LMFDB - Embedded newform 275.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.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:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +O(q^{10})$	\(q-2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} -11.0000 q^{11} -4.24871 q^{12} -5.35898 q^{13} +8.39230 q^{14} -59.4256 q^{16} +41.2154 q^{17} -97.9615 q^{18} +139.923 q^{19} -24.3538 q^{21} +30.0526 q^{22} +111.354 q^{23} +184.890 q^{24} +14.6410 q^{26} +70.2154 q^{27} +1.64617 q^{28} -24.9948 q^{29} +31.4974 q^{31} -24.2102 q^{32} -87.2102 q^{33} -112.603 q^{34} -19.2154 q^{36} -13.1436 q^{37} -382.277 q^{38} -42.4871 q^{39} +261.072 q^{41} +66.5359 q^{42} +57.7128 q^{43} +5.89488 q^{44} -304.224 q^{46} +343.846 q^{47} -471.138 q^{48} -333.564 q^{49} +326.764 q^{51} +2.87187 q^{52} +342.995 q^{53} -191.832 q^{54} -71.6359 q^{56} +1109.34 q^{57} +68.2872 q^{58} +88.3693 q^{59} +738.697 q^{61} -86.0526 q^{62} -110.144 q^{63} +541.549 q^{64} +238.263 q^{66} -342.359 q^{67} -22.0873 q^{68} +882.836 q^{69} -207.364 q^{71} +836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} -74.9845 q^{76} +33.7898 q^{77} +116.077 q^{78} +1294.23 q^{79} -411.441 q^{81} -713.261 q^{82} -441.846 q^{83} +13.0512 q^{84} -157.674 q^{86} -198.164 q^{87} -256.526 q^{88} -1489.11 q^{89} +16.4617 q^{91} -59.6743 q^{92} +249.718 q^{93} -939.405 q^{94} -191.944 q^{96} -1346.42 q^{97} +911.314 q^{98} -394.420 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 4 q^{14} - 8 q^{16} + 124 q^{17} - 92 q^{18} + 72 q^{19} + 76 q^{21} + 22 q^{22} + 98 q^{23} + 252 q^{24} - 40 q^{26} + 182 q^{27} + 128 q^{28} + 144 q^{29} - 34 q^{31} + 104 q^{32} - 22 q^{33} - 52 q^{34} - 80 q^{36} - 54 q^{37} - 432 q^{38} + 400 q^{39} + 536 q^{41} + 140 q^{42} + 60 q^{43} + 88 q^{44} - 314 q^{46} + 272 q^{47} - 776 q^{48} - 390 q^{49} - 164 q^{51} + 560 q^{52} + 492 q^{53} - 110 q^{54} + 120 q^{56} + 1512 q^{57} + 192 q^{58} + 634 q^{59} + 840 q^{61} - 134 q^{62} - 248 q^{63} + 224 q^{64} + 286 q^{66} - 754 q^{67} - 640 q^{68} + 962 q^{69} - 678 q^{71} + 744 q^{72} + 400 q^{73} + 6 q^{74} + 432 q^{76} + 220 q^{77} + 440 q^{78} + 316 q^{79} - 1294 q^{81} - 512 q^{82} - 468 q^{83} - 736 q^{84} - 156 q^{86} - 1200 q^{87} - 132 q^{88} - 1842 q^{89} + 1280 q^{91} + 40 q^{92} + 638 q^{93} - 992 q^{94} - 952 q^{96} - 2194 q^{97} + 870 q^{98} - 484 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 - 22 * q^11 + 40 * q^12 - 80 * q^13 - 4 * q^14 - 8 * q^16 + 124 * q^17 - 92 * q^18 + 72 * q^19 + 76 * q^21 + 22 * q^22 + 98 * q^23 + 252 * q^24 - 40 * q^26 + 182 * q^27 + 128 * q^28 + 144 * q^29 - 34 * q^31 + 104 * q^32 - 22 * q^33 - 52 * q^34 - 80 * q^36 - 54 * q^37 - 432 * q^38 + 400 * q^39 + 536 * q^41 + 140 * q^42 + 60 * q^43 + 88 * q^44 - 314 * q^46 + 272 * q^47 - 776 * q^48 - 390 * q^49 - 164 * q^51 + 560 * q^52 + 492 * q^53 - 110 * q^54 + 120 * q^56 + 1512 * q^57 + 192 * q^58 + 634 * q^59 + 840 * q^61 - 134 * q^62 - 248 * q^63 + 224 * q^64 + 286 * q^66 - 754 * q^67 - 640 * q^68 + 962 * q^69 - 678 * q^71 + 744 * q^72 + 400 * q^73 + 6 * q^74 + 432 * q^76 + 220 * q^77 + 440 * q^78 + 316 * q^79 - 1294 * q^81 - 512 * q^82 - 468 * q^83 - 736 * q^84 - 156 * q^86 - 1200 * q^87 - 132 * q^88 - 1842 * q^89 + 1280 * q^91 + 40 * q^92 + 638 * q^93 - 992 * q^94 - 952 * q^96 - 2194 * q^97 + 870 * q^98 - 484 * 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 $

$2$	−2.73205	−0.965926	−0.482963	−	0.875641i	$-0.660439\pi$
$2$	−2.73205	−0.965926	−0.482963	+	0.875641i	$0.660439\pi$
$3$	7.92820	1.52578	0.762892	−	0.646526i	$-0.223779\pi$
$3$	7.92820	1.52578	0.762892	+	0.646526i	$0.223779\pi$
$4$	−0.535898	−0.0669873
$5$	0	0
$5$	0	0
$6$	−21.6603	−1.47379
$7$	−3.07180	−0.165861	−0.0829307	−	0.996555i	$-0.526428\pi$
$7$	−3.07180	−0.165861	−0.0829307	+	0.996555i	$0.526428\pi$
$8$	23.3205	1.03063
$9$	35.8564	1.32802
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−4.24871	−0.102208
$13$	−5.35898	−0.114332	−0.0571659	−	0.998365i	$-0.518206\pi$
$13$	−5.35898	−0.114332	−0.0571659	+	0.998365i	$0.518206\pi$
$14$	8.39230	0.160210
$15$	0	0
$16$	−59.4256	−0.928525
$17$	41.2154	0.588012	0.294006	−	0.955804i	$-0.405011\pi$
$17$	41.2154	0.588012	0.294006	+	0.955804i	$0.405011\pi$
$18$	−97.9615	−1.28276
$19$	139.923	1.68950	0.844751	−	0.535159i	$-0.179748\pi$
$19$	139.923	1.68950	0.844751	+	0.535159i	$0.179748\pi$
$20$	0	0
$21$	−24.3538	−0.253069
$22$	30.0526	0.291238
$23$	111.354	1.00952	0.504758	−	0.863261i	$-0.331582\pi$
$23$	111.354	1.00952	0.504758	+	0.863261i	$0.331582\pi$
$24$	184.890	1.57252
$25$	0	0
$26$	14.6410	0.110436
$27$	70.2154	0.500480
$28$	1.64617	0.0111106
$29$	−24.9948	−0.160049	−0.0800246	−	0.996793i	$-0.525500\pi$
$29$	−24.9948	−0.160049	−0.0800246	+	0.996793i	$0.525500\pi$
$30$	0	0
$31$	31.4974	0.182487	0.0912436	−	0.995829i	$-0.470916\pi$
$31$	31.4974	0.182487	0.0912436	+	0.995829i	$0.470916\pi$
$32$	−24.2102	−0.133744
$33$	−87.2102	−0.460041
$34$	−112.603	−0.567976
$35$	0	0
$36$	−19.2154	−0.0889601
$37$	−13.1436	−0.0583998	−0.0291999	−	0.999574i	$-0.509296\pi$
$37$	−13.1436	−0.0583998	−0.0291999	+	0.999574i	$0.509296\pi$
$38$	−382.277	−1.63193
$39$	−42.4871	−0.174446
$40$	0	0
$41$	261.072	0.994453	0.497226	−	0.867621i	$-0.334352\pi$
$41$	261.072	0.994453	0.497226	+	0.867621i	$0.334352\pi$
$42$	66.5359	0.244446
$43$	57.7128	0.204677	0.102339	−	0.994750i	$-0.467367\pi$
$43$	57.7128	0.204677	0.102339	+	0.994750i	$0.467367\pi$
$44$	5.89488	0.0201974
$45$	0	0
$46$	−304.224	−0.975118
$47$	343.846	1.06713	0.533565	−	0.845759i	$-0.320852\pi$
$47$	343.846	1.06713	0.533565	+	0.845759i	$0.320852\pi$
$48$	−471.138	−1.41673
$49$	−333.564	−0.972490
$50$	0	0
$51$	326.764	0.897179
$52$	2.87187	0.00765879
$53$	342.995	0.888943	0.444471	−	0.895793i	$-0.353392\pi$
$53$	342.995	0.888943	0.444471	+	0.895793i	$0.353392\pi$
$54$	−191.832	−0.483426
$55$	0	0
$56$	−71.6359	−0.170942
$57$	1109.34	2.57782
$58$	68.2872	0.154596
$59$	88.3693	0.194995	0.0974975	−	0.995236i	$-0.468916\pi$
$59$	88.3693	0.194995	0.0974975	+	0.995236i	$0.468916\pi$
$60$	0	0
$61$	738.697	1.55050	0.775250	−	0.631654i	$-0.217624\pi$
$61$	738.697	1.55050	0.775250	+	0.631654i	$0.217624\pi$
$62$	−86.0526	−0.176269
$63$	−110.144	−0.220266
$64$	541.549	1.05771
$65$	0	0
$66$	238.263	0.444365
$67$	−342.359	−0.624266	−0.312133	−	0.950038i	$-0.601043\pi$
$67$	−342.359	−0.624266	−0.312133	+	0.950038i	$0.601043\pi$
$68$	−22.0873	−0.0393893
$69$	882.836	1.54030
$70$	0	0
$71$	−207.364	−0.346614	−0.173307	−	0.984868i	$-0.555445\pi$
$71$	−207.364	−0.346614	−0.173307	+	0.984868i	$0.555445\pi$
$72$	836.190	1.36869
$73$	1010.60	1.62030	0.810149	−	0.586224i	$-0.199386\pi$
$73$	1010.60	1.62030	0.810149	+	0.586224i	$0.199386\pi$
$74$	35.9090	0.0564099
$75$	0	0
$76$	−74.9845	−0.113175
$77$	33.7898	0.0500091
$78$	116.077	0.168502
$79$	1294.23	1.84319	0.921593	−	0.388157i	$-0.126888\pi$
$79$	1294.23	1.84319	0.921593	+	0.388157i	$0.126888\pi$
$80$	0	0
$81$	−411.441	−0.564391
$82$	−713.261	−0.960568
$83$	−441.846	−0.584324	−0.292162	−	0.956369i	$-0.594375\pi$
$83$	−441.846	−0.584324	−0.292162	+	0.956369i	$0.594375\pi$
$84$	13.0512	0.0169524
$85$	0	0
$86$	−157.674	−0.197703
$87$	−198.164	−0.244200
$88$	−256.526	−0.310747
$89$	−1489.11	−1.77355	−0.886773	−	0.462205i	$-0.847058\pi$
$89$	−1489.11	−1.77355	−0.886773	+	0.462205i	$0.847058\pi$
$90$	0	0
$91$	16.4617	0.0189633
$92$	−59.6743	−0.0676248
$93$	249.718	0.278436
$94$	−939.405	−1.03077
$95$	0	0
$96$	−191.944	−0.204064
$97$	−1346.42	−1.40936	−0.704679	−	0.709526i	$-0.748909\pi$
$97$	−1346.42	−1.40936	−0.704679	+	0.709526i	$0.748909\pi$
$98$	911.314	0.939353
$99$	−394.420	−0.400412
$100$	0	0
$101$	−161.461	−0.159069	−0.0795347	−	0.996832i	$-0.525343\pi$
$101$	−161.461	−0.159069	−0.0795347	+	0.996832i	$0.525343\pi$
$102$	−892.736	−0.866608
$103$	34.7592	0.0332517	0.0166259	−	0.999862i	$-0.494708\pi$
$103$	34.7592	0.0332517	0.0166259	+	0.999862i	$0.494708\pi$
$104$	−124.974	−0.117834
$105$	0	0
$106$	−937.079	−0.858653
$107$	−832.179	−0.751867	−0.375934	−	0.926647i	$-0.622678\pi$
$107$	−832.179	−0.751867	−0.375934	+	0.926647i	$0.622678\pi$
$108$	−37.6283	−0.0335258
$109$	1044.26	0.917629	0.458815	−	0.888532i	$-0.348274\pi$
$109$	1044.26	0.917629	0.458815	+	0.888532i	$0.348274\pi$
$110$	0	0
$111$	−104.205	−0.0891055
$112$	182.543	0.154007
$113$	−295.082	−0.245654	−0.122827	−	0.992428i	$-0.539196\pi$
$113$	−295.082	−0.245654	−0.122827	+	0.992428i	$0.539196\pi$
$114$	−3030.77	−2.48998
$115$	0	0
$116$	13.3947	0.0107213
$117$	−192.154	−0.151834
$118$	−241.429	−0.188351
$119$	−126.605	−0.0975285
$120$	0	0
$121$	121.000	0.0909091
$122$	−2018.16	−1.49767
$123$	2069.83	1.51732
$124$	−16.8794	−0.0122243
$125$	0	0
$126$	300.918	0.212761
$127$	1317.60	0.920618	0.460309	−	0.887759i	$-0.347739\pi$
$127$	1317.60	0.920618	0.460309	+	0.887759i	$0.347739\pi$
$128$	−1285.86	−0.887928
$129$	457.559	0.312293
$130$	0	0
$131$	−1600.71	−1.06759	−0.533797	−	0.845612i	$-0.679235\pi$
$131$	−1600.71	−1.06759	−0.533797	+	0.845612i	$0.679235\pi$
$132$	46.7358	0.0308169
$133$	−429.815	−0.280223
$134$	935.342	0.602994
$135$	0	0
$136$	961.164	0.606023
$137$	−1611.68	−1.00507	−0.502536	−	0.864556i	$-0.667600\pi$
$137$	−1611.68	−1.00507	−0.502536	+	0.864556i	$0.667600\pi$
$138$	−2411.95	−1.48782
$139$	−31.8619	−0.0194424	−0.00972120	−	0.999953i	$-0.503094\pi$
$139$	−31.8619	−0.0194424	−0.00972120	+	0.999953i	$0.503094\pi$
$140$	0	0
$141$	2726.08	1.62821
$142$	566.529	0.334803
$143$	58.9488	0.0344724
$144$	−2130.79	−1.23310
$145$	0	0
$146$	−2761.01	−1.56509
$147$	−2644.56	−1.48381
$148$	7.04363	0.00391205
$149$	−2428.34	−1.33515	−0.667576	−	0.744542i	$-0.732668\pi$
$149$	−2428.34	−1.33515	−0.667576	+	0.744542i	$0.732668\pi$
$150$	0	0
$151$	−2576.68	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$151$	−2576.68	−1.38866	−0.694328	+	0.719659i	$0.744298\pi$
$152$	3263.08	1.74125
$153$	1477.84	0.780889
$154$	−92.3154	−0.0483051
$155$	0	0
$156$	22.7688	0.0116856
$157$	−2475.94	−1.25861	−0.629305	−	0.777158i	$-0.716660\pi$
$157$	−2475.94	−1.25861	−0.629305	+	0.777158i	$0.716660\pi$
$158$	−3535.89	−1.78038
$159$	2719.33	1.35633
$160$	0	0
$161$	−342.056	−0.167440
$162$	1124.08	0.545160
$163$	2725.11	1.30949	0.654745	−	0.755850i	$-0.272776\pi$
$163$	2725.11	1.30949	0.654745	+	0.755850i	$0.272776\pi$
$164$	−139.908	−0.0666157
$165$	0	0
$166$	1207.15	0.564414
$167$	−2737.30	−1.26837	−0.634187	−	0.773180i	$-0.718665\pi$
$167$	−2737.30	−1.26837	−0.634187	+	0.773180i	$0.718665\pi$
$168$	−567.944	−0.260820
$169$	−2168.28	−0.986928
$170$	0	0
$171$	5017.14	2.24368
$172$	−30.9282	−0.0137108
$173$	−2307.42	−1.01404	−0.507022	−	0.861933i	$-0.669254\pi$
$173$	−2307.42	−1.01404	−0.507022	+	0.861933i	$0.669254\pi$
$174$	541.395	0.235879
$175$	0	0
$176$	653.682	0.279961
$177$	700.610	0.297520
$178$	4068.33	1.71311
$179$	−1312.15	−0.547905	−0.273953	−	0.961743i	$-0.588331\pi$
$179$	−1312.15	−0.547905	−0.273953	+	0.961743i	$0.588331\pi$
$180$	0	0
$181$	−803.174	−0.329831	−0.164916	−	0.986308i	$-0.552735\pi$
$181$	−803.174	−0.329831	−0.164916	+	0.986308i	$0.552735\pi$
$182$	−44.9742	−0.0183171
$183$	5856.54	2.36573
$184$	2596.83	1.04044
$185$	0	0
$186$	−682.242	−0.268949
$187$	−453.369	−0.177292
$188$	−184.267	−0.0714842
$189$	−215.687	−0.0830103
$190$	0	0
$191$	1718.25	0.650932	0.325466	−	0.945554i	$-0.394479\pi$
$191$	1718.25	0.650932	0.325466	+	0.945554i	$0.394479\pi$
$192$	4293.51	1.61384
$193$	−1340.18	−0.499837	−0.249919	−	0.968267i	$-0.580404\pi$
$193$	−1340.18	−0.499837	−0.249919	+	0.968267i	$0.580404\pi$
$194$	3678.48	1.36134
$195$	0	0
$196$	178.756	0.0651445
$197$	3518.33	1.27244	0.636220	−	0.771508i	$-0.280497\pi$
$197$	3518.33	1.27244	0.636220	+	0.771508i	$0.280497\pi$
$198$	1077.58	0.386768
$199$	823.692	0.293417	0.146709	−	0.989180i	$-0.453132\pi$
$199$	823.692	0.293417	0.146709	+	0.989180i	$0.453132\pi$
$200$	0	0
$201$	−2714.29	−0.952494
$202$	441.121	0.153649
$203$	76.7791	0.0265460
$204$	−175.112	−0.0600996
$205$	0	0
$206$	−94.9639	−0.0321187
$207$	3992.75	1.34065
$208$	318.461	0.106160
$209$	−1539.15	−0.509404
$210$	0	0
$211$	−107.343	−0.0350228	−0.0175114	−	0.999847i	$-0.505574\pi$
$211$	−107.343	−0.0350228	−0.0175114	+	0.999847i	$0.505574\pi$
$212$	−183.810	−0.0595479
$213$	−1644.03	−0.528858
$214$	2273.56	0.726248
$215$	0	0
$216$	1637.46	0.515810
$217$	−96.7537	−0.0302676
$218$	−2852.96	−0.886362
$219$	8012.24	2.47222
$220$	0	0
$221$	−220.873	−0.0672285
$222$	284.694	0.0860693
$223$	3933.68	1.18125	0.590625	−	0.806946i	$-0.298881\pi$
$223$	3933.68	1.18125	0.590625	+	0.806946i	$0.298881\pi$
$224$	74.3689	0.0221830
$225$	0	0
$226$	806.178	0.237284
$227$	1771.90	0.518085	0.259042	−	0.965866i	$-0.416593\pi$
$227$	1771.90	0.518085	0.259042	+	0.965866i	$0.416593\pi$
$228$	−594.493	−0.172681
$229$	1915.37	0.552713	0.276356	−	0.961055i	$-0.410873\pi$
$229$	1915.37	0.552713	0.276356	+	0.961055i	$0.410873\pi$
$230$	0	0
$231$	267.892	0.0763031
$232$	−582.892	−0.164952
$233$	−4396.32	−1.23610	−0.618052	−	0.786137i	$-0.712078\pi$
$233$	−4396.32	−1.23610	−0.618052	+	0.786137i	$0.712078\pi$
$234$	524.974	0.146661
$235$	0	0
$236$	−47.3570	−0.0130622
$237$	10260.9	2.81230
$238$	345.892	0.0942053
$239$	−4084.49	−1.10546	−0.552728	−	0.833362i	$-0.686413\pi$
$239$	−4084.49	−1.10546	−0.552728	+	0.833362i	$0.686413\pi$
$240$	0	0
$241$	3908.58	1.04471	0.522353	−	0.852730i	$-0.325054\pi$
$241$	3908.58	1.04471	0.522353	+	0.852730i	$0.325054\pi$
$242$	−330.578	−0.0878114
$243$	−5157.80	−1.36162
$244$	−395.867	−0.103864
$245$	0	0
$246$	−5654.88	−1.46562
$247$	−749.845	−0.193164
$248$	734.536	0.188077
$249$	−3503.05	−0.891552
$250$	0	0
$251$	1094.89	0.275335	0.137667	−	0.990479i	$-0.456040\pi$
$251$	1094.89	0.275335	0.137667	+	0.990479i	$0.456040\pi$
$252$	59.0258	0.0147551
$253$	−1224.89	−0.304381
$254$	−3599.76	−0.889249
$255$	0	0
$256$	−819.364	−0.200040
$257$	−783.179	−0.190091	−0.0950454	−	0.995473i	$-0.530300\pi$
$257$	−783.179	−0.190091	−0.0950454	+	0.995473i	$0.530300\pi$
$258$	−1250.07	−0.301652
$259$	40.3744	0.00968628
$260$	0	0
$261$	−896.225	−0.212548
$262$	4373.23	1.03122
$263$	−6180.06	−1.44897	−0.724484	−	0.689292i	$-0.757922\pi$
$263$	−6180.06	−1.44897	−0.724484	+	0.689292i	$0.757922\pi$
$264$	−2033.79	−0.474132
$265$	0	0
$266$	1174.28	0.270675
$267$	−11806.0	−2.70605
$268$	183.470	0.0418179
$269$	986.965	0.223704	0.111852	−	0.993725i	$-0.464322\pi$
$269$	986.965	0.223704	0.111852	+	0.993725i	$0.464322\pi$
$270$	0	0
$271$	4576.99	1.02595	0.512975	−	0.858404i	$-0.328543\pi$
$271$	4576.99	1.02595	0.512975	+	0.858404i	$0.328543\pi$
$272$	−2449.25	−0.545984
$273$	130.512	0.0289338
$274$	4403.18	0.970825
$275$	0	0
$276$	−473.110	−0.103181
$277$	−567.836	−0.123169	−0.0615847	−	0.998102i	$-0.519615\pi$
$277$	−567.836	−0.123169	−0.0615847	+	0.998102i	$0.519615\pi$
$278$	87.0484	0.0187799
$279$	1129.38	0.242346
$280$	0	0
$281$	5311.01	1.12750	0.563752	−	0.825944i	$-0.309357\pi$
$281$	5311.01	1.12750	0.563752	+	0.825944i	$0.309357\pi$
$282$	−7447.79	−1.57273
$283$	4728.44	0.993204	0.496602	−	0.867978i	$-0.334581\pi$
$283$	4728.44	0.993204	0.496602	+	0.867978i	$0.334581\pi$
$284$	111.126	0.0232187
$285$	0	0
$286$	−161.051	−0.0332977
$287$	−801.960	−0.164941
$288$	−868.092	−0.177614
$289$	−3214.29	−0.654242
$290$	0	0
$291$	−10674.7	−2.15038
$292$	−541.579	−0.108539
$293$	−2328.92	−0.464358	−0.232179	−	0.972673i	$-0.574585\pi$
$293$	−2328.92	−0.464358	−0.232179	+	0.972673i	$0.574585\pi$
$294$	7225.08	1.43325
$295$	0	0
$296$	−306.515	−0.0601886
$297$	−772.369	−0.150900
$298$	6634.36	1.28966
$299$	−596.743	−0.115420
$300$	0	0
$301$	−177.282	−0.0339481
$302$	7039.61	1.34134
$303$	−1280.10	−0.242705
$304$	−8315.01	−1.56875
$305$	0	0
$306$	−4037.52	−0.754280
$307$	1678.07	0.311962	0.155981	−	0.987760i	$-0.450146\pi$
$307$	1678.07	0.311962	0.155981	+	0.987760i	$0.450146\pi$
$308$	−18.1079	−0.00334997
$309$	275.578	0.0507349
$310$	0	0
$311$	3572.71	0.651413	0.325707	−	0.945471i	$-0.394398\pi$
$311$	3572.71	0.651413	0.325707	+	0.945471i	$0.394398\pi$
$312$	−990.821	−0.179789
$313$	−7184.36	−1.29739	−0.648697	−	0.761047i	$-0.724686\pi$
$313$	−7184.36	−1.29739	−0.648697	+	0.761047i	$0.724686\pi$
$314$	6764.40	1.21572
$315$	0	0
$316$	−693.573	−0.123470
$317$	15.7077	0.00278306	0.00139153	−	0.999999i	$-0.499557\pi$
$317$	15.7077	0.00278306	0.00139153	+	0.999999i	$0.499557\pi$
$318$	−7429.36	−1.31012
$319$	274.943	0.0482566
$320$	0	0
$321$	−6597.69	−1.14719
$322$	934.515	0.161734
$323$	5766.98	0.993447
$324$	220.491	0.0378070
$325$	0	0
$326$	−7445.13	−1.26487
$327$	8279.08	1.40010
$328$	6088.33	1.02491
$329$	−1056.23	−0.176996
$330$	0	0
$331$	−1318.95	−0.219022	−0.109511	−	0.993986i	$-0.534928\pi$
$331$	−1318.95	−0.219022	−0.109511	+	0.993986i	$0.534928\pi$
$332$	236.785	0.0391423
$333$	−471.282	−0.0775558
$334$	7478.43	1.22515
$335$	0	0
$336$	1447.24	0.234981
$337$	239.183	0.0386621	0.0193310	−	0.999813i	$-0.493846\pi$
$337$	239.183	0.0386621	0.0193310	+	0.999813i	$0.493846\pi$
$338$	5923.85	0.953299
$339$	−2339.47	−0.374816
$340$	0	0
$341$	−346.472	−0.0550220
$342$	−13707.1	−2.16723
$343$	2078.27	0.327160
$344$	1345.89	0.210947
$345$	0	0
$346$	6303.98	0.979491
$347$	5862.79	0.907006	0.453503	−	0.891255i	$-0.350174\pi$
$347$	5862.79	0.907006	0.453503	+	0.891255i	$0.350174\pi$
$348$	106.196	0.0163583
$349$	3491.73	0.535553	0.267776	−	0.963481i	$-0.413711\pi$
$349$	3491.73	0.535553	0.267776	+	0.963481i	$0.413711\pi$
$350$	0	0
$351$	−376.283	−0.0572208
$352$	266.313	0.0403253
$353$	10916.7	1.64600	0.822999	−	0.568043i	$-0.192299\pi$
$353$	10916.7	1.64600	0.822999	+	0.568043i	$0.192299\pi$
$354$	−1914.10	−0.287382
$355$	0	0
$356$	798.013	0.118805
$357$	−1003.75	−0.148807
$358$	3584.87	0.529236
$359$	−11500.7	−1.69077	−0.845384	−	0.534160i	$-0.820628\pi$
$359$	−11500.7	−1.69077	−0.845384	+	0.534160i	$0.820628\pi$
$360$	0	0
$361$	12719.5	1.85442
$362$	2194.31	0.318592
$363$	959.313	0.138708
$364$	−8.82180	−0.00127030
$365$	0	0
$366$	−16000.4	−2.28512
$367$	−6767.01	−0.962493	−0.481246	−	0.876585i	$-0.659816\pi$
$367$	−6767.01	−0.962493	−0.481246	+	0.876585i	$0.659816\pi$
$368$	−6617.27	−0.937362
$369$	9361.10	1.32065
$370$	0	0
$371$	−1053.61	−0.147441
$372$	−133.823	−0.0186517
$373$	5310.22	0.737139	0.368569	−	0.929600i	$-0.379848\pi$
$373$	5310.22	0.737139	0.368569	+	0.929600i	$0.379848\pi$
$374$	1238.63	0.171251
$375$	0	0
$376$	8018.67	1.09982
$377$	133.947	0.0182987
$378$	589.269	0.0801818
$379$	−838.267	−0.113612	−0.0568059	−	0.998385i	$-0.518092\pi$
$379$	−838.267	−0.113612	−0.0568059	+	0.998385i	$0.518092\pi$
$380$	0	0
$381$	10446.2	1.40466
$382$	−4694.34	−0.628752
$383$	2832.16	0.377851	0.188925	−	0.981991i	$-0.439500\pi$
$383$	2832.16	0.377851	0.188925	+	0.981991i	$0.439500\pi$
$384$	−10194.5	−1.35479
$385$	0	0
$386$	3661.45	0.482806
$387$	2069.37	0.271814
$388$	721.542	0.0944091
$389$	3111.25	0.405519	0.202759	−	0.979229i	$-0.435009\pi$
$389$	3111.25	0.405519	0.202759	+	0.979229i	$0.435009\pi$
$390$	0	0
$391$	4589.49	0.593608
$392$	−7778.88	−1.00228
$393$	−12690.8	−1.62892
$394$	−9612.25	−1.22908
$395$	0	0
$396$	211.369	0.0268225
$397$	−14208.7	−1.79626	−0.898131	−	0.439728i	$-0.855075\pi$
$397$	−14208.7	−1.79626	−0.898131	+	0.439728i	$0.855075\pi$
$398$	−2250.37	−0.283419
$399$	−3407.66	−0.427560
$400$	0	0
$401$	−6261.68	−0.779784	−0.389892	−	0.920861i	$-0.627488\pi$
$401$	−6261.68	−0.779784	−0.389892	+	0.920861i	$0.627488\pi$
$402$	7415.58	0.920039
$403$	−168.794	−0.0208641
$404$	86.5269	0.0106556
$405$	0	0
$406$	−209.764	−0.0256415
$407$	144.580	0.0176082
$408$	7620.30	0.924660
$409$	−4192.50	−0.506860	−0.253430	−	0.967354i	$-0.581559\pi$
$409$	−4192.50	−0.506860	−0.253430	+	0.967354i	$0.581559\pi$
$410$	0	0
$411$	−12777.7	−1.53352
$412$	−18.6274	−0.00222744
$413$	−271.453	−0.0323421
$414$	−10908.4	−1.29497
$415$	0	0
$416$	129.742	0.0152912
$417$	−252.608	−0.0296649
$418$	4205.05	0.492047
$419$	−9287.15	−1.08283	−0.541416	−	0.840755i	$-0.682112\pi$
$419$	−9287.15	−1.08283	−0.541416	+	0.840755i	$0.682112\pi$
$420$	0	0
$421$	13146.0	1.52185	0.760923	−	0.648842i	$-0.224746\pi$
$421$	13146.0	1.52185	0.760923	+	0.648842i	$0.224746\pi$
$422$	293.267	0.0338294
$423$	12329.1	1.41716
$424$	7998.81	0.916172
$425$	0	0
$426$	4491.56	0.510838
$427$	−2269.13	−0.257168
$428$	445.964	0.0503656
$429$	467.358	0.0525974
$430$	0	0
$431$	4909.67	0.548701	0.274351	−	0.961630i	$-0.411537\pi$
$431$	4909.67	0.548701	0.274351	+	0.961630i	$0.411537\pi$
$432$	−4172.59	−0.464708
$433$	11743.3	1.30334	0.651671	−	0.758502i	$-0.274068\pi$
$433$	11743.3	1.30334	0.651671	+	0.758502i	$0.274068\pi$
$434$	264.336	0.0292363
$435$	0	0
$436$	−559.615	−0.0614695
$437$	15581.0	1.70558
$438$	−21889.8	−2.38798
$439$	−11824.2	−1.28551	−0.642754	−	0.766073i	$-0.722208\pi$
$439$	−11824.2	−1.28551	−0.642754	+	0.766073i	$0.722208\pi$
$440$	0	0
$441$	−11960.4	−1.29148
$442$	603.435	0.0649377
$443$	−10102.1	−1.08344	−0.541722	−	0.840558i	$-0.682228\pi$
$443$	−10102.1	−1.08344	−0.541722	+	0.840558i	$0.682228\pi$
$444$	55.8433	0.00596894
$445$	0	0
$446$	−10747.0	−1.14100
$447$	−19252.4	−2.03715
$448$	−1663.53	−0.175434
$449$	−345.254	−0.0362885	−0.0181443	−	0.999835i	$-0.505776\pi$
$449$	−345.254	−0.0362885	−0.0181443	+	0.999835i	$0.505776\pi$
$450$	0	0
$451$	−2871.79	−0.299839
$452$	158.134	0.0164557
$453$	−20428.4	−2.11879
$454$	−4840.93	−0.500431
$455$	0	0
$456$	25870.3	2.65678
$457$	10567.1	1.08164	0.540821	−	0.841138i	$-0.318114\pi$
$457$	10567.1	1.08164	0.540821	+	0.841138i	$0.318114\pi$
$458$	−5232.89	−0.533879
$459$	2893.95	0.294288
$460$	0	0
$461$	4733.96	0.478270	0.239135	−	0.970986i	$-0.423136\pi$
$461$	4733.96	0.478270	0.239135	+	0.970986i	$0.423136\pi$
$462$	−731.895	−0.0737031
$463$	−3431.20	−0.344409	−0.172204	−	0.985061i	$-0.555089\pi$
$463$	−3431.20	−0.344409	−0.172204	+	0.985061i	$0.555089\pi$
$464$	1485.33	0.148610
$465$	0	0
$466$	12011.0	1.19399
$467$	−5116.96	−0.507034	−0.253517	−	0.967331i	$-0.581587\pi$
$467$	−5116.96	−0.507034	−0.253517	+	0.967331i	$0.581587\pi$
$468$	102.975	0.0101710
$469$	1051.66	0.103542
$470$	0	0
$471$	−19629.8	−1.92037
$472$	2060.82	0.200968
$473$	−634.841	−0.0617125
$474$	−28033.2	−2.71648
$475$	0	0
$476$	67.8476	0.00653317
$477$	12298.6	1.18053
$478$	11159.0	1.06779
$479$	11566.9	1.10335	0.551675	−	0.834059i	$-0.313989\pi$
$479$	11566.9	1.10335	0.551675	+	0.834059i	$0.313989\pi$
$480$	0	0
$481$	70.4363	0.00667696
$482$	−10678.4	−1.00911
$483$	−2711.89	−0.255477
$484$	−64.8437	−0.00608975
$485$	0	0
$486$	14091.4	1.31522
$487$	18326.5	1.70525	0.852623	−	0.522527i	$-0.175010\pi$
$487$	18326.5	1.70525	0.852623	+	0.522527i	$0.175010\pi$
$488$	17226.8	1.59799
$489$	21605.2	1.99800
$490$	0	0
$491$	−7617.58	−0.700156	−0.350078	−	0.936721i	$-0.613845\pi$
$491$	−7617.58	−0.700156	−0.350078	+	0.936721i	$0.613845\pi$
$492$	−1109.22	−0.101641
$493$	−1030.17	−0.0941108
$494$	2048.62	0.186582
$495$	0	0
$496$	−1871.75	−0.169444
$497$	636.980	0.0574899
$498$	9570.50	0.861173
$499$	12909.1	1.15810	0.579050	−	0.815292i	$-0.303424\pi$
$499$	12909.1	1.15810	0.579050	+	0.815292i	$0.303424\pi$
$500$	0	0
$501$	−21701.8	−1.93526
$502$	−2991.30	−0.265953
$503$	−10165.7	−0.901121	−0.450561	−	0.892746i	$-0.648776\pi$
$503$	−10165.7	−0.901121	−0.450561	+	0.892746i	$0.648776\pi$
$504$	−2568.60	−0.227013
$505$	0	0
$506$	3346.47	0.294009
$507$	−17190.6	−1.50584
$508$	−706.102	−0.0616697
$509$	6449.93	0.561666	0.280833	−	0.959757i	$-0.409389\pi$
$509$	6449.93	0.561666	0.280833	+	0.959757i	$0.409389\pi$
$510$	0	0
$511$	−3104.36	−0.268745
$512$	12525.4	1.08115
$513$	9824.75	0.845562
$514$	2139.68	0.183614
$515$	0	0
$516$	−245.205	−0.0209197
$517$	−3782.31	−0.321752
$518$	−110.305	−0.00935623
$519$	−18293.7	−1.54721
$520$	0	0
$521$	−19327.4	−1.62524	−0.812620	−	0.582794i	$-0.801959\pi$
$521$	−19327.4	−1.62524	−0.812620	+	0.582794i	$0.801959\pi$
$522$	2448.53	0.205305
$523$	−6259.09	−0.523310	−0.261655	−	0.965161i	$-0.584268\pi$
$523$	−6259.09	−0.523310	−0.261655	+	0.965161i	$0.584268\pi$
$524$	857.819	0.0715153
$525$	0	0
$526$	16884.2	1.39960
$527$	1298.18	0.107305
$528$	5182.52	0.427160
$529$	232.675	0.0191235
$530$	0	0
$531$	3168.61	0.258956
$532$	230.337	0.0187714
$533$	−1399.08	−0.113698
$534$	32254.6	2.61384
$535$	0	0
$536$	−7983.99	−0.643387
$537$	−10403.0	−0.835985
$538$	−2696.44	−0.216081
$539$	3669.20	0.293217
$540$	0	0
$541$	−14008.2	−1.11323	−0.556616	−	0.830770i	$-0.687900\pi$
$541$	−14008.2	−1.11323	−0.556616	+	0.830770i	$0.687900\pi$
$542$	−12504.6	−0.990991
$543$	−6367.72	−0.503251
$544$	−997.834	−0.0786430
$545$	0	0
$546$	−356.565	−0.0279479
$547$	4949.45	0.386879	0.193440	−	0.981112i	$-0.438036\pi$
$547$	4949.45	0.386879	0.193440	+	0.981112i	$0.438036\pi$
$548$	863.695	0.0673270
$549$	26487.0	2.05909
$550$	0	0
$551$	−3497.35	−0.270404
$552$	20588.2	1.58748
$553$	−3975.60	−0.305714
$554$	1551.36	0.118973
$555$	0	0
$556$	17.0748	0.00130239
$557$	3801.58	0.289188	0.144594	−	0.989491i	$-0.453812\pi$
$557$	3801.58	0.289188	0.144594	+	0.989491i	$0.453812\pi$
$558$	−3085.54	−0.234088
$559$	−309.282	−0.0234011
$560$	0	0
$561$	−3594.40	−0.270510
$562$	−14510.0	−1.08908
$563$	9900.11	0.741101	0.370551	−	0.928812i	$-0.379169\pi$
$563$	9900.11	0.741101	0.370551	+	0.928812i	$0.379169\pi$
$564$	−1460.90	−0.109069
$565$	0	0
$566$	−12918.3	−0.959361
$567$	1263.86	0.0936107
$568$	−4835.84	−0.357231
$569$	5329.16	0.392636	0.196318	−	0.980540i	$-0.437102\pi$
$569$	5329.16	0.392636	0.196318	+	0.980540i	$0.437102\pi$
$570$	0	0
$571$	−16962.6	−1.24319	−0.621597	−	0.783337i	$-0.713516\pi$
$571$	−16962.6	−1.24319	−0.621597	+	0.783337i	$0.713516\pi$
$572$	−31.5906	−0.00230921
$573$	13622.6	0.993181
$574$	2190.99	0.159321
$575$	0	0
$576$	19418.0	1.40466
$577$	15487.0	1.11738	0.558692	−	0.829375i	$-0.311303\pi$
$577$	15487.0	1.11738	0.558692	+	0.829375i	$0.311303\pi$
$578$	8781.61	0.631949
$579$	−10625.3	−0.762643
$580$	0	0
$581$	1357.26	0.0969169
$582$	29163.7	2.07710
$583$	−3772.94	−0.268026
$584$	23567.7	1.66993
$585$	0	0
$586$	6362.72	0.448535
$587$	−11084.2	−0.779373	−0.389686	−	0.920948i	$-0.627417\pi$
$587$	−11084.2	−0.779373	−0.389686	+	0.920948i	$0.627417\pi$
$588$	1417.22	0.0993964
$589$	4407.22	0.308313
$590$	0	0
$591$	27894.0	1.94147
$592$	781.066	0.0542257
$593$	−4349.68	−0.301214	−0.150607	−	0.988594i	$-0.548123\pi$
$593$	−4349.68	−0.301214	−0.150607	+	0.988594i	$0.548123\pi$
$594$	2110.15	0.145759
$595$	0	0
$596$	1301.34	0.0894382
$597$	6530.40	0.447691
$598$	1630.33	0.111487
$599$	13183.9	0.899299	0.449650	−	0.893205i	$-0.351549\pi$
$599$	13183.9	0.899299	0.449650	+	0.893205i	$0.351549\pi$
$600$	0	0
$601$	−18765.0	−1.27361	−0.636806	−	0.771024i	$-0.719745\pi$
$601$	−18765.0	−1.27361	−0.636806	+	0.771024i	$0.719745\pi$
$602$	484.344	0.0327913
$603$	−12275.8	−0.829034
$604$	1380.84	0.0930223
$605$	0	0
$606$	3497.29	0.234435
$607$	−21871.4	−1.46249	−0.731244	−	0.682116i	$-0.761060\pi$
$607$	−21871.4	−1.46249	−0.731244	+	0.682116i	$0.761060\pi$
$608$	−3387.57	−0.225961
$609$	608.720	0.0405034
$610$	0	0
$611$	−1842.67	−0.122007
$612$	−791.970	−0.0523096
$613$	3527.85	0.232445	0.116222	−	0.993223i	$-0.462921\pi$
$613$	3527.85	0.232445	0.116222	+	0.993223i	$0.462921\pi$
$614$	−4584.56	−0.301332
$615$	0	0
$616$	787.994	0.0515409
$617$	22728.1	1.48298	0.741490	−	0.670963i	$-0.234119\pi$
$617$	22728.1	1.48298	0.741490	+	0.670963i	$0.234119\pi$
$618$	−752.893	−0.0490062
$619$	−21443.3	−1.39237	−0.696187	−	0.717861i	$-0.745121\pi$
$619$	−21443.3	−1.39237	−0.696187	+	0.717861i	$0.745121\pi$
$620$	0	0
$621$	7818.75	0.505243
$622$	−9760.81	−0.629217
$623$	4574.25	0.294163
$624$	2524.82	0.161977
$625$	0	0
$626$	19628.0	1.25319
$627$	−12202.7	−0.777240
$628$	1326.85	0.0843109
$629$	−541.718	−0.0343398
$630$	0	0
$631$	21532.0	1.35844	0.679219	−	0.733936i	$-0.262319\pi$
$631$	21532.0	1.35844	0.679219	+	0.733936i	$0.262319\pi$
$632$	30182.0	1.89964
$633$	−851.038	−0.0534372
$634$	−42.9141	−0.00268823
$635$	0	0
$636$	−1457.29	−0.0908572
$637$	1787.56	0.111187
$638$	−751.159	−0.0466123
$639$	−7435.33	−0.460309
$640$	0	0
$641$	20148.3	1.24151	0.620756	−	0.784004i	$-0.286826\pi$
$641$	20148.3	1.24151	0.620756	+	0.784004i	$0.286826\pi$
$642$	18025.2	1.10810
$643$	−28869.7	−1.77062	−0.885310	−	0.465000i	$-0.846054\pi$
$643$	−28869.7	−1.77062	−0.885310	+	0.465000i	$0.846054\pi$
$644$	183.307	0.0112163
$645$	0	0
$646$	−15755.7	−0.959597
$647$	1590.02	0.0966155	0.0483077	−	0.998833i	$-0.484617\pi$
$647$	1590.02	0.0966155	0.0483077	+	0.998833i	$0.484617\pi$
$648$	−9595.02	−0.581679
$649$	−972.062	−0.0587932
$650$	0	0
$651$	−767.083	−0.0461818
$652$	−1460.38	−0.0877192
$653$	−20028.1	−1.20024	−0.600122	−	0.799909i	$-0.704881\pi$
$653$	−20028.1	−1.20024	−0.600122	+	0.799909i	$0.704881\pi$
$654$	−22618.9	−1.35240
$655$	0	0
$656$	−15514.4	−0.923375
$657$	36236.5	2.15178
$658$	2885.66	0.170965
$659$	−10520.7	−0.621897	−0.310948	−	0.950427i	$-0.600647\pi$
$659$	−10520.7	−0.621897	−0.310948	+	0.950427i	$0.600647\pi$
$660$	0	0
$661$	3295.83	0.193938	0.0969690	−	0.995287i	$-0.469085\pi$
$661$	3295.83	0.193938	0.0969690	+	0.995287i	$0.469085\pi$
$662$	3603.45	0.211559
$663$	−1751.12	−0.102576
$664$	−10304.1	−0.602222
$665$	0	0
$666$	1287.57	0.0749132
$667$	−2783.27	−0.161572
$668$	1466.91	0.0849649
$669$	31187.0	1.80233
$670$	0	0
$671$	−8125.67	−0.467493
$672$	589.612	0.0338464
$673$	1187.64	0.0680239	0.0340119	−	0.999421i	$-0.489172\pi$
$673$	1187.64	0.0680239	0.0340119	+	0.999421i	$0.489172\pi$
$674$	−653.460	−0.0373447
$675$	0	0
$676$	1161.98	0.0661117
$677$	−13221.4	−0.750574	−0.375287	−	0.926909i	$-0.622456\pi$
$677$	−13221.4	−0.750574	−0.375287	+	0.926909i	$0.622456\pi$
$678$	6391.55	0.362044
$679$	4135.91	0.233758
$680$	0	0
$681$	14048.0	0.790485
$682$	946.578	0.0531471
$683$	13831.4	0.774882	0.387441	−	0.921894i	$-0.373359\pi$
$683$	13831.4	0.774882	0.387441	+	0.921894i	$0.373359\pi$
$684$	−2688.68	−0.150298
$685$	0	0
$686$	−5677.93	−0.316012
$687$	15185.4	0.843320
$688$	−3429.62	−0.190048
$689$	−1838.10	−0.101635
$690$	0	0
$691$	−9817.07	−0.540462	−0.270231	−	0.962796i	$-0.587100\pi$
$691$	−9817.07	−0.540462	−0.270231	+	0.962796i	$0.587100\pi$
$692$	1236.54	0.0679280
$693$	1211.58	0.0664128
$694$	−16017.4	−0.876101
$695$	0	0
$696$	−4621.29	−0.251680
$697$	10760.2	0.584750
$698$	−9539.58	−0.517304
$699$	−34854.9	−1.88603
$700$	0	0
$701$	29949.8	1.61368	0.806838	−	0.590773i	$-0.201177\pi$
$701$	29949.8	1.61368	0.806838	+	0.590773i	$0.201177\pi$
$702$	1028.02	0.0552711
$703$	−1839.09	−0.0986667
$704$	−5957.03	−0.318912
$705$	0	0
$706$	−29825.0	−1.58991
$707$	495.976	0.0263835
$708$	−375.456	−0.0199301
$709$	11307.5	0.598959	0.299479	−	0.954103i	$-0.403187\pi$
$709$	11307.5	0.598959	0.299479	+	0.954103i	$0.403187\pi$
$710$	0	0
$711$	46406.3	2.44778
$712$	−34726.9	−1.82787
$713$	3507.36	0.184224
$714$	2742.30	0.143737
$715$	0	0
$716$	703.181	0.0367027
$717$	−32382.7	−1.68669
$718$	31420.6	1.63316
$719$	−32623.4	−1.69214	−0.846070	−	0.533071i	$-0.821038\pi$
$719$	−32623.4	−1.69214	−0.846070	+	0.533071i	$0.821038\pi$
$720$	0	0
$721$	−106.773	−0.00551518
$722$	−34750.2	−1.79123
$723$	30988.0	1.59399
$724$	430.420	0.0220945
$725$	0	0
$726$	−2620.89	−0.133981
$727$	502.545	0.0256373	0.0128187	−	0.999918i	$-0.495920\pi$
$727$	502.545	0.0256373	0.0128187	+	0.999918i	$0.495920\pi$
$728$	383.895	0.0195441
$729$	−29783.2	−1.51314
$730$	0	0
$731$	2378.66	0.120353
$732$	−3138.51	−0.158474
$733$	−8631.37	−0.434935	−0.217467	−	0.976068i	$-0.569780\pi$
$733$	−8631.37	−0.434935	−0.217467	+	0.976068i	$0.569780\pi$
$734$	18487.8	0.929697
$735$	0	0
$736$	−2695.90	−0.135017
$737$	3765.95	0.188223
$738$	−25575.0	−1.27565
$739$	−18357.5	−0.913792	−0.456896	−	0.889520i	$-0.651039\pi$
$739$	−18357.5	−0.913792	−0.456896	+	0.889520i	$0.651039\pi$
$740$	0	0
$741$	−5944.93	−0.294726
$742$	2878.52	0.142417
$743$	−11182.6	−0.552155	−0.276078	−	0.961135i	$-0.589035\pi$
$743$	−11182.6	−0.552155	−0.276078	+	0.961135i	$0.589035\pi$
$744$	5823.55	0.286965
$745$	0	0
$746$	−14507.8	−0.712021
$747$	−15843.0	−0.775991
$748$	242.960	0.0118763
$749$	2556.29	0.124706
$750$	0	0
$751$	16733.4	0.813063	0.406531	−	0.913637i	$-0.366738\pi$
$751$	16733.4	0.813063	0.406531	+	0.913637i	$0.366738\pi$
$752$	−20433.3	−0.990857
$753$	8680.53	0.420101
$754$	−365.950	−0.0176752
$755$	0	0
$756$	115.587	0.00556064
$757$	24402.4	1.17163	0.585813	−	0.810446i	$-0.300775\pi$
$757$	24402.4	1.17163	0.585813	+	0.810446i	$0.300775\pi$
$758$	2290.19	0.109741
$759$	−9711.19	−0.464419
$760$	0	0
$761$	8469.33	0.403434	0.201717	−	0.979444i	$-0.435348\pi$
$761$	8469.33	0.403434	0.201717	+	0.979444i	$0.435348\pi$
$762$	−28539.7	−1.35680
$763$	−3207.74	−0.152199
$764$	−920.805	−0.0436042
$765$	0	0
$766$	−7737.62	−0.364976
$767$	−473.570	−0.0222941
$768$	−6496.08	−0.305218
$769$	32834.7	1.53973	0.769864	−	0.638208i	$-0.220324\pi$
$769$	32834.7	1.53973	0.769864	+	0.638208i	$0.220324\pi$
$770$	0	0
$771$	−6209.20	−0.290038
$772$	718.202	0.0334827
$773$	35571.4	1.65513	0.827564	−	0.561371i	$-0.189726\pi$
$773$	35571.4	1.65513	0.827564	+	0.561371i	$0.189726\pi$
$774$	−5653.64	−0.262553
$775$	0	0
$776$	−31399.1	−1.45253
$777$	320.097	0.0147792
$778$	−8500.11	−0.391701
$779$	36530.0	1.68013
$780$	0	0
$781$	2281.01	0.104508
$782$	−12538.7	−0.573381
$783$	−1755.02	−0.0801014
$784$	19822.3	0.902982
$785$	0	0
$786$	34671.8	1.57341
$787$	−15729.6	−0.712452	−0.356226	−	0.934400i	$-0.615937\pi$
$787$	−15729.6	−0.712452	−0.356226	+	0.934400i	$0.615937\pi$
$788$	−1885.47	−0.0852373
$789$	−48996.7	−2.21081
$790$	0	0
$791$	906.431	0.0407446
$792$	−9198.09	−0.412676
$793$	−3958.67	−0.177272
$794$	38819.0	1.73506
$795$	0	0
$796$	−441.415	−0.0196552
$797$	−7888.07	−0.350577	−0.175288	−	0.984517i	$-0.556086\pi$
$797$	−7888.07	−0.350577	−0.175288	+	0.984517i	$0.556086\pi$
$798$	9309.91	0.412991
$799$	14171.8	0.627485
$800$	0	0
$801$	−53394.2	−2.35530
$802$	17107.2	0.753214
$803$	−11116.6	−0.488538
$804$	1454.58	0.0638050
$805$	0	0
$806$	461.154	0.0201532
$807$	7824.86	0.341323
$808$	−3765.36	−0.163942
$809$	5896.97	0.256275	0.128138	−	0.991756i	$-0.459100\pi$
$809$	5896.97	0.256275	0.128138	+	0.991756i	$0.459100\pi$
$810$	0	0
$811$	14197.9	0.614744	0.307372	−	0.951589i	$-0.400550\pi$
$811$	14197.9	0.614744	0.307372	+	0.951589i	$0.400550\pi$
$812$	−41.1458	−0.00177824
$813$	36287.3	1.56538
$814$	−394.999	−0.0170082
$815$	0	0
$816$	−19418.2	−0.833053
$817$	8075.35	0.345803
$818$	11454.1	0.489589
$819$	590.258	0.0251835
$820$	0	0
$821$	−19841.7	−0.843459	−0.421729	−	0.906722i	$-0.638577\pi$
$821$	−19841.7	−0.843459	−0.421729	+	0.906722i	$0.638577\pi$
$822$	34909.3	1.48127
$823$	28202.2	1.19449	0.597246	−	0.802058i	$-0.296262\pi$
$823$	28202.2	1.19449	0.597246	+	0.802058i	$0.296262\pi$
$824$	810.602	0.0342702
$825$	0	0
$826$	741.622	0.0312401
$827$	−34031.0	−1.43092	−0.715462	−	0.698651i	$-0.753784\pi$
$827$	−34031.0	−1.43092	−0.715462	+	0.698651i	$0.753784\pi$
$828$	−2139.71	−0.0898067
$829$	4931.55	0.206610	0.103305	−	0.994650i	$-0.467058\pi$
$829$	4931.55	0.206610	0.103305	+	0.994650i	$0.467058\pi$
$830$	0	0
$831$	−4501.92	−0.187930
$832$	−2902.15	−0.120930
$833$	−13748.0	−0.571836
$834$	690.138	0.0286541
$835$	0	0
$836$	824.830	0.0341236
$837$	2211.60	0.0913312
$838$	25373.0	1.04594
$839$	−38189.8	−1.57146	−0.785731	−	0.618568i	$-0.787713\pi$
$839$	−38189.8	−1.57146	−0.785731	+	0.618568i	$0.787713\pi$
$840$	0	0
$841$	−23764.3	−0.974384
$842$	−35915.6	−1.46999
$843$	42106.8	1.72033
$844$	57.5250	0.00234608
$845$	0	0
$846$	−33683.7	−1.36888
$847$	−371.687	−0.0150783
$848$	−20382.7	−0.825406
$849$	37488.0	1.51541
$850$	0	0
$851$	−1463.59	−0.0589556
$852$	881.030	0.0354268
$853$	−42966.8	−1.72469	−0.862343	−	0.506325i	$-0.831003\pi$
$853$	−42966.8	−1.72469	−0.862343	+	0.506325i	$0.831003\pi$
$854$	6199.37	0.248405
$855$	0	0
$856$	−19406.8	−0.774898
$857$	17281.5	0.688828	0.344414	−	0.938818i	$-0.388078\pi$
$857$	17281.5	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$858$	−1276.85	−0.0508052
$859$	9316.75	0.370062	0.185031	−	0.982733i	$-0.440761\pi$
$859$	9316.75	0.370062	0.185031	+	0.982733i	$0.440761\pi$
$860$	0	0
$861$	−6358.10	−0.251665
$862$	−13413.5	−0.530005
$863$	9647.65	0.380544	0.190272	−	0.981731i	$-0.439063\pi$
$863$	9647.65	0.380544	0.190272	+	0.981731i	$0.439063\pi$
$864$	−1699.93	−0.0669361
$865$	0	0
$866$	−32083.3	−1.25893
$867$	−25483.6	−0.998232
$868$	51.8501	0.00202754
$869$	−14236.5	−0.555742
$870$	0	0
$871$	1834.70	0.0713735
$872$	24352.6	0.945737
$873$	−48277.6	−1.87165
$874$	−42568.0	−1.64746
$875$	0	0
$876$	−4293.75	−0.165608
$877$	−19728.7	−0.759624	−0.379812	−	0.925064i	$-0.624011\pi$
$877$	−19728.7	−0.759624	−0.379812	+	0.925064i	$0.624011\pi$
$878$	32304.3	1.24171
$879$	−18464.1	−0.708509
$880$	0	0
$881$	19473.9	0.744712	0.372356	−	0.928090i	$-0.378550\pi$
$881$	19473.9	0.744712	0.372356	+	0.928090i	$0.378550\pi$
$882$	32676.4	1.24748
$883$	−49092.4	−1.87100	−0.935499	−	0.353329i	$-0.885050\pi$
$883$	−49092.4	−1.87100	−0.935499	+	0.353329i	$0.885050\pi$
$884$	118.365	0.00450346
$885$	0	0
$886$	27599.5	1.04653
$887$	−9292.86	−0.351774	−0.175887	−	0.984410i	$-0.556279\pi$
$887$	−9292.86	−0.351774	−0.175887	+	0.984410i	$0.556279\pi$
$888$	−2430.12	−0.0918348
$889$	−4047.41	−0.152695
$890$	0	0
$891$	4525.85	0.170170
$892$	−2108.05	−0.0791288
$893$	48112.0	1.80292
$894$	52598.5	1.96774
$895$	0	0
$896$	3949.89	0.147273
$897$	−4731.10	−0.176106
$898$	943.252	0.0350520
$899$	−787.273	−0.0292069
$900$	0	0
$901$	14136.7	0.522709
$902$	7845.88	0.289622
$903$	−1405.53	−0.0517974
$904$	−6881.46	−0.253179
$905$	0	0
$906$	55811.5	2.04659
$907$	−37688.7	−1.37975	−0.689875	−	0.723928i	$-0.742335\pi$
$907$	−37688.7	−1.37975	−0.689875	+	0.723928i	$0.742335\pi$
$908$	−949.559	−0.0347051
$909$	−5789.42	−0.211246
$910$	0	0
$911$	33049.6	1.20196	0.600979	−	0.799265i	$-0.294778\pi$
$911$	33049.6	1.20196	0.600979	+	0.799265i	$0.294778\pi$
$912$	−65923.1	−2.39357
$913$	4860.31	0.176180
$914$	−28870.0	−1.04479
$915$	0	0
$916$	−1026.44	−0.0370247
$917$	4917.06	0.177073
$918$	−7906.43	−0.284260
$919$	−23148.0	−0.830883	−0.415442	−	0.909620i	$-0.636373\pi$
$919$	−23148.0	−0.830883	−0.415442	+	0.909620i	$0.636373\pi$
$920$	0	0
$921$	13304.1	0.475986
$922$	−12933.4	−0.461973
$923$	1111.26	0.0396290
$924$	−143.563	−0.00511134
$925$	0	0
$926$	9374.21	0.332673
$927$	1246.34	0.0441588
$928$	605.131	0.0214056
$929$	−23177.9	−0.818561	−0.409280	−	0.912409i	$-0.634220\pi$
$929$	−23177.9	−0.818561	−0.409280	+	0.912409i	$0.634220\pi$
$930$	0	0
$931$	−46673.3	−1.64302
$932$	2355.98	0.0828033
$933$	28325.1	0.993916
$934$	13979.8	0.489757
$935$	0	0
$936$	−4481.13	−0.156485
$937$	34574.7	1.20545	0.602724	−	0.797950i	$-0.294082\pi$
$937$	34574.7	1.20545	0.602724	+	0.797950i	$0.294082\pi$
$938$	−2873.18	−0.100014
$939$	−56959.1	−1.97954
$940$	0	0
$941$	41831.2	1.44916	0.724578	−	0.689192i	$-0.242034\pi$
$941$	41831.2	1.44916	0.724578	+	0.689192i	$0.242034\pi$
$942$	53629.6	1.85493
$943$	29071.3	1.00392
$944$	−5251.40	−0.181058
$945$	0	0
$946$	1734.42	0.0596097
$947$	−27231.2	−0.934419	−0.467209	−	0.884147i	$-0.654741\pi$
$947$	−27231.2	−0.934419	−0.467209	+	0.884147i	$0.654741\pi$
$948$	−5498.79	−0.188389
$949$	−5415.79	−0.185252
$950$	0	0
$951$	124.534	0.00424635
$952$	−2952.50	−0.100516
$953$	−40939.4	−1.39156	−0.695781	−	0.718254i	$-0.744942\pi$
$953$	−40939.4	−1.39156	−0.695781	+	0.718254i	$0.744942\pi$
$954$	−33600.3	−1.14030
$955$	0	0
$956$	2188.87	0.0740515
$957$	2179.81	0.0736292
$958$	−31601.3	−1.06575
$959$	4950.74	0.166703
$960$	0	0
$961$	−28798.9	−0.966698
$962$	−192.436	−0.00644945
$963$	−29839.0	−0.998491
$964$	−2094.60	−0.0699820
$965$	0	0
$966$	7409.03	0.246772
$967$	46173.1	1.53550	0.767750	−	0.640750i	$-0.221376\pi$
$967$	46173.1	1.53550	0.767750	+	0.640750i	$0.221376\pi$
$968$	2821.78	0.0936937
$969$	45721.8	1.51579
$970$	0	0
$971$	−5153.91	−0.170337	−0.0851683	−	0.996367i	$-0.527143\pi$
$971$	−5153.91	−0.170337	−0.0851683	+	0.996367i	$0.527143\pi$
$972$	2764.06	0.0912111
$973$	97.8734	0.00322474
$974$	−50069.0	−1.64714
$975$	0	0
$976$	−43897.6	−1.43968
$977$	−9692.13	−0.317378	−0.158689	−	0.987329i	$-0.550727\pi$
$977$	−9692.13	−0.317378	−0.158689	+	0.987329i	$0.550727\pi$
$978$	−59026.5	−1.92992
$979$	16380.2	0.534744
$980$	0	0
$981$	37443.3	1.21863
$982$	20811.6	0.676299
$983$	−32915.7	−1.06800	−0.534002	−	0.845483i	$-0.679313\pi$
$983$	−32915.7	−1.06800	−0.534002	+	0.845483i	$0.679313\pi$
$984$	48269.5	1.56380
$985$	0	0
$986$	2814.48	0.0909041
$987$	−8373.97	−0.270057
$988$	401.841	0.0129395
$989$	6426.54	0.206625
$990$	0	0
$991$	29477.9	0.944901	0.472451	−	0.881357i	$-0.343370\pi$
$991$	29477.9	0.944901	0.472451	+	0.881357i	$0.343370\pi$
$992$	−762.560	−0.0244066
$993$	−10456.9	−0.334180
$994$	−1740.26	−0.0555310
$995$	0	0
$996$	1877.28	0.0597227
$997$	31944.4	1.01473	0.507366	−	0.861731i	$-0.330619\pi$
$997$	31944.4	1.01473	0.507366	+	0.861731i	$0.330619\pi$
$998$	−35268.4	−1.11864
$999$	−922.883	−0.0292279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.a.b.1.1		2
3.2	odd	2		2475.4.a.q.1.2		2
5.2	odd	4		275.4.b.c.199.1		4
5.3	odd	4		275.4.b.c.199.4		4
5.4	even	2		11.4.a.a.1.2	✓	2
15.14	odd	2		99.4.a.c.1.1		2
20.19	odd	2		176.4.a.i.1.2		2
35.34	odd	2		539.4.a.e.1.2		2
40.19	odd	2		704.4.a.n.1.1		2
40.29	even	2		704.4.a.p.1.2		2
55.4	even	10		121.4.c.c.27.2		8
55.9	even	10		121.4.c.c.81.1		8
55.14	even	10		121.4.c.c.9.2		8
55.19	odd	10		121.4.c.f.9.1		8
55.24	odd	10		121.4.c.f.81.2		8
55.29	odd	10		121.4.c.f.27.1		8
55.39	odd	10		121.4.c.f.3.2		8
55.49	even	10		121.4.c.c.3.1		8
55.54	odd	2		121.4.a.c.1.1		2
60.59	even	2		1584.4.a.bc.1.1		2
65.64	even	2		1859.4.a.a.1.1		2
165.164	even	2		1089.4.a.v.1.2		2
220.219	even	2		1936.4.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.2	✓	2	5.4	even	2
99.4.a.c.1.1		2	15.14	odd	2
121.4.a.c.1.1		2	55.54	odd	2
121.4.c.c.3.1		8	55.49	even	10
121.4.c.c.9.2		8	55.14	even	10
121.4.c.c.27.2		8	55.4	even	10
121.4.c.c.81.1		8	55.9	even	10
121.4.c.f.3.2		8	55.39	odd	10
121.4.c.f.9.1		8	55.19	odd	10
121.4.c.f.27.1		8	55.29	odd	10
121.4.c.f.81.2		8	55.24	odd	10
176.4.a.i.1.2		2	20.19	odd	2
275.4.a.b.1.1		2	1.1	even	1	trivial
275.4.b.c.199.1		4	5.2	odd	4
275.4.b.c.199.4		4	5.3	odd	4
539.4.a.e.1.2		2	35.34	odd	2
704.4.a.n.1.1		2	40.19	odd	2
704.4.a.p.1.2		2	40.29	even	2
1089.4.a.v.1.2		2	165.164	even	2
1584.4.a.bc.1.1		2	60.59	even	2
1859.4.a.a.1.1		2	65.64	even	2
1936.4.a.w.1.2		2	220.219	even	2
2475.4.a.q.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +O(q^{10})\)	\(q-2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} -11.0000 q^{11} -4.24871 q^{12} -5.35898 q^{13} +8.39230 q^{14} -59.4256 q^{16} +41.2154 q^{17} -97.9615 q^{18} +139.923 q^{19} -24.3538 q^{21} +30.0526 q^{22} +111.354 q^{23} +184.890 q^{24} +14.6410 q^{26} +70.2154 q^{27} +1.64617 q^{28} -24.9948 q^{29} +31.4974 q^{31} -24.2102 q^{32} -87.2102 q^{33} -112.603 q^{34} -19.2154 q^{36} -13.1436 q^{37} -382.277 q^{38} -42.4871 q^{39} +261.072 q^{41} +66.5359 q^{42} +57.7128 q^{43} +5.89488 q^{44} -304.224 q^{46} +343.846 q^{47} -471.138 q^{48} -333.564 q^{49} +326.764 q^{51} +2.87187 q^{52} +342.995 q^{53} -191.832 q^{54} -71.6359 q^{56} +1109.34 q^{57} +68.2872 q^{58} +88.3693 q^{59} +738.697 q^{61} -86.0526 q^{62} -110.144 q^{63} +541.549 q^{64} +238.263 q^{66} -342.359 q^{67} -22.0873 q^{68} +882.836 q^{69} -207.364 q^{71} +836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} -74.9845 q^{76} +33.7898 q^{77} +116.077 q^{78} +1294.23 q^{79} -411.441 q^{81} -713.261 q^{82} -441.846 q^{83} +13.0512 q^{84} -157.674 q^{86} -198.164 q^{87} -256.526 q^{88} -1489.11 q^{89} +16.4617 q^{91} -59.6743 q^{92} +249.718 q^{93} -939.405 q^{94} -191.944 q^{96} -1346.42 q^{97} +911.314 q^{98} -394.420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 4 q^{14} - 8 q^{16} + 124 q^{17} - 92 q^{18} + 72 q^{19} + 76 q^{21} + 22 q^{22} + 98 q^{23} + 252 q^{24} - 40 q^{26} + 182 q^{27} + 128 q^{28} + 144 q^{29} - 34 q^{31} + 104 q^{32} - 22 q^{33} - 52 q^{34} - 80 q^{36} - 54 q^{37} - 432 q^{38} + 400 q^{39} + 536 q^{41} + 140 q^{42} + 60 q^{43} + 88 q^{44} - 314 q^{46} + 272 q^{47} - 776 q^{48} - 390 q^{49} - 164 q^{51} + 560 q^{52} + 492 q^{53} - 110 q^{54} + 120 q^{56} + 1512 q^{57} + 192 q^{58} + 634 q^{59} + 840 q^{61} - 134 q^{62} - 248 q^{63} + 224 q^{64} + 286 q^{66} - 754 q^{67} - 640 q^{68} + 962 q^{69} - 678 q^{71} + 744 q^{72} + 400 q^{73} + 6 q^{74} + 432 q^{76} + 220 q^{77} + 440 q^{78} + 316 q^{79} - 1294 q^{81} - 512 q^{82} - 468 q^{83} - 736 q^{84} - 156 q^{86} - 1200 q^{87} - 132 q^{88} - 1842 q^{89} + 1280 q^{91} + 40 q^{92} + 638 q^{93} - 992 q^{94} - 952 q^{96} - 2194 q^{97} + 870 q^{98} - 484 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 - 22 * q^11 + 40 * q^12 - 80 * q^13 - 4 * q^14 - 8 * q^16 + 124 * q^17 - 92 * q^18 + 72 * q^19 + 76 * q^21 + 22 * q^22 + 98 * q^23 + 252 * q^24 - 40 * q^26 + 182 * q^27 + 128 * q^28 + 144 * q^29 - 34 * q^31 + 104 * q^32 - 22 * q^33 - 52 * q^34 - 80 * q^36 - 54 * q^37 - 432 * q^38 + 400 * q^39 + 536 * q^41 + 140 * q^42 + 60 * q^43 + 88 * q^44 - 314 * q^46 + 272 * q^47 - 776 * q^48 - 390 * q^49 - 164 * q^51 + 560 * q^52 + 492 * q^53 - 110 * q^54 + 120 * q^56 + 1512 * q^57 + 192 * q^58 + 634 * q^59 + 840 * q^61 - 134 * q^62 - 248 * q^63 + 224 * q^64 + 286 * q^66 - 754 * q^67 - 640 * q^68 + 962 * q^69 - 678 * q^71 + 744 * q^72 + 400 * q^73 + 6 * q^74 + 432 * q^76 + 220 * q^77 + 440 * q^78 + 316 * q^79 - 1294 * q^81 - 512 * q^82 - 468 * q^83 - 736 * q^84 - 156 * q^86 - 1200 * q^87 - 132 * q^88 - 1842 * q^89 + 1280 * q^91 + 40 * q^92 + 638 * q^93 - 992 * q^94 - 952 * q^96 - 2194 * q^97 + 870 * q^98 - 484 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more