^{13} +8.66756 q^{14} -2.29102 q^{16} -0.117580 q^{17} +5.11273 q^{18} +6.32619 q^{19} -3.18666 q^{21} -2.72594 q^{22} +2.32707 q^{23} -1.10324 q^{24} +10.5359 q^{26} -4.25707 q^{27} -10.6398 q^{28} +3.41746 q^{29} -5.96228 q^{31} +7.72253 q^{32} +1.00220 q^{33} +0.253437 q^{34} -6.27612 q^{36} +6.92150 q^{37} -13.6357 q^{38} -3.87354 q^{39} +12.2048 q^{41} +6.86863 q^{42} -2.15228 q^{43} +3.34622 q^{44} -5.01584 q^{46} -4.98498 q^{47} -1.81553 q^{48} +9.17051 q^{49} -0.0931768 q^{51} -12.9333 q^{52} +2.46248 q^{53} +9.17584 q^{54} +5.59835 q^{56} +5.01321 q^{57} -7.36612 q^{58} +10.4520 q^{59} -12.2639 q^{61} +12.8513 q^{62} +9.53850 q^{63} -12.0634 q^{64} -2.16018 q^{66} +10.8246 q^{67} -0.311105 q^{68} +1.84409 q^{69} -2.22311 q^{71} +3.30230 q^{72} -2.69526 q^{73} -14.9188 q^{74} +16.7384 q^{76} -5.08562 q^{77} +8.34917 q^{78} +5.79087 q^{79} +3.74253 q^{81} -26.3067 q^{82} -4.54310 q^{83} -8.43156 q^{84} +4.63910 q^{86} +2.70818 q^{87} -1.76068 q^{88} +10.1221 q^{89} +19.6561 q^{91} +6.15718 q^{92} -4.72483 q^{93} +10.7448 q^{94} +6.11974 q^{96} -4.37973 q^{97} -19.7664 q^{98} -2.99986 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.15543	−1.52412	−0.762061	−	0.647505i	$-0.775812\pi$
$2$	−2.15543	−1.52412	−0.762061	+	0.647505i	$0.775812\pi$
$3$	0.792453	0.457523	0.228761	−	0.973483i	$-0.426532\pi$
$3$	0.792453	0.457523	0.228761	+	0.973483i	$0.426532\pi$
$4$	2.64590	1.32295
$5$	0	0
$5$	0	0
$6$	−1.70808	−0.697321
$7$	−4.02126	−1.51989	−0.759946	−	0.649986i	$-0.774775\pi$
$7$	−4.02126	−1.51989	−0.759946	+	0.649986i	$0.774775\pi$
$8$	−1.39219	−0.492213
$9$	−2.37202	−0.790673
$10$	0	0
$11$	1.26468	0.381317	0.190658	−	0.981656i	$-0.438938\pi$
$11$	1.26468	0.381317	0.190658	+	0.981656i	$0.438938\pi$
$12$	2.09675	0.605279
$13$	−4.88804	−1.35570	−0.677849	−	0.735201i	$-0.737088\pi$
$13$	−4.88804	−1.35570	−0.677849	+	0.735201i	$0.737088\pi$
$14$	8.66756	2.31650
$15$	0	0
$16$	−2.29102	−0.572756
$17$	−0.117580	−0.0285174	−0.0142587	−	0.999898i	$-0.504539\pi$
$17$	−0.117580	−0.0285174	−0.0142587	+	0.999898i	$0.504539\pi$
$18$	5.11273	1.20508
$19$	6.32619	1.45133	0.725664	−	0.688050i	$-0.241533\pi$
$19$	6.32619	1.45133	0.725664	+	0.688050i	$0.241533\pi$
$20$	0	0
$21$	−3.18666	−0.695385
$22$	−2.72594	−0.581173
$23$	2.32707	0.485227	0.242614	−	0.970123i	$-0.421995\pi$
$23$	2.32707	0.485227	0.242614	+	0.970123i	$0.421995\pi$
$24$	−1.10324	−0.225199
$25$	0	0
$26$	10.5359	2.06625
$27$	−4.25707	−0.819274
$28$	−10.6398	−2.01074
$29$	3.41746	0.634607	0.317304	−	0.948324i	$-0.397223\pi$
$29$	3.41746	0.634607	0.317304	+	0.948324i	$0.397223\pi$
$30$	0	0
$31$	−5.96228	−1.07086	−0.535429	−	0.844580i	$-0.679850\pi$
$31$	−5.96228	−1.07086	−0.535429	+	0.844580i	$0.679850\pi$
$32$	7.72253	1.36516
$33$	1.00220	0.174461
$34$	0.253437	0.0434640
$35$	0	0
$36$	−6.27612	−1.04602
$37$	6.92150	1.13789	0.568944	−	0.822376i	$-0.307352\pi$
$37$	6.92150	1.13789	0.568944	+	0.822376i	$0.307352\pi$
$38$	−13.6357	−2.21200
$39$	−3.87354	−0.620263
$40$	0	0
$41$	12.2048	1.90607	0.953037	−	0.302854i	$-0.0979393\pi$
$41$	12.2048	1.90607	0.953037	+	0.302854i	$0.0979393\pi$
$42$	6.86863	1.05985
$43$	−2.15228	−0.328220	−0.164110	−	0.986442i	$-0.552475\pi$
$43$	−2.15228	−0.328220	−0.164110	+	0.986442i	$0.552475\pi$
$44$	3.34622	0.504462
$45$	0	0
$46$	−5.01584	−0.739546
$47$	−4.98498	−0.727134	−0.363567	−	0.931568i	$-0.618441\pi$
$47$	−4.98498	−0.727134	−0.363567	+	0.931568i	$0.618441\pi$
$48$	−1.81553	−0.262049
$49$	9.17051	1.31007
$50$	0	0
$51$	−0.0931768	−0.0130474
$52$	−12.9333	−1.79352
$53$	2.46248	0.338248	0.169124	−	0.985595i	$-0.445906\pi$
$53$	2.46248	0.338248	0.169124	+	0.985595i	$0.445906\pi$
$54$	9.17584	1.24867
$55$	0	0
$56$	5.59835	0.748111
$57$	5.01321	0.664015
$58$	−7.36612	−0.967219
$59$	10.4520	1.36073	0.680366	−	0.732873i	$-0.261821\pi$
$59$	10.4520	1.36073	0.680366	+	0.732873i	$0.261821\pi$
$60$	0	0
$61$	−12.2639	−1.57023	−0.785117	−	0.619348i	$-0.787397\pi$
$61$	−12.2639	−1.57023	−0.785117	+	0.619348i	$0.787397\pi$
$62$	12.8513	1.63212
$63$	9.53850	1.20174
$64$	−12.0634	−1.50792
$65$	0	0
$66$	−2.16018	−0.265900
$67$	10.8246	1.32244	0.661220	−	0.750192i	$-0.270039\pi$
$67$	10.8246	1.32244	0.661220	+	0.750192i	$0.270039\pi$
$68$	−0.311105	−0.0377271
$69$	1.84409	0.222003
$70$	0	0
$71$	−2.22311	−0.263835	−0.131918	−	0.991261i	$-0.542113\pi$
$71$	−2.22311	−0.263835	−0.131918	+	0.991261i	$0.542113\pi$
$72$	3.30230	0.389180
$73$	−2.69526	−0.315456	−0.157728	−	0.987483i	$-0.550417\pi$
$73$	−2.69526	−0.315456	−0.157728	+	0.987483i	$0.550417\pi$
$74$	−14.9188	−1.73428
$75$	0	0
$76$	16.7384	1.92003
$77$	−5.08562	−0.579560
$78$	8.34917	0.945357
$79$	5.79087	0.651524	0.325762	−	0.945452i	$-0.394379\pi$
$79$	5.79087	0.651524	0.325762	+	0.945452i	$0.394379\pi$
$80$	0	0
$81$	3.74253	0.415837
$82$	−26.3067	−2.90509
$83$	−4.54310	−0.498670	−0.249335	−	0.968417i	$-0.580212\pi$
$83$	−4.54310	−0.498670	−0.249335	+	0.968417i	$0.580212\pi$
$84$	−8.43156	−0.919959
$85$	0	0
$86$	4.63910	0.500247
$87$	2.70818	0.290347
$88$	−1.76068	−0.187689
$89$	10.1221	1.07294	0.536470	−	0.843920i	$-0.319758\pi$
$89$	10.1221	1.07294	0.536470	+	0.843920i	$0.319758\pi$
$90$	0	0
$91$	19.6561	2.06052
$92$	6.15718	0.641931
$93$	−4.72483	−0.489942
$94$	10.7448	1.10824
$95$	0	0
$96$	6.11974	0.624593
$97$	−4.37973	−0.444694	−0.222347	−	0.974968i	$-0.571372\pi$
$97$	−4.37973	−0.444694	−0.222347	+	0.974968i	$0.571372\pi$
$98$	−19.7664	−1.99671
$99$	−2.99986	−0.301497
$100$	0	0
$101$	−7.39846	−0.736174	−0.368087	−	0.929791i	$-0.619987\pi$
$101$	−7.39846	−0.736174	−0.368087	+	0.929791i	$0.619987\pi$
$102$	0.200836	0.0198858
$103$	−4.88099	−0.480938	−0.240469	−	0.970657i	$-0.577301\pi$
$103$	−4.88099	−0.480938	−0.240469	+	0.970657i	$0.577301\pi$
$104$	6.80508	0.667293
$105$	0	0
$106$	−5.30772	−0.515531
$107$	5.30418	0.512775	0.256387	−	0.966574i	$-0.417468\pi$
$107$	5.30418	0.512775	0.256387	+	0.966574i	$0.417468\pi$
$108$	−11.2638	−1.08386
$109$	17.7563	1.70075	0.850374	−	0.526179i	$-0.176376\pi$
$109$	17.7563	1.70075	0.850374	+	0.526179i	$0.176376\pi$
$110$	0	0
$111$	5.48496	0.520610
$112$	9.21279	0.870527
$113$	2.53400	0.238378	0.119189	−	0.992872i	$-0.461971\pi$
$113$	2.53400	0.238378	0.119189	+	0.992872i	$0.461971\pi$
$114$	−10.8056	−1.01204
$115$	0	0
$116$	9.04226	0.839553
$117$	11.5945	1.07191
$118$	−22.5286	−2.07392
$119$	0.472820	0.0433434
$120$	0	0
$121$	−9.40057	−0.854598
$122$	26.4341	2.39323
$123$	9.67175	0.872072
$124$	−15.7756	−1.41669
$125$	0	0
$126$	−20.5596	−1.83160
$127$	−18.7111	−1.66035	−0.830173	−	0.557506i	$-0.811758\pi$
$127$	−18.7111	−1.66035	−0.830173	+	0.557506i	$0.811758\pi$
$128$	10.5567	0.933091
$129$	−1.70558	−0.150168
$130$	0	0
$131$	3.74859	0.327516	0.163758	−	0.986501i	$-0.447638\pi$
$131$	3.74859	0.327516	0.163758	+	0.986501i	$0.447638\pi$
$132$	2.65172	0.230803
$133$	−25.4392	−2.20586
$134$	−23.3318	−2.01556
$135$	0	0
$136$	0.163694	0.0140366
$137$	3.09907	0.264771	0.132386	−	0.991198i	$-0.457736\pi$
$137$	3.09907	0.264771	0.132386	+	0.991198i	$0.457736\pi$
$138$	−3.97482	−0.338359
$139$	−4.99535	−0.423700	−0.211850	−	0.977302i	$-0.567949\pi$
$139$	−4.99535	−0.423700	−0.211850	+	0.977302i	$0.567949\pi$
$140$	0	0
$141$	−3.95036	−0.332681
$142$	4.79178	0.402117
$143$	−6.18183	−0.516951
$144$	5.43435	0.452862
$145$	0	0
$146$	5.80945	0.480793
$147$	7.26719	0.599388
$148$	18.3136	1.50537
$149$	5.18482	0.424757	0.212378	−	0.977188i	$-0.431879\pi$
$149$	5.18482	0.424757	0.212378	+	0.977188i	$0.431879\pi$
$150$	0	0
$151$	−12.2387	−0.995972	−0.497986	−	0.867185i	$-0.665927\pi$
$151$	−12.2387	−0.995972	−0.497986	+	0.867185i	$0.665927\pi$
$152$	−8.80725	−0.714363
$153$	0.278903	0.0225479
$154$	10.9617	0.883321
$155$	0	0
$156$	−10.2490	−0.820576
$157$	−11.1752	−0.891878	−0.445939	−	0.895063i	$-0.647130\pi$
$157$	−11.1752	−0.891878	−0.445939	+	0.895063i	$0.647130\pi$
$158$	−12.4818	−0.993002
$159$	1.95140	0.154756
$160$	0	0
$161$	−9.35774	−0.737493
$162$	−8.06678	−0.633786
$163$	−9.67146	−0.757527	−0.378764	−	0.925493i	$-0.623651\pi$
$163$	−9.67146	−0.757527	−0.378764	+	0.925493i	$0.623651\pi$
$164$	32.2927	2.52164
$165$	0	0
$166$	9.79235	0.760034
$167$	0.593804	0.0459500	0.0229750	−	0.999736i	$-0.492686\pi$
$167$	0.593804	0.0459500	0.0229750	+	0.999736i	$0.492686\pi$
$168$	4.43643	0.342278
$169$	10.8930	0.837920
$170$	0	0
$171$	−15.0058	−1.14753
$172$	−5.69471	−0.434218
$173$	−23.4099	−1.77982	−0.889910	−	0.456135i	$-0.849233\pi$
$173$	−23.4099	−1.77982	−0.889910	+	0.456135i	$0.849233\pi$
$174$	−5.83730	−0.442525
$175$	0	0
$176$	−2.89742	−0.218401
$177$	8.28270	0.622566
$178$	−21.8175	−1.63529
$179$	−14.4201	−1.07781	−0.538906	−	0.842366i	$-0.681162\pi$
$179$	−14.4201	−1.07781	−0.538906	+	0.842366i	$0.681162\pi$
$180$	0	0
$181$	−13.4405	−0.999025	−0.499513	−	0.866307i	$-0.666488\pi$
$181$	−13.4405	−0.999025	−0.499513	+	0.866307i	$0.666488\pi$
$182$	−42.3674	−3.14048
$183$	−9.71857	−0.718417
$184$	−3.23972	−0.238835
$185$	0	0
$186$	10.1841	0.746731
$187$	−0.148702	−0.0108742
$188$	−13.1898	−0.961961
$189$	17.1188	1.24521
$190$	0	0
$191$	−18.6555	−1.34986	−0.674931	−	0.737881i	$-0.735827\pi$
$191$	−18.6555	−1.34986	−0.674931	+	0.737881i	$0.735827\pi$
$192$	−9.55964	−0.689907
$193$	7.56894	0.544824	0.272412	−	0.962181i	$-0.412179\pi$
$193$	7.56894	0.544824	0.272412	+	0.962181i	$0.412179\pi$
$194$	9.44022	0.677769
$195$	0	0
$196$	24.2642	1.73316
$197$	−14.3006	−1.01888	−0.509438	−	0.860507i	$-0.670147\pi$
$197$	−14.3006	−1.01888	−0.509438	+	0.860507i	$0.670147\pi$
$198$	6.46599	0.459518
$199$	−0.777714	−0.0551307	−0.0275653	−	0.999620i	$-0.508775\pi$
$199$	−0.777714	−0.0551307	−0.0275653	+	0.999620i	$0.508775\pi$
$200$	0	0
$201$	8.57801	0.605046
$202$	15.9469	1.12202
$203$	−13.7425	−0.964535
$204$	−0.246536	−0.0172610
$205$	0	0
$206$	10.5206	0.733008
$207$	−5.51985	−0.383656
$208$	11.1986	0.776484
$209$	8.00063	0.553415
$210$	0	0
$211$	14.9509	1.02926	0.514630	−	0.857412i	$-0.327929\pi$
$211$	14.9509	1.02926	0.514630	+	0.857412i	$0.327929\pi$
$212$	6.51548	0.447485
$213$	−1.76171	−0.120711
$214$	−11.4328	−0.781531
$215$	0	0
$216$	5.92665	0.403257
$217$	23.9759	1.62759
$218$	−38.2726	−2.59215
$219$	−2.13586	−0.144328
$220$	0	0
$221$	0.574737	0.0386610
$222$	−11.8225	−0.793473
$223$	1.60462	0.107453	0.0537265	−	0.998556i	$-0.482890\pi$
$223$	1.60462	0.107453	0.0537265	+	0.998556i	$0.482890\pi$
$224$	−31.0543	−2.07490
$225$	0	0
$226$	−5.46186	−0.363318
$227$	3.91203	0.259650	0.129825	−	0.991537i	$-0.458558\pi$
$227$	3.91203	0.259650	0.129825	+	0.991537i	$0.458558\pi$
$228$	13.2644	0.878458
$229$	4.30379	0.284403	0.142201	−	0.989838i	$-0.454582\pi$
$229$	4.30379	0.284403	0.142201	+	0.989838i	$0.454582\pi$
$230$	0	0
$231$	−4.03011	−0.265162
$232$	−4.75776	−0.312362
$233$	5.59962	0.366843	0.183422	−	0.983034i	$-0.441283\pi$
$233$	5.59962	0.366843	0.183422	+	0.983034i	$0.441283\pi$
$234$	−24.9912	−1.63373
$235$	0	0
$236$	27.6549	1.80018
$237$	4.58899	0.298087
$238$	−1.01913	−0.0660606
$239$	−18.9647	−1.22672	−0.613361	−	0.789803i	$-0.710183\pi$
$239$	−18.9647	−1.22672	−0.613361	+	0.789803i	$0.710183\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	20.2623	1.30251
$243$	15.7370	1.00953
$244$	−32.4491	−2.07734
$245$	0	0
$246$	−20.8468	−1.32914
$247$	−30.9227	−1.96756
$248$	8.30062	0.527090
$249$	−3.60019	−0.228153
$250$	0	0
$251$	21.3888	1.35005	0.675026	−	0.737794i	$-0.264133\pi$
$251$	21.3888	1.35005	0.675026	+	0.737794i	$0.264133\pi$
$252$	25.2379	1.58984
$253$	2.94301	0.185025
$254$	40.3306	2.53057
$255$	0	0
$256$	1.37240	0.0857750
$257$	−31.0354	−1.93593	−0.967967	−	0.251076i	$-0.919216\pi$
$257$	−31.0354	−1.93593	−0.967967	+	0.251076i	$0.919216\pi$
$258$	3.67626	0.228874
$259$	−27.8331	−1.72947
$260$	0	0
$261$	−8.10629	−0.501767
$262$	−8.07984	−0.499174
$263$	25.3404	1.56256	0.781279	−	0.624182i	$-0.214568\pi$
$263$	25.3404	1.56256	0.781279	+	0.624182i	$0.214568\pi$
$264$	−1.39526	−0.0858720
$265$	0	0
$266$	54.8326	3.36200
$267$	8.02128	0.490894
$268$	28.6409	1.74952
$269$	−7.07666	−0.431472	−0.215736	−	0.976452i	$-0.569215\pi$
$269$	−7.07666	−0.431472	−0.215736	+	0.976452i	$0.569215\pi$
$270$	0	0
$271$	2.89647	0.175948	0.0879739	−	0.996123i	$-0.471961\pi$
$271$	2.89647	0.175948	0.0879739	+	0.996123i	$0.471961\pi$
$272$	0.269379	0.0163335
$273$	15.5765	0.942733
$274$	−6.67984	−0.403544
$275$	0	0
$276$	4.87928	0.293698
$277$	8.94662	0.537550	0.268775	−	0.963203i	$-0.413381\pi$
$277$	8.94662	0.537550	0.268775	+	0.963203i	$0.413381\pi$
$278$	10.7671	0.645771
$279$	14.1426	0.846698
$280$	0	0
$281$	−27.0481	−1.61356	−0.806778	−	0.590855i	$-0.798790\pi$
$281$	−27.0481	−1.61356	−0.806778	+	0.590855i	$0.798790\pi$
$282$	8.51475	0.507046
$283$	18.6511	1.10869	0.554347	−	0.832286i	$-0.312968\pi$
$283$	18.6511	1.10869	0.554347	+	0.832286i	$0.312968\pi$
$284$	−5.88213	−0.349040
$285$	0	0
$286$	13.3245	0.787896
$287$	−49.0788	−2.89703
$288$	−18.3180	−1.07940
$289$	−16.9862	−0.999187
$290$	0	0
$291$	−3.47073	−0.203458
$292$	−7.13137	−0.417332
$293$	8.34015	0.487237	0.243618	−	0.969871i	$-0.421666\pi$
$293$	8.34015	0.487237	0.243618	+	0.969871i	$0.421666\pi$
$294$	−15.6640	−0.913541
$295$	0	0
$296$	−9.63604	−0.560084
$297$	−5.38385	−0.312403
$298$	−11.1755	−0.647381
$299$	−11.3748	−0.657822
$300$	0	0
$301$	8.65487	0.498858
$302$	26.3797	1.51798
$303$	−5.86293	−0.336816
$304$	−14.4934	−0.831256
$305$	0	0
$306$	−0.601156	−0.0343658
$307$	−18.8024	−1.07311	−0.536555	−	0.843865i	$-0.680275\pi$
$307$	−18.8024	−1.07311	−0.536555	+	0.843865i	$0.680275\pi$
$308$	−13.4560	−0.766729
$309$	−3.86795	−0.220040
$310$	0	0
$311$	25.6111	1.45227	0.726136	−	0.687551i	$-0.241314\pi$
$311$	25.6111	1.45227	0.726136	+	0.687551i	$0.241314\pi$
$312$	5.39270	0.305302
$313$	−24.6616	−1.39396	−0.696979	−	0.717092i	$-0.745473\pi$
$313$	−24.6616	−1.39396	−0.696979	+	0.717092i	$0.745473\pi$
$314$	24.0874	1.35933
$315$	0	0
$316$	15.3221	0.861933
$317$	−29.3220	−1.64689	−0.823443	−	0.567399i	$-0.807950\pi$
$317$	−29.3220	−1.64689	−0.823443	+	0.567399i	$0.807950\pi$
$318$	−4.20612	−0.235867
$319$	4.32201	0.241986
$320$	0	0
$321$	4.20331	0.234606
$322$	20.1700	1.12403
$323$	−0.743835	−0.0413881
$324$	9.90235	0.550131
$325$	0	0
$326$	20.8462	1.15456
$327$	14.0711	0.778131
$328$	−16.9914	−0.938195
$329$	20.0459	1.10517
$330$	0	0
$331$	−7.69020	−0.422691	−0.211346	−	0.977411i	$-0.567785\pi$
$331$	−7.69020	−0.422691	−0.211346	+	0.977411i	$0.567785\pi$
$332$	−12.0206	−0.659715
$333$	−16.4179	−0.899697
$334$	−1.27991	−0.0700334
$335$	0	0
$336$	7.30070	0.398286
$337$	31.9251	1.73907	0.869537	−	0.493869i	$-0.164418\pi$
$337$	31.9251	1.73907	0.869537	+	0.493869i	$0.164418\pi$
$338$	−23.4790	−1.27709
$339$	2.00807	0.109063
$340$	0	0
$341$	−7.54040	−0.408336
$342$	32.3441	1.74897
$343$	−8.72818	−0.471277
$344$	2.99638	0.161554
$345$	0	0
$346$	50.4585	2.71266
$347$	−3.55329	−0.190750	−0.0953752	−	0.995441i	$-0.530405\pi$
$347$	−3.55329	−0.190750	−0.0953752	+	0.995441i	$0.530405\pi$
$348$	7.16556	0.384114
$349$	10.3190	0.552366	0.276183	−	0.961105i	$-0.410930\pi$
$349$	10.3190	0.552366	0.276183	+	0.961105i	$0.410930\pi$
$350$	0	0
$351$	20.8087	1.11069
$352$	9.76656	0.520559
$353$	15.7625	0.838951	0.419475	−	0.907767i	$-0.362214\pi$
$353$	15.7625	0.838951	0.419475	+	0.907767i	$0.362214\pi$
$354$	−17.8528	−0.948866
$355$	0	0
$356$	26.7820	1.41944
$357$	0.374688	0.0198306
$358$	31.0817	1.64272
$359$	−9.93399	−0.524296	−0.262148	−	0.965028i	$-0.584431\pi$
$359$	−9.93399	−0.524296	−0.262148	+	0.965028i	$0.584431\pi$
$360$	0	0
$361$	21.0207	1.10635
$362$	28.9701	1.52264
$363$	−7.44951	−0.390998
$364$	52.0080	2.72596
$365$	0	0
$366$	20.9477	1.09496
$367$	33.7262	1.76049	0.880247	−	0.474516i	$-0.157377\pi$
$367$	33.7262	1.76049	0.880247	+	0.474516i	$0.157377\pi$
$368$	−5.33136	−0.277917
$369$	−28.9501	−1.50708
$370$	0	0
$371$	−9.90228	−0.514101
$372$	−12.5014	−0.648168
$373$	−27.2813	−1.41257	−0.706286	−	0.707926i	$-0.749631\pi$
$373$	−27.2813	−1.41257	−0.706286	+	0.707926i	$0.749631\pi$
$374$	0.320517	0.0165735
$375$	0	0
$376$	6.94004	0.357905
$377$	−16.7047	−0.860336
$378$	−36.8984	−1.89785
$379$	−24.0664	−1.23621	−0.618105	−	0.786095i	$-0.712099\pi$
$379$	−24.0664	−1.23621	−0.618105	+	0.786095i	$0.712099\pi$
$380$	0	0
$381$	−14.8277	−0.759646
$382$	40.2106	2.05735
$383$	−22.7133	−1.16059	−0.580297	−	0.814405i	$-0.697064\pi$
$383$	−22.7133	−1.16059	−0.580297	+	0.814405i	$0.697064\pi$
$384$	8.36570	0.426910
$385$	0	0
$386$	−16.3144	−0.830379
$387$	5.10525	0.259514
$388$	−11.5883	−0.588308
$389$	−21.3440	−1.08218	−0.541092	−	0.840963i	$-0.681989\pi$
$389$	−21.3440	−1.08218	−0.541092	+	0.840963i	$0.681989\pi$
$390$	0	0
$391$	−0.273617	−0.0138374
$392$	−12.7671	−0.644835
$393$	2.97058	0.149846
$394$	30.8240	1.55289
$395$	0	0
$396$	−7.93731	−0.398865
$397$	17.9316	0.899964	0.449982	−	0.893038i	$-0.351430\pi$
$397$	17.9316	0.899964	0.449982	+	0.893038i	$0.351430\pi$
$398$	1.67631	0.0840259
$399$	−20.1594	−1.00923
$400$	0	0
$401$	0.944898	0.0471860	0.0235930	−	0.999722i	$-0.492489\pi$
$401$	0.944898	0.0471860	0.0235930	+	0.999722i	$0.492489\pi$
$402$	−18.4893	−0.922165
$403$	29.1439	1.45176
$404$	−19.5756	−0.973920
$405$	0	0
$406$	29.6211	1.47007
$407$	8.75352	0.433896
$408$	0.129720	0.00642208
$409$	−7.06324	−0.349255	−0.174627	−	0.984635i	$-0.555872\pi$
$409$	−7.06324	−0.349255	−0.174627	+	0.984635i	$0.555872\pi$
$410$	0	0
$411$	2.45586	0.121139
$412$	−12.9146	−0.636256
$413$	−42.0301	−2.06817
$414$	11.8977	0.584739
$415$	0	0
$416$	−37.7480	−1.85075
$417$	−3.95858	−0.193852
$418$	−17.2448	−0.843473
$419$	−3.87614	−0.189362	−0.0946809	−	0.995508i	$-0.530183\pi$
$419$	−3.87614	−0.189362	−0.0946809	+	0.995508i	$0.530183\pi$
$420$	0	0
$421$	2.19057	0.106762	0.0533810	−	0.998574i	$-0.483000\pi$
$421$	2.19057	0.106762	0.0533810	+	0.998574i	$0.483000\pi$
$422$	−32.2256	−1.56872
$423$	11.8245	0.574925
$424$	−3.42824	−0.166490
$425$	0	0
$426$	3.79726	0.183978
$427$	49.3164	2.38659
$428$	14.0343	0.678374
$429$	−4.89881	−0.236517
$430$	0	0
$431$	−21.0623	−1.01454	−0.507268	−	0.861789i	$-0.669344\pi$
$431$	−21.0623	−1.01454	−0.507268	+	0.861789i	$0.669344\pi$
$432$	9.75304	0.469243
$433$	30.8310	1.48164	0.740821	−	0.671702i	$-0.234437\pi$
$433$	30.8310	1.48164	0.740821	+	0.671702i	$0.234437\pi$
$434$	−51.6784	−2.48064
$435$	0	0
$436$	46.9814	2.25000
$437$	14.7215	0.704224
$438$	4.60371	0.219974
$439$	17.4647	0.833543	0.416772	−	0.909011i	$-0.363161\pi$
$439$	17.4647	0.833543	0.416772	+	0.909011i	$0.363161\pi$
$440$	0	0
$441$	−21.7526	−1.03584
$442$	−1.23881	−0.0589241
$443$	−3.30586	−0.157066	−0.0785330	−	0.996912i	$-0.525024\pi$
$443$	−3.30586	−0.157066	−0.0785330	+	0.996912i	$0.525024\pi$
$444$	14.5127	0.688740
$445$	0	0
$446$	−3.45864	−0.163771
$447$	4.10872	0.194336
$448$	48.5099	2.29188
$449$	9.92029	0.468167	0.234084	−	0.972216i	$-0.424791\pi$
$449$	9.92029	0.468167	0.234084	+	0.972216i	$0.424791\pi$
$450$	0	0
$451$	15.4353	0.726818
$452$	6.70469	0.315362
$453$	−9.69860	−0.455680
$454$	−8.43211	−0.395739
$455$	0	0
$456$	−6.97933	−0.326837
$457$	19.3141	0.903477	0.451739	−	0.892150i	$-0.350804\pi$
$457$	19.3141	0.903477	0.451739	+	0.892150i	$0.350804\pi$
$458$	−9.27654	−0.433464
$459$	0.500547	0.0233636
$460$	0	0
$461$	15.7925	0.735528	0.367764	−	0.929919i	$-0.380123\pi$
$461$	15.7925	0.735528	0.367764	+	0.929919i	$0.380123\pi$
$462$	8.68665	0.404139
$463$	28.9904	1.34730	0.673650	−	0.739051i	$-0.264726\pi$
$463$	28.9904	1.34730	0.673650	+	0.739051i	$0.264726\pi$
$464$	−7.82949	−0.363475
$465$	0	0
$466$	−12.0696	−0.559114
$467$	6.08723	0.281683	0.140842	−	0.990032i	$-0.455019\pi$
$467$	6.08723	0.281683	0.140842	+	0.990032i	$0.455019\pi$
$468$	30.6779	1.41809
$469$	−43.5287	−2.00997
$470$	0	0
$471$	−8.85582	−0.408055
$472$	−14.5511	−0.669770
$473$	−2.72195	−0.125156
$474$	−9.89127	−0.454321
$475$	0	0
$476$	1.25103	0.0573411
$477$	−5.84106	−0.267444
$478$	40.8771	1.86967
$479$	−14.5655	−0.665514	−0.332757	−	0.943013i	$-0.607979\pi$
$479$	−14.5655	−0.665514	−0.332757	+	0.943013i	$0.607979\pi$
$480$	0	0
$481$	−33.8326	−1.54263
$482$	−2.15543	−0.0981773
$483$	−7.41557	−0.337420
$484$	−24.8730	−1.13059
$485$	0	0
$486$	−33.9200	−1.53864
$487$	−23.8258	−1.07965	−0.539824	−	0.841778i	$-0.681509\pi$
$487$	−23.8258	−1.07965	−0.539824	+	0.841778i	$0.681509\pi$
$488$	17.0737	0.772890
$489$	−7.66417	−0.346586
$490$	0	0
$491$	0.744841	0.0336142	0.0168071	−	0.999859i	$-0.494650\pi$
$491$	0.744841	0.0336142	0.0168071	+	0.999859i	$0.494650\pi$
$492$	25.5905	1.15371
$493$	−0.401826	−0.0180973
$494$	66.6518	2.99881
$495$	0	0
$496$	13.6597	0.613339
$497$	8.93971	0.401001
$498$	7.75998	0.347733
$499$	6.33790	0.283723	0.141862	−	0.989886i	$-0.454691\pi$
$499$	6.33790	0.283723	0.141862	+	0.989886i	$0.454691\pi$
$500$	0	0
$501$	0.470562	0.0210232
$502$	−46.1022	−2.05764
$503$	36.9729	1.64854	0.824271	−	0.566195i	$-0.191585\pi$
$503$	36.9729	1.64854	0.824271	+	0.566195i	$0.191585\pi$
$504$	−13.2794	−0.591511
$505$	0	0
$506$	−6.34346	−0.282001
$507$	8.63215	0.383367
$508$	−49.5078	−2.19655
$509$	−10.2485	−0.454255	−0.227128	−	0.973865i	$-0.572933\pi$
$509$	−10.2485	−0.454255	−0.227128	+	0.973865i	$0.572933\pi$
$510$	0	0
$511$	10.8383	0.479459
$512$	−24.0716	−1.06382
$513$	−26.9310	−1.18903
$514$	66.8948	2.95060
$515$	0	0
$516$	−4.51279	−0.198664
$517$	−6.30443	−0.277268
$518$	59.9925	2.63592
$519$	−18.5512	−0.814308
$520$	0	0
$521$	−42.0322	−1.84146	−0.920731	−	0.390197i	$-0.872407\pi$
$521$	−42.0322	−1.84146	−0.920731	+	0.390197i	$0.872407\pi$
$522$	17.4726	0.764754
$523$	3.14151	0.137369	0.0686843	−	0.997638i	$-0.478120\pi$
$523$	3.14151	0.137369	0.0686843	+	0.997638i	$0.478120\pi$
$524$	9.91839	0.433287
$525$	0	0
$526$	−54.6196	−2.38153
$527$	0.701047	0.0305381
$528$	−2.29607	−0.0999235
$529$	−17.5848	−0.764554
$530$	0	0
$531$	−24.7923	−1.07589
$532$	−67.3096	−2.91824
$533$	−59.6577	−2.58406
$534$	−17.2893	−0.748183
$535$	0	0
$536$	−15.0699	−0.650922
$537$	−11.4273	−0.493124
$538$	15.2533	0.657616
$539$	11.5978	0.499553
$540$	0	0
$541$	38.2592	1.64489	0.822445	−	0.568844i	$-0.192609\pi$
$541$	38.2592	1.64489	0.822445	+	0.568844i	$0.192609\pi$
$542$	−6.24314	−0.268166
$543$	−10.6510	−0.457077
$544$	−0.908017	−0.0389309
$545$	0	0
$546$	−33.5741	−1.43684
$547$	−10.3788	−0.443765	−0.221882	−	0.975073i	$-0.571220\pi$
$547$	−10.3788	−0.443765	−0.221882	+	0.975073i	$0.571220\pi$
$548$	8.19982	0.350279
$549$	29.0902	1.24154
$550$	0	0
$551$	21.6195	0.921023
$552$	−2.56732	−0.109273
$553$	−23.2866	−0.990247
$554$	−19.2838	−0.819292
$555$	0	0
$556$	−13.2172	−0.560533
$557$	1.11070	0.0470620	0.0235310	−	0.999723i	$-0.492509\pi$
$557$	1.11070	0.0470620	0.0235310	+	0.999723i	$0.492509\pi$
$558$	−30.4835	−1.29047
$559$	10.5204	0.444967
$560$	0	0
$561$	−0.117839	−0.00497518
$562$	58.3004	2.45926
$563$	17.9377	0.755985	0.377993	−	0.925809i	$-0.376614\pi$
$563$	17.9377	0.755985	0.377993	+	0.925809i	$0.376614\pi$
$564$	−10.4523	−0.440119
$565$	0	0
$566$	−40.2013	−1.68979
$567$	−15.0497	−0.632027
$568$	3.09500	0.129863
$569$	20.3524	0.853218	0.426609	−	0.904436i	$-0.359708\pi$
$569$	20.3524	0.853218	0.426609	+	0.904436i	$0.359708\pi$
$570$	0	0
$571$	−21.0026	−0.878930	−0.439465	−	0.898260i	$-0.644832\pi$
$571$	−21.0026	−0.878930	−0.439465	+	0.898260i	$0.644832\pi$
$572$	−16.3565	−0.683899
$573$	−14.7836	−0.617593
$574$	105.786	4.41542
$575$	0	0
$576$	28.6145	1.19227
$577$	−2.54443	−0.105926	−0.0529630	−	0.998596i	$-0.516867\pi$
$577$	−2.54443	−0.105926	−0.0529630	+	0.998596i	$0.516867\pi$
$578$	36.6126	1.52288
$579$	5.99803	0.249269
$580$	0	0
$581$	18.2690	0.757925
$582$	7.48093	0.310095
$583$	3.11426	0.128980
$584$	3.75231	0.155272
$585$	0	0
$586$	−17.9766	−0.742608
$587$	6.44267	0.265918	0.132959	−	0.991122i	$-0.457552\pi$
$587$	6.44267	0.265918	0.132959	+	0.991122i	$0.457552\pi$
$588$	19.2283	0.792960
$589$	−37.7185	−1.55416
$590$	0	0
$591$	−11.3326	−0.466159
$592$	−15.8573	−0.651732
$593$	−38.0711	−1.56339	−0.781695	−	0.623660i	$-0.785645\pi$
$593$	−38.0711	−1.56339	−0.781695	+	0.623660i	$0.785645\pi$
$594$	11.6045	0.476140
$595$	0	0
$596$	13.7185	0.561931
$597$	−0.616302	−0.0252235
$598$	24.5177	1.00260
$599$	−0.446996	−0.0182638	−0.00913189	−	0.999958i	$-0.502907\pi$
$599$	−0.446996	−0.0182638	−0.00913189	+	0.999958i	$0.502907\pi$
$600$	0	0
$601$	−13.8915	−0.566644	−0.283322	−	0.959025i	$-0.591437\pi$
$601$	−13.8915	−0.566644	−0.283322	+	0.959025i	$0.591437\pi$
$602$	−18.6550	−0.760321
$603$	−25.6762	−1.04562
$604$	−32.3824	−1.31762
$605$	0	0
$606$	12.6372	0.513349
$607$	−10.7515	−0.436391	−0.218195	−	0.975905i	$-0.570017\pi$
$607$	−10.7515	−0.436391	−0.218195	+	0.975905i	$0.570017\pi$
$608$	48.8542	1.98130
$609$	−10.8903	−0.441297
$610$	0	0
$611$	24.3668	0.985775
$612$	0.737948	0.0298298
$613$	30.0657	1.21434	0.607171	−	0.794571i	$-0.292304\pi$
$613$	30.0657	1.21434	0.607171	+	0.794571i	$0.292304\pi$
$614$	40.5274	1.63555
$615$	0	0
$616$	7.08015	0.285267
$617$	−37.9912	−1.52947	−0.764735	−	0.644345i	$-0.777130\pi$
$617$	−37.9912	−1.52947	−0.764735	+	0.644345i	$0.777130\pi$
$618$	8.33711	0.335368
$619$	−39.7270	−1.59676	−0.798382	−	0.602151i	$-0.794311\pi$
$619$	−39.7270	−1.59676	−0.798382	+	0.602151i	$0.794311\pi$
$620$	0	0
$621$	−9.90649	−0.397534
$622$	−55.2030	−2.21344
$623$	−40.7035	−1.63075
$624$	8.87437	0.355259
$625$	0	0
$626$	53.1565	2.12456
$627$	6.34012	0.253200
$628$	−29.5684	−1.17991
$629$	−0.813832	−0.0324496
$630$	0	0
$631$	−38.5664	−1.53530	−0.767652	−	0.640867i	$-0.778575\pi$
$631$	−38.5664	−1.53530	−0.767652	+	0.640867i	$0.778575\pi$
$632$	−8.06199	−0.320689
$633$	11.8479	0.470910
$634$	63.2016	2.51006
$635$	0	0
$636$	5.16321	0.204734
$637$	−44.8258	−1.77606
$638$	−9.31582	−0.368817
$639$	5.27327	0.208607
$640$	0	0
$641$	−22.1038	−0.873046	−0.436523	−	0.899693i	$-0.643790\pi$
$641$	−22.1038	−0.873046	−0.436523	+	0.899693i	$0.643790\pi$
$642$	−9.05996	−0.357568
$643$	38.9962	1.53786	0.768931	−	0.639332i	$-0.220789\pi$
$643$	38.9962	1.53786	0.768931	+	0.639332i	$0.220789\pi$
$644$	−24.7596	−0.975666
$645$	0	0
$646$	1.60329	0.0630805
$647$	−3.31365	−0.130273	−0.0651365	−	0.997876i	$-0.520748\pi$
$647$	−3.31365	−0.130273	−0.0651365	+	0.997876i	$0.520748\pi$
$648$	−5.21031	−0.204680
$649$	13.2185	0.518870
$650$	0	0
$651$	18.9997	0.744659
$652$	−25.5897	−1.00217
$653$	40.0976	1.56914	0.784570	−	0.620040i	$-0.212884\pi$
$653$	40.0976	1.56914	0.784570	+	0.620040i	$0.212884\pi$
$654$	−30.3292	−1.18597
$655$	0	0
$656$	−27.9615	−1.09171
$657$	6.39320	0.249422
$658$	−43.2076	−1.68441
$659$	−35.3910	−1.37864	−0.689318	−	0.724458i	$-0.742090\pi$
$659$	−35.3910	−1.37864	−0.689318	+	0.724458i	$0.742090\pi$
$660$	0	0
$661$	−32.9142	−1.28021	−0.640107	−	0.768285i	$-0.721110\pi$
$661$	−32.9142	−1.28021	−0.640107	+	0.768285i	$0.721110\pi$
$662$	16.5757	0.644233
$663$	0.455452	0.0176883
$664$	6.32485	0.245452
$665$	0	0
$666$	35.3878	1.37125
$667$	7.95267	0.307929
$668$	1.57115	0.0607894
$669$	1.27158	0.0491622
$670$	0	0
$671$	−15.5100	−0.598756
$672$	−24.6090	−0.949314
$673$	17.6626	0.680845	0.340422	−	0.940273i	$-0.389430\pi$
$673$	17.6626	0.680845	0.340422	+	0.940273i	$0.389430\pi$
$674$	−68.8125	−2.65056
$675$	0	0
$676$	28.8216	1.10852
$677$	1.94789	0.0748634	0.0374317	−	0.999299i	$-0.488082\pi$
$677$	1.94789	0.0748634	0.0374317	+	0.999299i	$0.488082\pi$
$678$	−4.32827	−0.166226
$679$	17.6120	0.675888
$680$	0	0
$681$	3.10010	0.118796
$682$	16.2528	0.622354
$683$	−17.0284	−0.651572	−0.325786	−	0.945444i	$-0.605629\pi$
$683$	−17.0284	−0.651572	−0.325786	+	0.945444i	$0.605629\pi$
$684$	−39.7039	−1.51812
$685$	0	0
$686$	18.8130	0.718284
$687$	3.41055	0.130121
$688$	4.93092	0.187990
$689$	−12.0367	−0.458563
$690$	0	0
$691$	1.61770	0.0615403	0.0307701	−	0.999526i	$-0.490204\pi$
$691$	1.61770	0.0615403	0.0307701	+	0.999526i	$0.490204\pi$
$692$	−61.9402	−2.35461
$693$	12.0632	0.458243
$694$	7.65887	0.290727
$695$	0	0
$696$	−3.77030	−0.142913
$697$	−1.43505	−0.0543563
$698$	−22.2420	−0.841873
$699$	4.43743	0.167839
$700$	0	0
$701$	−41.2862	−1.55936	−0.779680	−	0.626178i	$-0.784618\pi$
$701$	−41.2862	−1.55936	−0.779680	+	0.626178i	$0.784618\pi$
$702$	−44.8519	−1.69282
$703$	43.7867	1.65145
$704$	−15.2563	−0.574995
$705$	0	0
$706$	−33.9749	−1.27866
$707$	29.7511	1.11891
$708$	21.9152	0.823623
$709$	9.77605	0.367147	0.183574	−	0.983006i	$-0.441233\pi$
$709$	9.77605	0.367147	0.183574	+	0.983006i	$0.441233\pi$
$710$	0	0
$711$	−13.7361	−0.515143
$712$	−14.0919	−0.528115
$713$	−13.8746	−0.519609
$714$	−0.807615	−0.0302242
$715$	0	0
$716$	−38.1542	−1.42589
$717$	−15.0286	−0.561253
$718$	21.4121	0.799091
$719$	28.1718	1.05063	0.525315	−	0.850908i	$-0.323947\pi$
$719$	28.1718	1.05063	0.525315	+	0.850908i	$0.323947\pi$
$720$	0	0
$721$	19.6277	0.730974
$722$	−45.3087	−1.68621
$723$	0.792453	0.0294716
$724$	−35.5622	−1.32166
$725$	0	0
$726$	16.0569	0.595929
$727$	−33.9563	−1.25937	−0.629685	−	0.776851i	$-0.716816\pi$
$727$	−33.9563	−1.25937	−0.629685	+	0.776851i	$0.716816\pi$
$728$	−27.3650	−1.01421
$729$	1.24323	0.0460455
$730$	0	0
$731$	0.253066	0.00935997
$732$	−25.7143	−0.950429
$733$	−20.8587	−0.770434	−0.385217	−	0.922826i	$-0.625873\pi$
$733$	−20.8587	−0.770434	−0.385217	+	0.922826i	$0.625873\pi$
$734$	−72.6946	−2.68321
$735$	0	0
$736$	17.9708	0.662414
$737$	13.6898	0.504268
$738$	62.4000	2.29698
$739$	−23.2375	−0.854807	−0.427404	−	0.904061i	$-0.640572\pi$
$739$	−23.2375	−0.854807	−0.427404	+	0.904061i	$0.640572\pi$
$740$	0	0
$741$	−24.5048	−0.900205
$742$	21.3437	0.783552
$743$	−49.5179	−1.81664	−0.908319	−	0.418279i	$-0.862633\pi$
$743$	−49.5179	−1.81664	−0.908319	+	0.418279i	$0.862633\pi$
$744$	6.57785	0.241156
$745$	0	0
$746$	58.8031	2.15293
$747$	10.7763	0.394285
$748$	−0.393450	−0.0143860
$749$	−21.3295	−0.779362
$750$	0	0
$751$	−17.8649	−0.651899	−0.325949	−	0.945387i	$-0.605684\pi$
$751$	−17.8649	−0.651899	−0.325949	+	0.945387i	$0.605684\pi$
$752$	11.4207	0.416470
$753$	16.9496	0.617679
$754$	36.0059	1.31126
$755$	0	0
$756$	45.2945	1.64735
$757$	−36.9576	−1.34325	−0.671624	−	0.740892i	$-0.734403\pi$
$757$	−36.9576	−1.34325	−0.671624	+	0.740892i	$0.734403\pi$
$758$	51.8736	1.88414
$759$	2.33219	0.0846533
$760$	0	0
$761$	15.6910	0.568799	0.284400	−	0.958706i	$-0.408206\pi$
$761$	15.6910	0.568799	0.284400	+	0.958706i	$0.408206\pi$
$762$	31.9601	1.15779
$763$	−71.4028	−2.58495
$764$	−49.3604	−1.78580
$765$	0	0
$766$	48.9570	1.76889
$767$	−51.0897	−1.84474
$768$	1.08756	0.0392440
$769$	−47.2791	−1.70493	−0.852463	−	0.522787i	$-0.824892\pi$
$769$	−47.2791	−1.70493	−0.852463	+	0.522787i	$0.824892\pi$
$770$	0	0
$771$	−24.5941	−0.885734
$772$	20.0266	0.720774
$773$	−39.4942	−1.42051	−0.710253	−	0.703947i	$-0.751419\pi$
$773$	−39.4942	−1.42051	−0.710253	+	0.703947i	$0.751419\pi$
$774$	−11.0040	−0.395532
$775$	0	0
$776$	6.09741	0.218884
$777$	−22.0565	−0.791271
$778$	46.0056	1.64938
$779$	77.2101	2.76634
$780$	0	0
$781$	−2.81154	−0.100605
$782$	0.589764	0.0210899
$783$	−14.5484	−0.519917
$784$	−21.0098	−0.750351
$785$	0	0
$786$	−6.40289	−0.228384
$787$	4.69663	0.167417	0.0837084	−	0.996490i	$-0.473324\pi$
$787$	4.69663	0.167417	0.0837084	+	0.996490i	$0.473324\pi$
$788$	−37.8379	−1.34792
$789$	20.0811	0.714906
$790$	0	0
$791$	−10.1899	−0.362309
$792$	4.17637	0.148401
$793$	59.9465	2.12876
$794$	−38.6505	−1.37165
$795$	0	0
$796$	−2.05775	−0.0729351
$797$	29.1051	1.03096	0.515478	−	0.856903i	$-0.327615\pi$
$797$	29.1051	1.03096	0.515478	+	0.856903i	$0.327615\pi$
$798$	43.4522	1.53819
$799$	0.586135	0.0207360
$800$	0	0
$801$	−24.0098	−0.848344
$802$	−2.03667	−0.0719172
$803$	−3.40865	−0.120289
$804$	22.6965	0.800445
$805$	0	0
$806$	−62.8177	−2.21266
$807$	−5.60792	−0.197408
$808$	10.3001	0.362355
$809$	34.8797	1.22631	0.613153	−	0.789965i	$-0.289901\pi$
$809$	34.8797	1.22631	0.613153	+	0.789965i	$0.289901\pi$
$810$	0	0
$811$	−38.2504	−1.34315	−0.671576	−	0.740936i	$-0.734382\pi$
$811$	−38.2504	−1.34315	−0.671576	+	0.740936i	$0.734382\pi$
$812$	−36.3613	−1.27603
$813$	2.29531	0.0805001
$814$	−18.8676	−0.661310
$815$	0	0
$816$	0.213470	0.00747295
$817$	−13.6157	−0.476354
$818$	15.2243	0.532307
$819$	−46.6246	−1.62919
$820$	0	0
$821$	−24.8149	−0.866045	−0.433023	−	0.901383i	$-0.642553\pi$
$821$	−24.8149	−0.866045	−0.433023	+	0.901383i	$0.642553\pi$
$822$	−5.29346	−0.184630
$823$	−33.7812	−1.17754	−0.588769	−	0.808301i	$-0.700387\pi$
$823$	−33.7812	−1.17754	−0.588769	+	0.808301i	$0.700387\pi$
$824$	6.79526	0.236724
$825$	0	0
$826$	90.5931	3.15214
$827$	−52.0648	−1.81047	−0.905235	−	0.424911i	$-0.860305\pi$
$827$	−52.0648	−1.81047	−0.905235	+	0.424911i	$0.860305\pi$
$828$	−14.6050	−0.507557
$829$	34.3782	1.19400	0.597002	−	0.802240i	$-0.296358\pi$
$829$	34.3782	1.19400	0.597002	+	0.802240i	$0.296358\pi$
$830$	0	0
$831$	7.08977	0.245941
$832$	58.9662	2.04428
$833$	−1.07827	−0.0373599
$834$	8.53246	0.295455
$835$	0	0
$836$	21.1689	0.732140
$837$	25.3819	0.877325
$838$	8.35477	0.288611
$839$	32.0635	1.10695	0.553477	−	0.832864i	$-0.313301\pi$
$839$	32.0635	1.10695	0.553477	+	0.832864i	$0.313301\pi$
$840$	0	0
$841$	−17.3209	−0.597274
$842$	−4.72164	−0.162718
$843$	−21.4343	−0.738238
$844$	39.5585	1.36166
$845$	0	0
$846$	−25.4869	−0.876257
$847$	37.8021	1.29890
$848$	−5.64160	−0.193733
$849$	14.7801	0.507253
$850$	0	0
$851$	16.1068	0.552134
$852$	−4.66131	−0.159694
$853$	8.94259	0.306188	0.153094	−	0.988212i	$-0.451076\pi$
$853$	8.94259	0.306188	0.153094	+	0.988212i	$0.451076\pi$
$854$	−106.298	−3.63745
$855$	0	0
$856$	−7.38442	−0.252394
$857$	29.1403	0.995413	0.497706	−	0.867346i	$-0.334176\pi$
$857$	29.1403	0.995413	0.497706	+	0.867346i	$0.334176\pi$
$858$	10.5591	0.360480
$859$	18.7246	0.638874	0.319437	−	0.947607i	$-0.396506\pi$
$859$	18.7246	0.638874	0.319437	+	0.947607i	$0.396506\pi$
$860$	0	0
$861$	−38.8926	−1.32546
$862$	45.3984	1.54628
$863$	17.3469	0.590496	0.295248	−	0.955421i	$-0.404598\pi$
$863$	17.3469	0.590496	0.295248	+	0.955421i	$0.404598\pi$
$864$	−32.8753	−1.11844
$865$	0	0
$866$	−66.4542	−2.25820
$867$	−13.4607	−0.457151
$868$	63.4377	2.15322
$869$	7.32363	0.248437
$870$	0	0
$871$	−52.9113	−1.79283
$872$	−24.7202	−0.837131
$873$	10.3888	0.351608
$874$	−31.7312	−1.07332
$875$	0	0
$876$	−5.65127	−0.190939
$877$	12.3117	0.415738	0.207869	−	0.978157i	$-0.433347\pi$
$877$	12.3117	0.415738	0.207869	+	0.978157i	$0.433347\pi$
$878$	−37.6440	−1.27042
$879$	6.60917	0.222922
$880$	0	0
$881$	−28.2850	−0.952946	−0.476473	−	0.879189i	$-0.658085\pi$
$881$	−28.2850	−0.952946	−0.476473	+	0.879189i	$0.658085\pi$
$882$	46.8863	1.57875
$883$	20.0774	0.675657	0.337829	−	0.941208i	$-0.390308\pi$
$883$	20.0774	0.675657	0.337829	+	0.941208i	$0.390308\pi$
$884$	1.52070	0.0511465
$885$	0	0
$886$	7.12556	0.239388
$887$	28.1703	0.945865	0.472932	−	0.881099i	$-0.343195\pi$
$887$	28.1703	0.945865	0.472932	+	0.881099i	$0.343195\pi$
$888$	−7.63611	−0.256251
$889$	75.2423	2.52355
$890$	0	0
$891$	4.73312	0.158565
$892$	4.24565	0.142155
$893$	−31.5359	−1.05531
$894$	−8.85608	−0.296192
$895$	0	0
$896$	−42.4513	−1.41820
$897$	−9.01400	−0.300969
$898$	−21.3825	−0.713544
$899$	−20.3759	−0.679574
$900$	0	0
$901$	−0.289539	−0.00964596
$902$	−33.2697	−1.10776
$903$	6.85857	0.228239
$904$	−3.52780	−0.117333
$905$	0	0
$906$	20.9047	0.694512
$907$	−25.5497	−0.848362	−0.424181	−	0.905577i	$-0.639438\pi$
$907$	−25.5497	−0.848362	−0.424181	+	0.905577i	$0.639438\pi$
$908$	10.3508	0.343504
$909$	17.5493	0.582073
$910$	0	0
$911$	1.11939	0.0370871	0.0185436	−	0.999828i	$-0.494097\pi$
$911$	1.11939	0.0370871	0.0185436	+	0.999828i	$0.494097\pi$
$912$	−11.4854	−0.380318
$913$	−5.74559	−0.190151
$914$	−41.6304	−1.37701
$915$	0	0
$916$	11.3874	0.376250
$917$	−15.0741	−0.497789
$918$	−1.07890	−0.0356089
$919$	−39.5953	−1.30613	−0.653063	−	0.757303i	$-0.726517\pi$
$919$	−39.5953	−1.30613	−0.653063	+	0.757303i	$0.726517\pi$
$920$	0	0
$921$	−14.9000	−0.490972
$922$	−34.0396	−1.12103
$923$	10.8667	0.357681
$924$	−10.6633	−0.350796
$925$	0	0
$926$	−62.4870	−2.05345
$927$	11.5778	0.380265
$928$	26.3915	0.866342
$929$	2.97814	0.0977097	0.0488549	−	0.998806i	$-0.484443\pi$
$929$	2.97814	0.0977097	0.0488549	+	0.998806i	$0.484443\pi$
$930$	0	0
$931$	58.0144	1.90134
$932$	14.8160	0.485315
$933$	20.2956	0.664448
$934$	−13.1206	−0.429320
$935$	0	0
$936$	−16.1418	−0.527610
$937$	47.1582	1.54059	0.770296	−	0.637687i	$-0.220108\pi$
$937$	47.1582	1.54059	0.770296	+	0.637687i	$0.220108\pi$
$938$	93.8232	3.06343
$939$	−19.5432	−0.637767
$940$	0	0
$941$	−18.5041	−0.603217	−0.301609	−	0.953432i	$-0.597524\pi$
$941$	−18.5041	−0.603217	−0.301609	+	0.953432i	$0.597524\pi$
$942$	19.0881	0.621925
$943$	28.4015	0.924879
$944$	−23.9457	−0.779367
$945$	0	0
$946$	5.86699	0.190752
$947$	5.64621	0.183477	0.0917385	−	0.995783i	$-0.470758\pi$
$947$	5.64621	0.183477	0.0917385	+	0.995783i	$0.470758\pi$
$948$	12.1420	0.394354
$949$	13.1745	0.427663
$950$	0	0
$951$	−23.2363	−0.753488
$952$	−0.658256	−0.0213342
$953$	−29.5827	−0.958278	−0.479139	−	0.877739i	$-0.659051\pi$
$953$	−29.5827	−0.958278	−0.479139	+	0.877739i	$0.659051\pi$
$954$	12.5900	0.407617
$955$	0	0
$956$	−50.1785	−1.62289
$957$	3.42499	0.110714
$958$	31.3949	1.01432
$959$	−12.4622	−0.402424
$960$	0	0
$961$	4.54881	0.146736
$962$	72.9240	2.35116
$963$	−12.5816	−0.405437
$964$	2.64590	0.0852186
$965$	0	0
$966$	15.9838	0.514269
$967$	−40.4545	−1.30093	−0.650464	−	0.759537i	$-0.725426\pi$
$967$	−40.4545	−1.30093	−0.650464	+	0.759537i	$0.725426\pi$
$968$	13.0874	0.420644
$969$	−0.589454	−0.0189360
$970$	0	0
$971$	−54.2419	−1.74070	−0.870352	−	0.492430i	$-0.836109\pi$
$971$	−54.2419	−1.74070	−0.870352	+	0.492430i	$0.836109\pi$
$972$	41.6385	1.33555
$973$	20.0876	0.643978
$974$	51.3549	1.64552
$975$	0	0
$976$	28.0969	0.899360
$977$	−53.8321	−1.72224	−0.861120	−	0.508402i	$-0.830237\pi$
$977$	−53.8321	−1.72224	−0.861120	+	0.508402i	$0.830237\pi$
$978$	16.5196	0.528239
$979$	12.8012	0.409130
$980$	0	0
$981$	−42.1183	−1.34474
$982$	−1.60546	−0.0512322
$983$	−21.3045	−0.679509	−0.339754	−	0.940514i	$-0.610344\pi$
$983$	−21.3045	−0.679509	−0.339754	+	0.940514i	$0.610344\pi$
$984$	−13.4649	−0.429246
$985$	0	0
$986$	0.866110	0.0275826
$987$	15.8854	0.505639
$988$	−81.8182	−2.60298
$989$	−5.00850	−0.159261
$990$	0	0
$991$	17.5486	0.557450	0.278725	−	0.960371i	$-0.410088\pi$
$991$	17.5486	0.557450	0.278725	+	0.960371i	$0.410088\pi$
$992$	−46.0439	−1.46189
$993$	−6.09412	−0.193391
$994$	−19.2690	−0.611175
$995$	0	0
$996$	−9.52574	−0.301835
$997$	39.9614	1.26559	0.632794	−	0.774320i	$-0.281908\pi$
$997$	39.9614	1.26559	0.632794	+	0.774320i	$0.281908\pi$
$998$	−13.6609	−0.432429
$999$	−29.4653	−0.932242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.m.1.8	✓	40
5.4	even	2		6025.2.a.n.1.33	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.8	✓	40	1.1	even	1	trivial
6025.2.a.n.1.33	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)

\(f(q)\)	\(=\)	\(q-2.15543 q^{2} +0.792453 q^{3} +2.64590 q^{4} -1.70808 q^{6} -4.02126 q^{7} -1.39219 q^{8} -2.37202 q^{9} +O(q^{10})\)	\(q-2.15543 q^{2} +0.792453 q^{3} +2.64590 q^{4} -1.70808 q^{6} -4.02126 q^{7} -1.39219 q^{8} -2.37202 q^{9} +1.26468 q^{11} +2.09675 q^{12} -4.88804 q^{13} +8.66756 q^{14} -2.29102 q^{16} -0.117580 q^{17} +5.11273 q^{18} +6.32619 q^{19} -3.18666 q^{21} -2.72594 q^{22} +2.32707 q^{23} -1.10324 q^{24} +10.5359 q^{26} -4.25707 q^{27} -10.6398 q^{28} +3.41746 q^{29} -5.96228 q^{31} +7.72253 q^{32} +1.00220 q^{33} +0.253437 q^{34} -6.27612 q^{36} +6.92150 q^{37} -13.6357 q^{38} -3.87354 q^{39} +12.2048 q^{41} +6.86863 q^{42} -2.15228 q^{43} +3.34622 q^{44} -5.01584 q^{46} -4.98498 q^{47} -1.81553 q^{48} +9.17051 q^{49} -0.0931768 q^{51} -12.9333 q^{52} +2.46248 q^{53} +9.17584 q^{54} +5.59835 q^{56} +5.01321 q^{57} -7.36612 q^{58} +10.4520 q^{59} -12.2639 q^{61} +12.8513 q^{62} +9.53850 q^{63} -12.0634 q^{64} -2.16018 q^{66} +10.8246 q^{67} -0.311105 q^{68} +1.84409 q^{69} -2.22311 q^{71} +3.30230 q^{72} -2.69526 q^{73} -14.9188 q^{74} +16.7384 q^{76} -5.08562 q^{77} +8.34917 q^{78} +5.79087 q^{79} +3.74253 q^{81} -26.3067 q^{82} -4.54310 q^{83} -8.43156 q^{84} +4.63910 q^{86} +2.70818 q^{87} -1.76068 q^{88} +10.1221 q^{89} +19.6561 q^{91} +6.15718 q^{92} -4.72483 q^{93} +10.7448 q^{94} +6.11974 q^{96} -4.37973 q^{97} -19.7664 q^{98} -2.99986 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9}+O(q^{10}) \) 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

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