LMFDB - Embedded newform 275.6.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,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, 6, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
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:	$44.1055504486$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 129x^{3} + 45x^{2} + 2924x - 5216 $ x^5 - x^4 - 129x^3 + 45x^2 + 2924*x - 5216
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.13497$ 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)

$f(q)$	$=$	$q+0.134973 q^{2} +22.6392 q^{3} -31.9818 q^{4} +3.05567 q^{6} +118.127 q^{7} -8.63578 q^{8} +269.534 q^{9} +O(q^{10})$	\(q+0.134973 q^{2} +22.6392 q^{3} -31.9818 q^{4} +3.05567 q^{6} +118.127 q^{7} -8.63578 q^{8} +269.534 q^{9} -121.000 q^{11} -724.042 q^{12} -1115.26 q^{13} +15.9439 q^{14} +1022.25 q^{16} -1761.28 q^{17} +36.3797 q^{18} -2122.55 q^{19} +2674.31 q^{21} -16.3317 q^{22} +3653.78 q^{23} -195.507 q^{24} -150.529 q^{26} +600.711 q^{27} -3777.92 q^{28} -2916.19 q^{29} +9021.17 q^{31} +414.321 q^{32} -2739.35 q^{33} -237.724 q^{34} -8620.18 q^{36} +800.006 q^{37} -286.485 q^{38} -25248.6 q^{39} -6490.42 q^{41} +360.958 q^{42} -12652.4 q^{43} +3869.80 q^{44} +493.160 q^{46} -26957.8 q^{47} +23143.0 q^{48} -2852.93 q^{49} -39874.0 q^{51} +35668.0 q^{52} -9998.59 q^{53} +81.0795 q^{54} -1020.12 q^{56} -48052.8 q^{57} -393.605 q^{58} +21134.3 q^{59} +11087.2 q^{61} +1217.61 q^{62} +31839.3 q^{63} -32656.1 q^{64} -369.736 q^{66} -39791.8 q^{67} +56328.8 q^{68} +82718.7 q^{69} +38238.6 q^{71} -2327.64 q^{72} -15983.0 q^{73} +107.979 q^{74} +67882.8 q^{76} -14293.4 q^{77} -3407.87 q^{78} -44791.2 q^{79} -51897.2 q^{81} -876.028 q^{82} -54122.1 q^{83} -85529.2 q^{84} -1707.73 q^{86} -66020.2 q^{87} +1044.93 q^{88} -23144.7 q^{89} -131743. q^{91} -116854. q^{92} +204232. q^{93} -3638.57 q^{94} +9379.90 q^{96} -10056.4 q^{97} -385.067 q^{98} -32613.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9}+O(q^{10}) $ 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9	\( 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9} - 605 q^{11} + 1605 q^{12} - 1498 q^{13} + 2113 q^{14} + 4883 q^{16} - 3874 q^{17} - 5838 q^{18} + 882 q^{19} + 1092 q^{21} + 1089 q^{22} + 5344 q^{23} - 13119 q^{24} - 4478 q^{26} + 2160 q^{27} - 12565 q^{28} + 5318 q^{29} - 7916 q^{31} - 21385 q^{32} - 18605 q^{34} + 5628 q^{36} + 1788 q^{37} + 34421 q^{38} - 29760 q^{39} + 5854 q^{41} + 46725 q^{42} + 4364 q^{43} - 13915 q^{44} - 33834 q^{46} - 46452 q^{47} + 127545 q^{48} - 34217 q^{49} + 19842 q^{51} - 3222 q^{52} - 4412 q^{53} - 86535 q^{54} + 115575 q^{56} - 137160 q^{57} + 58221 q^{58} + 17896 q^{59} - 35930 q^{61} + 19627 q^{62} - 100980 q^{63} + 14779 q^{64} + 28677 q^{66} - 73136 q^{67} + 83409 q^{68} + 34296 q^{69} + 43612 q^{71} - 372276 q^{72} - 142306 q^{73} - 95609 q^{74} - 6617 q^{76} + 8470 q^{77} - 15750 q^{78} - 46504 q^{79} + 79101 q^{81} - 175798 q^{82} - 81604 q^{83} - 532533 q^{84} - 101788 q^{86} - 219750 q^{87} + 91113 q^{88} + 8664 q^{89} - 203380 q^{91} - 251174 q^{92} + 46470 q^{93} - 71458 q^{94} - 925479 q^{96} + 22230 q^{97} - 59962 q^{98} - 128139 q^{99}+O(q^{100}) \) 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9 - 605 * q^11 + 1605 * q^12 - 1498 * q^13 + 2113 * q^14 + 4883 * q^16 - 3874 * q^17 - 5838 * q^18 + 882 * q^19 + 1092 * q^21 + 1089 * q^22 + 5344 * q^23 - 13119 * q^24 - 4478 * q^26 + 2160 * q^27 - 12565 * q^28 + 5318 * q^29 - 7916 * q^31 - 21385 * q^32 - 18605 * q^34 + 5628 * q^36 + 1788 * q^37 + 34421 * q^38 - 29760 * q^39 + 5854 * q^41 + 46725 * q^42 + 4364 * q^43 - 13915 * q^44 - 33834 * q^46 - 46452 * q^47 + 127545 * q^48 - 34217 * q^49 + 19842 * q^51 - 3222 * q^52 - 4412 * q^53 - 86535 * q^54 + 115575 * q^56 - 137160 * q^57 + 58221 * q^58 + 17896 * q^59 - 35930 * q^61 + 19627 * q^62 - 100980 * q^63 + 14779 * q^64 + 28677 * q^66 - 73136 * q^67 + 83409 * q^68 + 34296 * q^69 + 43612 * q^71 - 372276 * q^72 - 142306 * q^73 - 95609 * q^74 - 6617 * q^76 + 8470 * q^77 - 15750 * q^78 - 46504 * q^79 + 79101 * q^81 - 175798 * q^82 - 81604 * q^83 - 532533 * q^84 - 101788 * q^86 - 219750 * q^87 + 91113 * q^88 + 8664 * q^89 - 203380 * q^91 - 251174 * q^92 + 46470 * q^93 - 71458 * q^94 - 925479 * q^96 + 22230 * q^97 - 59962 * q^98 - 128139 * 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 $

$2$	0.134973	0.0238600	0.0119300	−	0.999929i	$-0.496202\pi$
$2$	0.134973	0.0238600	0.0119300	+	0.999929i	$0.496202\pi$
$3$	22.6392	1.45231	0.726153	−	0.687533i	$-0.241306\pi$
$3$	22.6392	1.45231	0.726153	+	0.687533i	$0.241306\pi$
$4$	−31.9818	−0.999431
$5$	0	0
$5$	0	0
$6$	3.05567	0.0346520
$7$	118.127	0.911183	0.455591	−	0.890189i	$-0.349428\pi$
$7$	118.127	0.911183	0.455591	+	0.890189i	$0.349428\pi$
$8$	−8.63578	−0.0477064
$9$	269.534	1.10919
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−724.042	−1.45148
$13$	−1115.26	−1.83028	−0.915140	−	0.403136i	$-0.867920\pi$
$13$	−1115.26	−1.83028	−0.915140	+	0.403136i	$0.867920\pi$
$14$	15.9439	0.0217408
$15$	0	0
$16$	1022.25	0.998292
$17$	−1761.28	−1.47811	−0.739053	−	0.673647i	$-0.764727\pi$
$17$	−1761.28	−1.47811	−0.739053	+	0.673647i	$0.764727\pi$
$18$	36.3797	0.0264654
$19$	−2122.55	−1.34888	−0.674440	−	0.738330i	$-0.735615\pi$
$19$	−2122.55	−1.34888	−0.674440	+	0.738330i	$0.735615\pi$
$20$	0	0
$21$	2674.31	1.32332
$22$	−16.3317	−0.00719406
$23$	3653.78	1.44020	0.720100	−	0.693871i	$-0.244096\pi$
$23$	3653.78	1.44020	0.720100	+	0.693871i	$0.244096\pi$
$24$	−195.507	−0.0692843
$25$	0	0
$26$	−150.529	−0.0436705
$27$	600.711	0.158583
$28$	−3777.92	−0.910664
$29$	−2916.19	−0.643903	−0.321952	−	0.946756i	$-0.604339\pi$
$29$	−2916.19	−0.643903	−0.321952	+	0.946756i	$0.604339\pi$
$30$	0	0
$31$	9021.17	1.68600	0.843002	−	0.537911i	$-0.180786\pi$
$31$	9021.17	1.68600	0.843002	+	0.537911i	$0.180786\pi$
$32$	414.321	0.0715257
$33$	−2739.35	−0.437887
$34$	−237.724	−0.0352676
$35$	0	0
$36$	−8620.18	−1.10856
$37$	800.006	0.0960703	0.0480351	−	0.998846i	$-0.484704\pi$
$37$	800.006	0.0960703	0.0480351	+	0.998846i	$0.484704\pi$
$38$	−286.485	−0.0321843
$39$	−25248.6	−2.65813
$40$	0	0
$41$	−6490.42	−0.602994	−0.301497	−	0.953467i	$-0.597486\pi$
$41$	−6490.42	−0.602994	−0.301497	+	0.953467i	$0.597486\pi$
$42$	360.958	0.0315743
$43$	−12652.4	−1.04352	−0.521761	−	0.853092i	$-0.674725\pi$
$43$	−12652.4	−1.04352	−0.521761	+	0.853092i	$0.674725\pi$
$44$	3869.80	0.301340
$45$	0	0
$46$	493.160	0.0343632
$47$	−26957.8	−1.78008	−0.890042	−	0.455878i	$-0.849325\pi$
$47$	−26957.8	−1.78008	−0.890042	+	0.455878i	$0.849325\pi$
$48$	23143.0	1.44983
$49$	−2852.93	−0.169746
$50$	0	0
$51$	−39874.0	−2.14666
$52$	35668.0	1.82924
$53$	−9998.59	−0.488933	−0.244466	−	0.969658i	$-0.578613\pi$
$53$	−9998.59	−0.488933	−0.244466	+	0.969658i	$0.578613\pi$
$54$	81.0795	0.00378379
$55$	0	0
$56$	−1020.12	−0.0434692
$57$	−48052.8	−1.95899
$58$	−393.605	−0.0153635
$59$	21134.3	0.790421	0.395211	−	0.918591i	$-0.370672\pi$
$59$	21134.3	0.790421	0.395211	+	0.918591i	$0.370672\pi$
$60$	0	0
$61$	11087.2	0.381502	0.190751	−	0.981638i	$-0.438908\pi$
$61$	11087.2	0.381502	0.190751	+	0.981638i	$0.438908\pi$
$62$	1217.61	0.0402280
$63$	31839.3	1.01068
$64$	−32656.1	−0.996586
$65$	0	0
$66$	−369.736	−0.0104480
$67$	−39791.8	−1.08295	−0.541473	−	0.840718i	$-0.682133\pi$
$67$	−39791.8	−1.08295	−0.541473	+	0.840718i	$0.682133\pi$
$68$	56328.8	1.47727
$69$	82718.7	2.09161
$70$	0	0
$71$	38238.6	0.900235	0.450118	−	0.892969i	$-0.351382\pi$
$71$	38238.6	0.900235	0.450118	+	0.892969i	$0.351382\pi$
$72$	−2327.64	−0.0529157
$73$	−15983.0	−0.351036	−0.175518	−	0.984476i	$-0.556160\pi$
$73$	−15983.0	−0.351036	−0.175518	+	0.984476i	$0.556160\pi$
$74$	107.979	0.00229224
$75$	0	0
$76$	67882.8	1.34811
$77$	−14293.4	−0.274732
$78$	−3407.87	−0.0634229
$79$	−44791.2	−0.807467	−0.403734	−	0.914877i	$-0.632288\pi$
$79$	−44791.2	−0.807467	−0.403734	+	0.914877i	$0.632288\pi$
$80$	0	0
$81$	−51897.2	−0.878883
$82$	−876.028	−0.0143874
$83$	−54122.1	−0.862342	−0.431171	−	0.902270i	$-0.641900\pi$
$83$	−54122.1	−0.862342	−0.431171	+	0.902270i	$0.641900\pi$
$84$	−85529.2	−1.32256
$85$	0	0
$86$	−1707.73	−0.0248984
$87$	−66020.2	−0.935145
$88$	1044.93	0.0143840
$89$	−23144.7	−0.309725	−0.154863	−	0.987936i	$-0.549494\pi$
$89$	−23144.7	−0.309725	−0.154863	+	0.987936i	$0.549494\pi$
$90$	0	0
$91$	−131743.	−1.66772
$92$	−116854.	−1.43938
$93$	204232.	2.44859
$94$	−3638.57	−0.0424728
$95$	0	0
$96$	9379.90	0.103877
$97$	−10056.4	−0.108520	−0.0542602	−	0.998527i	$-0.517280\pi$
$97$	−10056.4	−0.108520	−0.0542602	+	0.998527i	$0.517280\pi$
$98$	−385.067	−0.00405015
$99$	−32613.6	−0.334435
$100$	0	0
$101$	88081.8	0.859177	0.429589	−	0.903025i	$-0.358659\pi$
$101$	88081.8	0.859177	0.429589	+	0.903025i	$0.358659\pi$
$102$	−5381.89	−0.0512194
$103$	112989.	1.04940	0.524702	−	0.851286i	$-0.324177\pi$
$103$	112989.	1.04940	0.524702	+	0.851286i	$0.324177\pi$
$104$	9631.14	0.0873161
$105$	0	0
$106$	−1349.53	−0.0116659
$107$	53202.7	0.449235	0.224618	−	0.974447i	$-0.427887\pi$
$107$	53202.7	0.449235	0.224618	+	0.974447i	$0.427887\pi$
$108$	−19211.8	−0.158493
$109$	−57029.0	−0.459758	−0.229879	−	0.973219i	$-0.573833\pi$
$109$	−57029.0	−0.459758	−0.229879	+	0.973219i	$0.573833\pi$
$110$	0	0
$111$	18111.5	0.139523
$112$	120756.	0.909627
$113$	−161138.	−1.18714	−0.593572	−	0.804781i	$-0.702283\pi$
$113$	−161138.	−1.18714	−0.593572	+	0.804781i	$0.702283\pi$
$114$	−6485.81	−0.0467414
$115$	0	0
$116$	93264.9	0.643537
$117$	−300601.	−2.03014
$118$	2852.55	0.0188594
$119$	−208055.	−1.34682
$120$	0	0
$121$	14641.0	0.0909091
$122$	1496.47	0.00910264
$123$	−146938.	−0.875732
$124$	−288513.	−1.68504
$125$	0	0
$126$	4297.44	0.0241148
$127$	−22343.5	−0.122925	−0.0614627	−	0.998109i	$-0.519577\pi$
$127$	−22343.5	−0.122925	−0.0614627	+	0.998109i	$0.519577\pi$
$128$	−17665.9	−0.0953042
$129$	−286440.	−1.51551
$130$	0	0
$131$	−122484.	−0.623595	−0.311798	−	0.950149i	$-0.600931\pi$
$131$	−122484.	−0.623595	−0.311798	+	0.950149i	$0.600931\pi$
$132$	87609.1	0.437638
$133$	−250731.	−1.22908
$134$	−5370.80	−0.0258391
$135$	0	0
$136$	15210.0	0.0705152
$137$	61551.4	0.280180	0.140090	−	0.990139i	$-0.455261\pi$
$137$	61551.4	0.280180	0.140090	+	0.990139i	$0.455261\pi$
$138$	11164.7	0.0499058
$139$	219541.	0.963782	0.481891	−	0.876231i	$-0.339950\pi$
$139$	219541.	0.963782	0.481891	+	0.876231i	$0.339950\pi$
$140$	0	0
$141$	−610304.	−2.58523
$142$	5161.16	0.0214796
$143$	134946.	0.551850
$144$	275532.	1.10730
$145$	0	0
$146$	−2157.27	−0.00837572
$147$	−64588.1	−0.246524
$148$	−25585.6	−0.0960156
$149$	393179.	1.45086	0.725428	−	0.688298i	$-0.241642\pi$
$149$	393179.	1.45086	0.725428	+	0.688298i	$0.241642\pi$
$150$	0	0
$151$	147257.	0.525572	0.262786	−	0.964854i	$-0.415359\pi$
$151$	147257.	0.525572	0.262786	+	0.964854i	$0.415359\pi$
$152$	18329.8	0.0643502
$153$	−474725.	−1.63951
$154$	−1929.22	−0.00655510
$155$	0	0
$156$	807495.	2.65661
$157$	337144.	1.09161	0.545803	−	0.837914i	$-0.316225\pi$
$157$	337144.	1.09161	0.545803	+	0.837914i	$0.316225\pi$
$158$	−6045.58	−0.0192662
$159$	−226360.	−0.710080
$160$	0	0
$161$	431611.	1.31228
$162$	−7004.69	−0.0209701
$163$	313729.	0.924882	0.462441	−	0.886650i	$-0.346974\pi$
$163$	313729.	0.924882	0.462441	+	0.886650i	$0.346974\pi$
$164$	207575.	0.602651
$165$	0	0
$166$	−7305.00	−0.0205755
$167$	−411548.	−1.14190	−0.570951	−	0.820984i	$-0.693425\pi$
$167$	−411548.	−1.14190	−0.570951	+	0.820984i	$0.693425\pi$
$168$	−23094.8	−0.0631307
$169$	872511.	2.34993
$170$	0	0
$171$	−572099.	−1.49617
$172$	404646.	1.04293
$173$	83623.5	0.212429	0.106214	−	0.994343i	$-0.466127\pi$
$173$	83623.5	0.212429	0.106214	+	0.994343i	$0.466127\pi$
$174$	−8910.91	−0.0223125
$175$	0	0
$176$	−123692.	−0.300996
$177$	478465.	1.14793
$178$	−3123.90	−0.00739005
$179$	−143482.	−0.334706	−0.167353	−	0.985897i	$-0.553522\pi$
$179$	−143482.	−0.334706	−0.167353	+	0.985897i	$0.553522\pi$
$180$	0	0
$181$	−159148.	−0.361082	−0.180541	−	0.983567i	$-0.557785\pi$
$181$	−159148.	−0.361082	−0.180541	+	0.983567i	$0.557785\pi$
$182$	−17781.6	−0.0397918
$183$	251005.	0.554058
$184$	−31553.2	−0.0687067
$185$	0	0
$186$	27565.7	0.0584234
$187$	213115.	0.445666
$188$	862160.	1.77907
$189$	70960.4	0.144498
$190$	0	0
$191$	96671.0	0.191740	0.0958700	−	0.995394i	$-0.469437\pi$
$191$	96671.0	0.191740	0.0958700	+	0.995394i	$0.469437\pi$
$192$	−739309.	−1.44735
$193$	715512.	1.38269	0.691343	−	0.722527i	$-0.257019\pi$
$193$	715512.	1.38269	0.691343	+	0.722527i	$0.257019\pi$
$194$	−1357.33	−0.00258930
$195$	0	0
$196$	91241.7	0.169650
$197$	149817.	0.275039	0.137520	−	0.990499i	$-0.456087\pi$
$197$	149817.	0.275039	0.137520	+	0.990499i	$0.456087\pi$
$198$	−4401.94	−0.00797961
$199$	374182.	0.669808	0.334904	−	0.942252i	$-0.391296\pi$
$199$	374182.	0.669808	0.334904	+	0.942252i	$0.391296\pi$
$200$	0	0
$201$	−900856.	−1.57277
$202$	11888.6	0.0205000
$203$	−344482.	−0.586713
$204$	1.27524e6	2.14544
$205$	0	0
$206$	15250.4	0.0250388
$207$	984818.	1.59746
$208$	−1.14008e6	−1.82716
$209$	256828.	0.406703
$210$	0	0
$211$	−352779.	−0.545502	−0.272751	−	0.962085i	$-0.587934\pi$
$211$	−352779.	−0.545502	−0.272751	+	0.962085i	$0.587934\pi$
$212$	319773.	0.488654
$213$	865692.	1.30742
$214$	7180.90	0.0107188
$215$	0	0
$216$	−5187.61	−0.00756542
$217$	1.06565e6	1.53626
$218$	−7697.34	−0.0109698
$219$	−361843.	−0.509812
$220$	0	0
$221$	1.96428e6	2.70535
$222$	2444.56	0.00332903
$223$	−1.40562e6	−1.89280	−0.946401	−	0.322995i	$-0.895310\pi$
$223$	−1.40562e6	−1.89280	−0.946401	+	0.322995i	$0.895310\pi$
$224$	48942.6	0.0651729
$225$	0	0
$226$	−21749.3	−0.0283252
$227$	243905.	0.314163	0.157082	−	0.987586i	$-0.449791\pi$
$227$	243905.	0.314163	0.157082	+	0.987586i	$0.449791\pi$
$228$	1.53681e6	1.95787
$229$	−783940.	−0.987857	−0.493928	−	0.869503i	$-0.664440\pi$
$229$	−783940.	−0.987857	−0.493928	+	0.869503i	$0.664440\pi$
$230$	0	0
$231$	−323592.	−0.398995
$232$	25183.6	0.0307183
$233$	−1.49455e6	−1.80352	−0.901759	−	0.432239i	$-0.857724\pi$
$233$	−1.49455e6	−1.80352	−0.901759	+	0.432239i	$0.857724\pi$
$234$	−40572.8	−0.0484390
$235$	0	0
$236$	−675914.	−0.789971
$237$	−1.01404e6	−1.17269
$238$	−28081.7	−0.0321352
$239$	31598.2	0.0357822	0.0178911	−	0.999840i	$-0.494305\pi$
$239$	31598.2	0.0357822	0.0178911	+	0.999840i	$0.494305\pi$
$240$	0	0
$241$	424884.	0.471225	0.235612	−	0.971847i	$-0.424290\pi$
$241$	424884.	0.471225	0.235612	+	0.971847i	$0.424290\pi$
$242$	1976.13	0.00216909
$243$	−1.32088e6	−1.43499
$244$	−354588.	−0.381285
$245$	0	0
$246$	−19832.6	−0.0208950
$247$	2.36719e6	2.46883
$248$	−77904.8	−0.0804332
$249$	−1.22528e6	−1.25239
$250$	0	0
$251$	257979.	0.258464	0.129232	−	0.991614i	$-0.458749\pi$
$251$	257979.	0.258464	0.129232	+	0.991614i	$0.458749\pi$
$252$	−1.01828e6	−1.01010
$253$	−442107.	−0.434236
$254$	−3015.76	−0.00293300
$255$	0	0
$256$	1.04261e6	0.994312
$257$	703522.	0.664423	0.332212	−	0.943205i	$-0.392205\pi$
$257$	703522.	0.664423	0.332212	+	0.943205i	$0.392205\pi$
$258$	−38661.6	−0.0361602
$259$	94502.6	0.0875375
$260$	0	0
$261$	−786012.	−0.714213
$262$	−16532.0	−0.0148790
$263$	−1.19967e6	−1.06948	−0.534742	−	0.845015i	$-0.679591\pi$
$263$	−1.19967e6	−1.06948	−0.534742	+	0.845015i	$0.679591\pi$
$264$	23656.4	0.0208900
$265$	0	0
$266$	−33841.8	−0.0293257
$267$	−523978.	−0.449816
$268$	1.27261e6	1.08233
$269$	448785.	0.378144	0.189072	−	0.981963i	$-0.439452\pi$
$269$	448785.	0.378144	0.189072	+	0.981963i	$0.439452\pi$
$270$	0	0
$271$	2.23617e6	1.84962	0.924808	−	0.380433i	$-0.124225\pi$
$271$	2.23617e6	1.84962	0.924808	+	0.380433i	$0.124225\pi$
$272$	−1.80047e6	−1.47558
$273$	−2.98255e6	−2.42204
$274$	8307.75	0.00668509
$275$	0	0
$276$	−2.64549e6	−2.09042
$277$	−865906.	−0.678065	−0.339032	−	0.940775i	$-0.610100\pi$
$277$	−865906.	−0.678065	−0.339032	+	0.940775i	$0.610100\pi$
$278$	29632.0	0.0229958
$279$	2.43151e6	1.87010
$280$	0	0
$281$	526213.	0.397554	0.198777	−	0.980045i	$-0.436303\pi$
$281$	526213.	0.397554	0.198777	+	0.980045i	$0.436303\pi$
$282$	−82374.3	−0.0616835
$283$	984369.	0.730620	0.365310	−	0.930886i	$-0.380963\pi$
$283$	984369.	0.730620	0.365310	+	0.930886i	$0.380963\pi$
$284$	−1.22294e6	−0.899723
$285$	0	0
$286$	18214.1	0.0131671
$287$	−766696.	−0.549438
$288$	111674.	0.0793358
$289$	1.68225e6	1.18480
$290$	0	0
$291$	−227668.	−0.157605
$292$	511165.	0.350836
$293$	61527.8	0.0418699	0.0209350	−	0.999781i	$-0.493336\pi$
$293$	61527.8	0.0418699	0.0209350	+	0.999781i	$0.493336\pi$
$294$	−8717.61	−0.00588206
$295$	0	0
$296$	−6908.68	−0.00458317
$297$	−72686.0	−0.0478145
$298$	53068.3	0.0346174
$299$	−4.07491e6	−2.63597
$300$	0	0
$301$	−1.49459e6	−0.950839
$302$	19875.6	0.0125401
$303$	1.99410e6	1.24779
$304$	−2.16978e6	−1.34658
$305$	0	0
$306$	−64074.8	−0.0391186
$307$	−526315.	−0.318713	−0.159357	−	0.987221i	$-0.550942\pi$
$307$	−526315.	−0.318713	−0.159357	+	0.987221i	$0.550942\pi$
$308$	457129.	0.274575
$309$	2.55798e6	1.52406
$310$	0	0
$311$	−587250.	−0.344288	−0.172144	−	0.985072i	$-0.555070\pi$
$311$	−587250.	−0.344288	−0.172144	+	0.985072i	$0.555070\pi$
$312$	218041.	0.126810
$313$	1.62294e6	0.936355	0.468178	−	0.883634i	$-0.344911\pi$
$313$	1.62294e6	0.936355	0.468178	+	0.883634i	$0.344911\pi$
$314$	45505.1	0.0260457
$315$	0	0
$316$	1.43250e6	0.807008
$317$	1.29486e6	0.723729	0.361864	−	0.932231i	$-0.382140\pi$
$317$	1.29486e6	0.723729	0.361864	+	0.932231i	$0.382140\pi$
$318$	−30552.4	−0.0169425
$319$	352859.	0.194144
$320$	0	0
$321$	1.20447e6	0.652428
$322$	58255.7	0.0313111
$323$	3.73840e6	1.99379
$324$	1.65976e6	0.878383
$325$	0	0
$326$	42344.9	0.0220677
$327$	−1.29109e6	−0.667709
$328$	56049.8	0.0287667
$329$	−3.18446e6	−1.62198
$330$	0	0
$331$	−3.67690e6	−1.84464	−0.922320	−	0.386427i	$-0.873709\pi$
$331$	−3.67690e6	−1.84464	−0.922320	+	0.386427i	$0.873709\pi$
$332$	1.73092e6	0.861851
$333$	215629.	0.106561
$334$	−55547.6	−0.0272458
$335$	0	0
$336$	2.73382e6	1.32106
$337$	−2.08071e6	−0.998013	−0.499006	−	0.866598i	$-0.666302\pi$
$337$	−2.08071e6	−0.998013	−0.499006	+	0.866598i	$0.666302\pi$
$338$	117765.	0.0560692
$339$	−3.64805e6	−1.72410
$340$	0	0
$341$	−1.09156e6	−0.508349
$342$	−77217.6	−0.0356986
$343$	−2.32238e6	−1.06585
$344$	109263.	0.0497827
$345$	0	0
$346$	11286.9	0.00506855
$347$	1.31000e6	0.584047	0.292024	−	0.956411i	$-0.405671\pi$
$347$	1.31000e6	0.584047	0.292024	+	0.956411i	$0.405671\pi$
$348$	2.11144e6	0.934612
$349$	1.91442e6	0.841344	0.420672	−	0.907213i	$-0.361794\pi$
$349$	1.91442e6	0.841344	0.420672	+	0.907213i	$0.361794\pi$
$350$	0	0
$351$	−669949.	−0.290251
$352$	−50132.8	−0.0215658
$353$	2.85037e6	1.21749	0.608744	−	0.793367i	$-0.291674\pi$
$353$	2.85037e6	1.21749	0.608744	+	0.793367i	$0.291674\pi$
$354$	64579.6	0.0273897
$355$	0	0
$356$	740209.	0.309549
$357$	−4.71021e6	−1.95600
$358$	−19366.1	−0.00798609
$359$	3.18520e6	1.30437	0.652185	−	0.758060i	$-0.273853\pi$
$359$	3.18520e6	1.30437	0.652185	+	0.758060i	$0.273853\pi$
$360$	0	0
$361$	2.02911e6	0.819477
$362$	−21480.7	−0.00861542
$363$	331461.	0.132028
$364$	4.21337e6	1.66677
$365$	0	0
$366$	33878.8	0.0132198
$367$	415201.	0.160914	0.0804569	−	0.996758i	$-0.474362\pi$
$367$	415201.	0.160914	0.0804569	+	0.996758i	$0.474362\pi$
$368$	3.73508e6	1.43774
$369$	−1.74939e6	−0.668837
$370$	0	0
$371$	−1.18111e6	−0.445507
$372$	−6.53171e6	−2.44720
$373$	936364.	0.348476	0.174238	−	0.984704i	$-0.444254\pi$
$373$	936364.	0.348476	0.174238	+	0.984704i	$0.444254\pi$
$374$	28764.6	0.0106336
$375$	0	0
$376$	232802.	0.0849214
$377$	3.25231e6	1.17852
$378$	9577.71	0.00344772
$379$	−4.97179e6	−1.77793	−0.888966	−	0.457974i	$-0.848575\pi$
$379$	−4.97179e6	−1.77793	−0.888966	+	0.457974i	$0.848575\pi$
$380$	0	0
$381$	−505839.	−0.178525
$382$	13047.9	0.00457492
$383$	−1.64596e6	−0.573355	−0.286677	−	0.958027i	$-0.592551\pi$
$383$	−1.64596e6	−0.573355	−0.286677	+	0.958027i	$0.592551\pi$
$384$	−399943.	−0.138411
$385$	0	0
$386$	96574.4	0.0329909
$387$	−3.41025e6	−1.15747
$388$	321620.	0.108459
$389$	5.30240e6	1.77664	0.888318	−	0.459228i	$-0.151874\pi$
$389$	5.30240e6	1.77664	0.888318	+	0.459228i	$0.151874\pi$
$390$	0	0
$391$	−6.43532e6	−2.12877
$392$	24637.3	0.00809799
$393$	−2.77295e6	−0.905651
$394$	20221.1	0.00656243
$395$	0	0
$396$	1.04304e6	0.334244
$397$	−632564.	−0.201432	−0.100716	−	0.994915i	$-0.532113\pi$
$397$	−632564.	−0.201432	−0.100716	+	0.994915i	$0.532113\pi$
$398$	50504.3	0.0159816
$399$	−5.67635e6	−1.78499
$400$	0	0
$401$	3.40479e6	1.05738	0.528688	−	0.848816i	$-0.322684\pi$
$401$	3.40479e6	1.05738	0.528688	+	0.848816i	$0.322684\pi$
$402$	−121591.	−0.0375263
$403$	−1.00609e7	−3.08586
$404$	−2.81701e6	−0.858688
$405$	0	0
$406$	−46495.5	−0.0139990
$407$	−96800.7	−0.0289663
$408$	344343.	0.102410
$409$	−907404.	−0.268221	−0.134110	−	0.990966i	$-0.542818\pi$
$409$	−907404.	−0.268221	−0.134110	+	0.990966i	$0.542818\pi$
$410$	0	0
$411$	1.39348e6	0.406907
$412$	−3.61359e6	−1.04881
$413$	2.49654e6	0.720218
$414$	132923.	0.0381154
$415$	0	0
$416$	−462075.	−0.130912
$417$	4.97024e6	1.39971
$418$	34664.7	0.00970392
$419$	−4.91409e6	−1.36744	−0.683719	−	0.729745i	$-0.739639\pi$
$419$	−4.91409e6	−1.36744	−0.683719	+	0.729745i	$0.739639\pi$
$420$	0	0
$421$	−3.80407e6	−1.04603	−0.523015	−	0.852324i	$-0.675193\pi$
$421$	−3.80407e6	−1.04603	−0.523015	+	0.852324i	$0.675193\pi$
$422$	−47615.4	−0.0130157
$423$	−7.26606e6	−1.97446
$424$	86345.6	0.0233252
$425$	0	0
$426$	116845.	0.0311950
$427$	1.30970e6	0.347618
$428$	−1.70152e6	−0.448980
$429$	3.05508e6	0.801456
$430$	0	0
$431$	3.11946e6	0.808883	0.404441	−	0.914564i	$-0.367466\pi$
$431$	3.11946e6	0.808883	0.404441	+	0.914564i	$0.367466\pi$
$432$	614078.	0.158312
$433$	−393705.	−0.100914	−0.0504570	−	0.998726i	$-0.516068\pi$
$433$	−393705.	−0.100914	−0.0504570	+	0.998726i	$0.516068\pi$
$434$	143833.	0.0366551
$435$	0	0
$436$	1.82389e6	0.459496
$437$	−7.75532e6	−1.94266
$438$	−48838.9	−0.0121641
$439$	−7.28832e6	−1.80495	−0.902477	−	0.430739i	$-0.858253\pi$
$439$	−7.28832e6	−1.80495	−0.902477	+	0.430739i	$0.858253\pi$
$440$	0	0
$441$	−768962.	−0.188282
$442$	265124.	0.0645496
$443$	−1.21705e6	−0.294646	−0.147323	−	0.989088i	$-0.547066\pi$
$443$	−1.21705e6	−0.294646	−0.147323	+	0.989088i	$0.547066\pi$
$444$	−579238.	−0.139444
$445$	0	0
$446$	−189720.	−0.0451622
$447$	8.90126e6	2.10709
$448$	−3.85758e6	−0.908072
$449$	6.40219e6	1.49869	0.749346	−	0.662178i	$-0.230368\pi$
$449$	6.40219e6	1.49869	0.749346	+	0.662178i	$0.230368\pi$
$450$	0	0
$451$	785341.	0.181810
$452$	5.15349e6	1.18647
$453$	3.33377e6	0.763292
$454$	32920.4	0.00749594
$455$	0	0
$456$	414973.	0.0934562
$457$	−5.42746e6	−1.21564	−0.607822	−	0.794073i	$-0.707957\pi$
$457$	−5.42746e6	−1.21564	−0.607822	+	0.794073i	$0.707957\pi$
$458$	−105810.	−0.0235703
$459$	−1.05802e6	−0.234402
$460$	0	0
$461$	−6.78139e6	−1.48616	−0.743081	−	0.669201i	$-0.766636\pi$
$461$	−6.78139e6	−1.48616	−0.743081	+	0.669201i	$0.766636\pi$
$462$	−43676.0	−0.00952001
$463$	−1.29395e6	−0.280520	−0.140260	−	0.990115i	$-0.544794\pi$
$463$	−1.29395e6	−0.280520	−0.140260	+	0.990115i	$0.544794\pi$
$464$	−2.98108e6	−0.642804
$465$	0	0
$466$	−201723.	−0.0430319
$467$	1.79969e6	0.381860	0.190930	−	0.981604i	$-0.438850\pi$
$467$	1.79969e6	0.381860	0.190930	+	0.981604i	$0.438850\pi$
$468$	9.61374e6	2.02898
$469$	−4.70050e6	−0.986762
$470$	0	0
$471$	7.63267e6	1.58535
$472$	−182512.	−0.0377082
$473$	1.53094e6	0.314634
$474$	−136867.	−0.0279804
$475$	0	0
$476$	6.65398e6	1.34606
$477$	−2.69496e6	−0.542321
$478$	4264.89	0.000853764 0
$479$	6.71162e6	1.33656	0.668280	−	0.743910i	$-0.267031\pi$
$479$	6.71162e6	1.33656	0.668280	+	0.743910i	$0.267031\pi$
$480$	0	0
$481$	−892215.	−0.175836
$482$	57347.7	0.0112434
$483$	9.77134e6	1.90584
$484$	−468245.	−0.0908573
$485$	0	0
$486$	−178283.	−0.0342389
$487$	−5.32244e6	−1.01692	−0.508462	−	0.861084i	$-0.669786\pi$
$487$	−5.32244e6	−1.01692	−0.508462	+	0.861084i	$0.669786\pi$
$488$	−95746.6	−0.0182001
$489$	7.10259e6	1.34321
$490$	0	0
$491$	−5.43908e6	−1.01817	−0.509087	−	0.860715i	$-0.670017\pi$
$491$	−5.43908e6	−1.01817	−0.509087	+	0.860715i	$0.670017\pi$
$492$	4.69934e6	0.875234
$493$	5.13622e6	0.951758
$494$	319506.	0.0589062
$495$	0	0
$496$	9.22190e6	1.68312
$497$	4.51702e6	0.820279
$498$	−165380.	−0.0298819
$499$	−2.16501e6	−0.389232	−0.194616	−	0.980880i	$-0.562346\pi$
$499$	−2.16501e6	−0.389232	−0.194616	+	0.980880i	$0.562346\pi$
$500$	0	0
$501$	−9.31712e6	−1.65839
$502$	34820.0	0.00616694
$503$	−5.50791e6	−0.970659	−0.485329	−	0.874331i	$-0.661300\pi$
$503$	−5.50791e6	−0.970659	−0.485329	+	0.874331i	$0.661300\pi$
$504$	−274958.	−0.0482158
$505$	0	0
$506$	−59672.3	−0.0103609
$507$	1.97530e7	3.41281
$508$	714585.	0.122855
$509$	796560.	0.136277	0.0681387	−	0.997676i	$-0.478294\pi$
$509$	796560.	0.136277	0.0681387	+	0.997676i	$0.478294\pi$
$510$	0	0
$511$	−1.88803e6	−0.319858
$512$	706034.	0.119028
$513$	−1.27504e6	−0.213909
$514$	94956.1	0.0158531
$515$	0	0
$516$	9.16087e6	1.51465
$517$	3.26190e6	0.536716
$518$	12755.3	0.00208865
$519$	1.89317e6	0.308512
$520$	0	0
$521$	−3.66495e6	−0.591526	−0.295763	−	0.955261i	$-0.595574\pi$
$521$	−3.66495e6	−0.591526	−0.295763	+	0.955261i	$0.595574\pi$
$522$	−106090.	−0.0170411
$523$	−6.07276e6	−0.970805	−0.485403	−	0.874291i	$-0.661327\pi$
$523$	−6.07276e6	−0.970805	−0.485403	+	0.874291i	$0.661327\pi$
$524$	3.91727e6	0.623240
$525$	0	0
$526$	−161923.	−0.0255179
$527$	−1.58888e7	−2.49209
$528$	−2.80030e6	−0.437139
$529$	6.91376e6	1.07417
$530$	0	0
$531$	5.69643e6	0.876730
$532$	8.01882e6	1.22838
$533$	7.23850e6	1.10365
$534$	−70722.7	−0.0107326
$535$	0	0
$536$	343634.	0.0516635
$537$	−3.24831e6	−0.486096
$538$	60573.6	0.00902252
$539$	345204.	0.0511805
$540$	0	0
$541$	−1.21762e7	−1.78863	−0.894313	−	0.447442i	$-0.852335\pi$
$541$	−1.21762e7	−1.78863	−0.894313	+	0.447442i	$0.852335\pi$
$542$	301822.	0.0441318
$543$	−3.60300e6	−0.524402
$544$	−729735.	−0.105723
$545$	0	0
$546$	−402562.	−0.0577899
$547$	3.53391e6	0.504995	0.252497	−	0.967598i	$-0.418748\pi$
$547$	3.53391e6	0.504995	0.252497	+	0.967598i	$0.418748\pi$
$548$	−1.96852e6	−0.280020
$549$	2.98838e6	0.423160
$550$	0	0
$551$	6.18975e6	0.868548
$552$	−714341.	−0.0997832
$553$	−5.29107e6	−0.735750
$554$	−116873.	−0.0161786
$555$	0	0
$556$	−7.02131e6	−0.963233
$557$	7.13926e6	0.975024	0.487512	−	0.873116i	$-0.337905\pi$
$557$	7.13926e6	0.975024	0.487512	+	0.873116i	$0.337905\pi$
$558$	328187.	0.0446207
$559$	1.41107e7	1.90994
$560$	0	0
$561$	4.82475e6	0.647243
$562$	71024.3	0.00948563
$563$	−4.77394e6	−0.634755	−0.317378	−	0.948299i	$-0.602802\pi$
$563$	−4.77394e6	−0.634755	−0.317378	+	0.948299i	$0.602802\pi$
$564$	1.95186e7	2.58376
$565$	0	0
$566$	132863.	0.0174326
$567$	−6.13047e6	−0.800823
$568$	−330220.	−0.0429470
$569$	1.33496e7	1.72857	0.864284	−	0.503003i	$-0.167772\pi$
$569$	1.33496e7	1.72857	0.864284	+	0.503003i	$0.167772\pi$
$570$	0	0
$571$	930724.	0.119462	0.0597311	−	0.998215i	$-0.480976\pi$
$571$	930724.	0.119462	0.0597311	+	0.998215i	$0.480976\pi$
$572$	−4.31583e6	−0.551536
$573$	2.18856e6	0.278465
$574$	−103483.	−0.0131096
$575$	0	0
$576$	−8.80194e6	−1.10541
$577$	−1.59301e6	−0.199195	−0.0995975	−	0.995028i	$-0.531756\pi$
$577$	−1.59301e6	−0.199195	−0.0995975	+	0.995028i	$0.531756\pi$
$578$	227057.	0.0282693
$579$	1.61986e7	2.00808
$580$	0	0
$581$	−6.39331e6	−0.785751
$582$	−30728.9	−0.00376045
$583$	1.20983e6	0.147419
$584$	138026.	0.0167467
$585$	0	0
$586$	8304.56	0.000999017 0
$587$	−2.71292e6	−0.324969	−0.162485	−	0.986711i	$-0.551951\pi$
$587$	−2.71292e6	−0.324969	−0.162485	+	0.986711i	$0.551951\pi$
$588$	2.06564e6	0.246383
$589$	−1.91478e7	−2.27422
$590$	0	0
$591$	3.39173e6	0.399441
$592$	817807.	0.0959062
$593$	1.02808e7	1.20058	0.600291	−	0.799782i	$-0.295051\pi$
$593$	1.02808e7	1.20058	0.600291	+	0.799782i	$0.295051\pi$
$594$	−9810.62	−0.00114085
$595$	0	0
$596$	−1.25746e7	−1.45003
$597$	8.47119e6	0.972767
$598$	−550001.	−0.0628942
$599$	−1.49992e7	−1.70805	−0.854026	−	0.520230i	$-0.825846\pi$
$599$	−1.49992e7	−1.70805	−0.854026	+	0.520230i	$0.825846\pi$
$600$	0	0
$601$	−168031.	−0.0189759	−0.00948796	−	0.999955i	$-0.503020\pi$
$601$	−168031.	−0.0189759	−0.00948796	+	0.999955i	$0.503020\pi$
$602$	−201729.	−0.0226870
$603$	−1.07253e7	−1.20120
$604$	−4.70953e6	−0.525273
$605$	0	0
$606$	269149.	0.0297722
$607$	1.53058e7	1.68610	0.843052	−	0.537832i	$-0.180757\pi$
$607$	1.53058e7	1.68610	0.843052	+	0.537832i	$0.180757\pi$
$608$	−879415.	−0.0964795
$609$	−7.79879e6	−0.852088
$610$	0	0
$611$	3.00650e7	3.25805
$612$	1.51825e7	1.63857
$613$	4.94209e6	0.531202	0.265601	−	0.964083i	$-0.414430\pi$
$613$	4.94209e6	0.531202	0.265601	+	0.964083i	$0.414430\pi$
$614$	−71038.1	−0.00760449
$615$	0	0
$616$	123435.	0.0131065
$617$	805785.	0.0852131	0.0426065	−	0.999092i	$-0.486434\pi$
$617$	805785.	0.0852131	0.0426065	+	0.999092i	$0.486434\pi$
$618$	345257.	0.0363640
$619$	9.20583e6	0.965686	0.482843	−	0.875707i	$-0.339604\pi$
$619$	9.20583e6	0.965686	0.482843	+	0.875707i	$0.339604\pi$
$620$	0	0
$621$	2.19487e6	0.228391
$622$	−79262.7	−0.00821472
$623$	−2.73402e6	−0.282216
$624$	−2.58104e7	−2.65359
$625$	0	0
$626$	219052.	0.0223414
$627$	5.81439e6	0.590657
$628$	−1.07825e7	−1.09098
$629$	−1.40903e6	−0.142002
$630$	0	0
$631$	−9.87340e6	−0.987173	−0.493587	−	0.869697i	$-0.664314\pi$
$631$	−9.87340e6	−0.987173	−0.493587	+	0.869697i	$0.664314\pi$
$632$	386807.	0.0385214
$633$	−7.98663e6	−0.792236
$634$	174771.	0.0172682
$635$	0	0
$636$	7.23940e6	0.709676
$637$	3.18176e6	0.310684
$638$	47626.2	0.00463228
$639$	1.03066e7	0.998536
$640$	0	0
$641$	4.09762e6	0.393901	0.196950	−	0.980413i	$-0.436896\pi$
$641$	4.09762e6	0.393901	0.196950	+	0.980413i	$0.436896\pi$
$642$	162570.	0.0155669
$643$	−1.16722e7	−1.11333	−0.556664	−	0.830737i	$-0.687919\pi$
$643$	−1.16722e7	−1.11333	−0.556664	+	0.830737i	$0.687919\pi$
$644$	−1.38037e7	−1.31154
$645$	0	0
$646$	504581.	0.0475718
$647$	−5.20905e6	−0.489213	−0.244607	−	0.969622i	$-0.578659\pi$
$647$	−5.20905e6	−0.489213	−0.244607	+	0.969622i	$0.578659\pi$
$648$	448173.	0.0419283
$649$	−2.55726e6	−0.238321
$650$	0	0
$651$	2.41254e7	2.23112
$652$	−1.00336e7	−0.924355
$653$	−1.90677e7	−1.74991	−0.874954	−	0.484207i	$-0.839108\pi$
$653$	−1.90677e7	−1.74991	−0.874954	+	0.484207i	$0.839108\pi$
$654$	−174262.	−0.0159315
$655$	0	0
$656$	−6.63484e6	−0.601964
$657$	−4.30797e6	−0.389367
$658$	−429814.	−0.0387005
$659$	1.19782e7	1.07443	0.537213	−	0.843447i	$-0.319477\pi$
$659$	1.19782e7	1.07443	0.537213	+	0.843447i	$0.319477\pi$
$660$	0	0
$661$	−1.60950e7	−1.43281	−0.716405	−	0.697685i	$-0.754214\pi$
$661$	−1.60950e7	−1.43281	−0.716405	+	0.697685i	$0.754214\pi$
$662$	−496280.	−0.0440131
$663$	4.44698e7	3.92900
$664$	467387.	0.0411393
$665$	0	0
$666$	29104.0	0.00254253
$667$	−1.06551e7	−0.927349
$668$	1.31620e7	1.14125
$669$	−3.18221e7	−2.74893
$670$	0	0
$671$	−1.34155e6	−0.115027
$672$	1.10802e6	0.0946511
$673$	2.12991e6	0.181269	0.0906343	−	0.995884i	$-0.471111\pi$
$673$	2.12991e6	0.181269	0.0906343	+	0.995884i	$0.471111\pi$
$674$	−280838.	−0.0238126
$675$	0	0
$676$	−2.79045e7	−2.34859
$677$	−2.17699e7	−1.82551	−0.912757	−	0.408503i	$-0.866051\pi$
$677$	−2.17699e7	−1.82551	−0.912757	+	0.408503i	$0.866051\pi$
$678$	−492386.	−0.0411369
$679$	−1.18793e6	−0.0988819
$680$	0	0
$681$	5.52181e6	0.456262
$682$	−147331.	−0.0121292
$683$	−1.33130e7	−1.09200	−0.546000	−	0.837785i	$-0.683850\pi$
$683$	−1.33130e7	−1.09200	−0.546000	+	0.837785i	$0.683850\pi$
$684$	1.82967e7	1.49532
$685$	0	0
$686$	−313457.	−0.0254312
$687$	−1.77478e7	−1.43467
$688$	−1.29339e7	−1.04174
$689$	1.11510e7	0.894884
$690$	0	0
$691$	−1.88639e7	−1.50292	−0.751460	−	0.659779i	$-0.770650\pi$
$691$	−1.88639e7	−1.50292	−0.751460	+	0.659779i	$0.770650\pi$
$692$	−2.67443e6	−0.212308
$693$	−3.85256e6	−0.304731
$694$	176814.	0.0139354
$695$	0	0
$696$	570136.	0.0446124
$697$	1.14314e7	0.891290
$698$	258394.	0.0200745
$699$	−3.38354e7	−2.61926
$700$	0	0
$701$	3.10378e6	0.238559	0.119279	−	0.992861i	$-0.461942\pi$
$701$	3.10378e6	0.238559	0.119279	+	0.992861i	$0.461942\pi$
$702$	−90424.7	−0.00692539
$703$	−1.69805e6	−0.129587
$704$	3.95139e6	0.300482
$705$	0	0
$706$	384722.	0.0290493
$707$	1.04049e7	0.782867
$708$	−1.53022e7	−1.14728
$709$	−3.73810e6	−0.279277	−0.139639	−	0.990203i	$-0.544594\pi$
$709$	−3.73810e6	−0.279277	−0.139639	+	0.990203i	$0.544594\pi$
$710$	0	0
$711$	−1.20728e7	−0.895638
$712$	199873.	0.0147759
$713$	3.29613e7	2.42818
$714$	−635748.	−0.0466702
$715$	0	0
$716$	4.58880e6	0.334516
$717$	715358.	0.0519668
$718$	429915.	0.0311223
$719$	−3.69716e6	−0.266714	−0.133357	−	0.991068i	$-0.542576\pi$
$719$	−3.69716e6	−0.266714	−0.133357	+	0.991068i	$0.542576\pi$
$720$	0	0
$721$	1.33471e7	0.956199
$722$	273873.	0.0195527
$723$	9.61905e6	0.684363
$724$	5.08985e6	0.360876
$725$	0	0
$726$	44738.1	0.00315018
$727$	3.48674e6	0.244671	0.122336	−	0.992489i	$-0.460962\pi$
$727$	3.48674e6	0.244671	0.122336	+	0.992489i	$0.460962\pi$
$728$	1.13770e6	0.0795609
$729$	−1.72928e7	−1.20516
$730$	0	0
$731$	2.22844e7	1.54244
$732$	−8.02760e6	−0.553743
$733$	−749439.	−0.0515201	−0.0257600	−	0.999668i	$-0.508201\pi$
$733$	−749439.	−0.0515201	−0.0257600	+	0.999668i	$0.508201\pi$
$734$	56040.8	0.00383940
$735$	0	0
$736$	1.51384e6	0.103011
$737$	4.81481e6	0.326521
$738$	−236119.	−0.0159585
$739$	2.27099e7	1.52969	0.764845	−	0.644215i	$-0.222816\pi$
$739$	2.27099e7	1.52969	0.764845	+	0.644215i	$0.222816\pi$
$740$	0	0
$741$	5.35913e7	3.58550
$742$	−159417.	−0.0106298
$743$	2.05231e7	1.36386	0.681932	−	0.731415i	$-0.261140\pi$
$743$	2.05231e7	1.36386	0.681932	+	0.731415i	$0.261140\pi$
$744$	−1.76370e6	−0.116814
$745$	0	0
$746$	126383.	0.00831463
$747$	−1.45878e7	−0.956505
$748$	−6.81579e6	−0.445412
$749$	6.28469e6	0.409336
$750$	0	0
$751$	−1.25415e7	−0.811425	−0.405713	−	0.914001i	$-0.632977\pi$
$751$	−1.25415e7	−0.811425	−0.405713	+	0.914001i	$0.632977\pi$
$752$	−2.75577e7	−1.77704
$753$	5.84043e6	0.375368
$754$	438972.	0.0281196
$755$	0	0
$756$	−2.26944e6	−0.144416
$757$	−829844.	−0.0526328	−0.0263164	−	0.999654i	$-0.508378\pi$
$757$	−829844.	−0.0526328	−0.0263164	+	0.999654i	$0.508378\pi$
$758$	−671055.	−0.0424214
$759$	−1.00090e7	−0.630644
$760$	0	0
$761$	−8.42388e6	−0.527291	−0.263646	−	0.964620i	$-0.584925\pi$
$761$	−8.42388e6	−0.527291	−0.263646	+	0.964620i	$0.584925\pi$
$762$	−68274.4	−0.00425962
$763$	−6.73668e6	−0.418923
$764$	−3.09171e6	−0.191631
$765$	0	0
$766$	−222160.	−0.0136802
$767$	−2.35703e7	−1.44669
$768$	2.36039e7	1.44405
$769$	−2.74632e6	−0.167469	−0.0837346	−	0.996488i	$-0.526685\pi$
$769$	−2.74632e6	−0.167469	−0.0837346	+	0.996488i	$0.526685\pi$
$770$	0	0
$771$	1.59272e7	0.964946
$772$	−2.28833e7	−1.38190
$773$	−1.91063e7	−1.15008	−0.575041	−	0.818125i	$-0.695014\pi$
$773$	−1.91063e7	−1.15008	−0.575041	+	0.818125i	$0.695014\pi$
$774$	−460290.	−0.0276172
$775$	0	0
$776$	86844.5	0.00517712
$777$	2.13946e6	0.127131
$778$	715678.	0.0423905
$779$	1.37762e7	0.813367
$780$	0	0
$781$	−4.62687e6	−0.271431
$782$	−868592.	−0.0507924
$783$	−1.75179e6	−0.102112
$784$	−2.91641e6	−0.169457
$785$	0	0
$786$	−374272.	−0.0216088
$787$	−2.47363e6	−0.142363	−0.0711817	−	0.997463i	$-0.522677\pi$
$787$	−2.47363e6	−0.142363	−0.0711817	+	0.997463i	$0.522677\pi$
$788$	−4.79140e6	−0.274883
$789$	−2.71597e7	−1.55322
$790$	0	0
$791$	−1.90349e7	−1.08170
$792$	281644.	0.0159547
$793$	−1.23651e7	−0.698256
$794$	−85378.7	−0.00480616
$795$	0	0
$796$	−1.19670e7	−0.669427
$797$	6.70091e6	0.373670	0.186835	−	0.982391i	$-0.440177\pi$
$797$	6.70091e6	0.373670	0.186835	+	0.982391i	$0.440177\pi$
$798$	−766151.	−0.0425900
$799$	4.74803e7	2.63115
$800$	0	0
$801$	−6.23829e6	−0.343546
$802$	459553.	0.0252290
$803$	1.93395e6	0.105841
$804$	2.88110e7	1.57187
$805$	0	0
$806$	−1.35795e6	−0.0736286
$807$	1.01601e7	0.549181
$808$	−760655.	−0.0409883
$809$	−3.26172e6	−0.175217	−0.0876083	−	0.996155i	$-0.527922\pi$
$809$	−3.26172e6	−0.175217	−0.0876083	+	0.996155i	$0.527922\pi$
$810$	0	0
$811$	−3.18531e7	−1.70059	−0.850294	−	0.526307i	$-0.823576\pi$
$811$	−3.18531e7	−1.70059	−0.850294	+	0.526307i	$0.823576\pi$
$812$	1.10171e7	0.586379
$813$	5.06252e7	2.68621
$814$	−13065.4	−0.000691135 0
$815$	0	0
$816$	−4.07612e7	−2.14300
$817$	2.68553e7	1.40759
$818$	−122475.	−0.00639975
$819$	−3.55091e7	−1.84982
$820$	0	0
$821$	2.40684e7	1.24620	0.623102	−	0.782140i	$-0.285872\pi$
$821$	2.40684e7	1.24620	0.623102	+	0.782140i	$0.285872\pi$
$822$	188081.	0.00970879
$823$	1.86832e7	0.961505	0.480752	−	0.876856i	$-0.340364\pi$
$823$	1.86832e7	0.961505	0.480752	+	0.876856i	$0.340364\pi$
$824$	−975748.	−0.0500633
$825$	0	0
$826$	336965.	0.0171844
$827$	−2.00821e7	−1.02105	−0.510523	−	0.859864i	$-0.670548\pi$
$827$	−2.00821e7	−1.02105	−0.510523	+	0.859864i	$0.670548\pi$
$828$	−3.14962e7	−1.59655
$829$	−3.25162e7	−1.64329	−0.821644	−	0.570001i	$-0.806943\pi$
$829$	−3.25162e7	−1.64329	−0.821644	+	0.570001i	$0.806943\pi$
$830$	0	0
$831$	−1.96034e7	−0.984758
$832$	3.64201e7	1.82403
$833$	5.02480e6	0.250903
$834$	670845.	0.0333970
$835$	0	0
$836$	−8.21382e6	−0.406471
$837$	5.41912e6	0.267371
$838$	−663267.	−0.0326271
$839$	1.23891e7	0.607623	0.303812	−	0.952732i	$-0.401741\pi$
$839$	1.23891e7	0.607623	0.303812	+	0.952732i	$0.401741\pi$
$840$	0	0
$841$	−1.20070e7	−0.585389
$842$	−513446.	−0.0249583
$843$	1.19130e7	0.577370
$844$	1.12825e7	0.545191
$845$	0	0
$846$	−980718.	−0.0471106
$847$	1.72950e6	0.0828348
$848$	−1.02211e7	−0.488098
$849$	2.22853e7	1.06108
$850$	0	0
$851$	2.92304e6	0.138360
$852$	−2.76864e7	−1.30667
$853$	−1.69664e7	−0.798392	−0.399196	−	0.916866i	$-0.630711\pi$
$853$	−1.69664e7	−0.798392	−0.399196	+	0.916866i	$0.630711\pi$
$854$	176774.	0.00829417
$855$	0	0
$856$	−459447.	−0.0214314
$857$	−2.65677e7	−1.23567	−0.617835	−	0.786308i	$-0.711990\pi$
$857$	−2.65677e7	−1.23567	−0.617835	+	0.786308i	$0.711990\pi$
$858$	412352.	0.0191227
$859$	−1.94648e7	−0.900050	−0.450025	−	0.893016i	$-0.648585\pi$
$859$	−1.94648e7	−0.900050	−0.450025	+	0.893016i	$0.648585\pi$
$860$	0	0
$861$	−1.73574e7	−0.797952
$862$	421041.	0.0192999
$863$	2.61455e7	1.19501	0.597503	−	0.801866i	$-0.296159\pi$
$863$	2.61455e7	1.19501	0.597503	+	0.801866i	$0.296159\pi$
$864$	248887.	0.0113427
$865$	0	0
$866$	−53139.4	−0.00240781
$867$	3.80847e7	1.72069
$868$	−3.40813e7	−1.53538
$869$	5.41974e6	0.243461
$870$	0	0
$871$	4.43782e7	1.98210
$872$	492490.	0.0219334
$873$	−2.71053e6	−0.120370
$874$	−1.04675e6	−0.0463518
$875$	0	0
$876$	1.15724e7	0.509522
$877$	−3.19581e7	−1.40308	−0.701539	−	0.712631i	$-0.747503\pi$
$877$	−3.19581e7	−1.40308	−0.701539	+	0.712631i	$0.747503\pi$
$878$	−983723.	−0.0430662
$879$	1.39294e6	0.0608080
$880$	0	0
$881$	1.74845e7	0.758949	0.379474	−	0.925202i	$-0.376105\pi$
$881$	1.74845e7	0.758949	0.379474	+	0.925202i	$0.376105\pi$
$882$	−103789.	−0.00449240
$883$	−5.77865e6	−0.249416	−0.124708	−	0.992193i	$-0.539799\pi$
$883$	−5.77865e6	−0.249416	−0.124708	+	0.992193i	$0.539799\pi$
$884$	−6.28213e7	−2.70381
$885$	0	0
$886$	−164269.	−0.00703025
$887$	3.26984e7	1.39546	0.697731	−	0.716360i	$-0.254193\pi$
$887$	3.26984e7	1.39546	0.697731	+	0.716360i	$0.254193\pi$
$888$	−156407.	−0.00665616
$889$	−2.63938e6	−0.112008
$890$	0	0
$891$	6.27956e6	0.264993
$892$	4.49542e7	1.89172
$893$	5.72193e7	2.40112
$894$	1.20143e6	0.0502751
$895$	0	0
$896$	−2.08683e6	−0.0868395
$897$	−9.22528e7	−3.82823
$898$	864119.	0.0357588
$899$	−2.63074e7	−1.08562
$900$	0	0
$901$	1.76103e7	0.722695
$902$	105999.	0.00433798
$903$	−3.38364e7	−1.38091
$904$	1.39156e6	0.0566343
$905$	0	0
$906$	449968.	0.0182121
$907$	6.03198e6	0.243468	0.121734	−	0.992563i	$-0.461155\pi$
$907$	6.03198e6	0.243468	0.121734	+	0.992563i	$0.461155\pi$
$908$	−7.80051e6	−0.313985
$909$	2.37410e7	0.952994
$910$	0	0
$911$	−1.61928e7	−0.646436	−0.323218	−	0.946325i	$-0.604765\pi$
$911$	−1.61928e7	−0.646436	−0.323218	+	0.946325i	$0.604765\pi$
$912$	−4.91220e7	−1.95564
$913$	6.54878e6	0.260006
$914$	−732559.	−0.0290053
$915$	0	0
$916$	2.50718e7	0.987295
$917$	−1.44688e7	−0.568209
$918$	−142804.	−0.00559284
$919$	−1.59851e7	−0.624349	−0.312174	−	0.950025i	$-0.601057\pi$
$919$	−1.59851e7	−0.624349	−0.312174	+	0.950025i	$0.601057\pi$
$920$	0	0
$921$	−1.19154e7	−0.462869
$922$	−915301.	−0.0354598
$923$	−4.26460e7	−1.64768
$924$	1.03490e7	0.398768
$925$	0	0
$926$	−174647.	−0.00669321
$927$	3.04544e7	1.16399
$928$	−1.20824e6	−0.0460556
$929$	−3.20196e7	−1.21724	−0.608621	−	0.793461i	$-0.708277\pi$
$929$	−3.20196e7	−1.21724	−0.608621	+	0.793461i	$0.708277\pi$
$930$	0	0
$931$	6.05547e6	0.228968
$932$	4.77984e7	1.80249
$933$	−1.32949e7	−0.500012
$934$	242908.	0.00911118
$935$	0	0
$936$	2.59592e6	0.0968505
$937$	9.02510e6	0.335817	0.167909	−	0.985803i	$-0.446299\pi$
$937$	9.02510e6	0.335817	0.167909	+	0.985803i	$0.446299\pi$
$938$	−634439.	−0.0235441
$939$	3.67420e7	1.35987
$940$	0	0
$941$	−2.52174e7	−0.928382	−0.464191	−	0.885735i	$-0.653655\pi$
$941$	−2.52174e7	−0.928382	−0.464191	+	0.885735i	$0.653655\pi$
$942$	1.03020e6	0.0378263
$943$	−2.37146e7	−0.868432
$944$	2.16046e7	0.789072
$945$	0	0
$946$	206635.	0.00750716
$947$	4.78514e7	1.73388	0.866941	−	0.498412i	$-0.166083\pi$
$947$	4.78514e7	1.73388	0.866941	+	0.498412i	$0.166083\pi$
$948$	3.24307e7	1.17202
$949$	1.78252e7	0.642494
$950$	0	0
$951$	2.93147e7	1.05108
$952$	1.79672e6	0.0642522
$953$	3.48413e7	1.24269	0.621344	−	0.783538i	$-0.286587\pi$
$953$	3.48413e7	1.24269	0.621344	+	0.783538i	$0.286587\pi$
$954$	−363746.	−0.0129398
$955$	0	0
$956$	−1.01057e6	−0.0357619
$957$	7.98845e6	0.281957
$958$	905884.	0.0318903
$959$	7.27091e6	0.255295
$960$	0	0
$961$	5.27523e7	1.84261
$962$	−120424.	−0.00419543
$963$	1.43399e7	0.498289
$964$	−1.35886e7	−0.470957
$965$	0	0
$966$	1.31886e6	0.0454733
$967$	1.57556e6	0.0541839	0.0270919	−	0.999633i	$-0.491375\pi$
$967$	1.57556e6	0.0541839	0.0270919	+	0.999633i	$0.491375\pi$
$968$	−126436.	−0.00433695
$969$	8.46344e7	2.89559
$970$	0	0
$971$	2.01598e7	0.686181	0.343091	−	0.939302i	$-0.388526\pi$
$971$	2.01598e7	0.686181	0.343091	+	0.939302i	$0.388526\pi$
$972$	4.22442e7	1.43417
$973$	2.59338e7	0.878181
$974$	−718383.	−0.0242638
$975$	0	0
$976$	1.13339e7	0.380851
$977$	−2.27423e7	−0.762250	−0.381125	−	0.924524i	$-0.624463\pi$
$977$	−2.27423e7	−0.762250	−0.381125	+	0.924524i	$0.624463\pi$
$978$	958654.	0.0320490
$979$	2.80051e6	0.0933857
$980$	0	0
$981$	−1.53712e7	−0.509961
$982$	−734126.	−0.0242936
$983$	1.89151e7	0.624345	0.312172	−	0.950026i	$-0.398943\pi$
$983$	1.89151e7	0.624345	0.312172	+	0.950026i	$0.398943\pi$
$984$	1.26892e6	0.0417780
$985$	0	0
$986$	693249.	0.0227089
$987$	−7.20937e7	−2.35561
$988$	−7.57070e7	−2.46742
$989$	−4.62291e7	−1.50288
$990$	0	0
$991$	−1.43913e7	−0.465496	−0.232748	−	0.972537i	$-0.574772\pi$
$991$	−1.43913e7	−0.465496	−0.232748	+	0.972537i	$0.574772\pi$
$992$	3.73766e6	0.120593
$993$	−8.32421e7	−2.67898
$994$	609674.	0.0195718
$995$	0	0
$996$	3.91867e7	1.25167
$997$	2.76917e7	0.882290	0.441145	−	0.897436i	$-0.354572\pi$
$997$	2.76917e7	0.882290	0.441145	+	0.897436i	$0.354572\pi$
$998$	−292217.	−0.00928707
$999$	480573.	0.0152351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.6.a.e.1.3		5
5.2	odd	4		275.6.b.e.199.6		10
5.3	odd	4		275.6.b.e.199.5		10
5.4	even	2		55.6.a.c.1.3	✓	5
15.14	odd	2		495.6.a.h.1.3		5
20.19	odd	2		880.6.a.r.1.4		5
55.54	odd	2		605.6.a.d.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.c.1.3	✓	5	5.4	even	2
275.6.a.e.1.3		5	1.1	even	1	trivial
275.6.b.e.199.5		10	5.3	odd	4
275.6.b.e.199.6		10	5.2	odd	4
495.6.a.h.1.3		5	15.14	odd	2
605.6.a.d.1.3		5	55.54	odd	2
880.6.a.r.1.4		5	20.19	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q+0.134973 q^{2} +22.6392 q^{3} -31.9818 q^{4} +3.05567 q^{6} +118.127 q^{7} -8.63578 q^{8} +269.534 q^{9} +O(q^{10})\)	\(q+0.134973 q^{2} +22.6392 q^{3} -31.9818 q^{4} +3.05567 q^{6} +118.127 q^{7} -8.63578 q^{8} +269.534 q^{9} -121.000 q^{11} -724.042 q^{12} -1115.26 q^{13} +15.9439 q^{14} +1022.25 q^{16} -1761.28 q^{17} +36.3797 q^{18} -2122.55 q^{19} +2674.31 q^{21} -16.3317 q^{22} +3653.78 q^{23} -195.507 q^{24} -150.529 q^{26} +600.711 q^{27} -3777.92 q^{28} -2916.19 q^{29} +9021.17 q^{31} +414.321 q^{32} -2739.35 q^{33} -237.724 q^{34} -8620.18 q^{36} +800.006 q^{37} -286.485 q^{38} -25248.6 q^{39} -6490.42 q^{41} +360.958 q^{42} -12652.4 q^{43} +3869.80 q^{44} +493.160 q^{46} -26957.8 q^{47} +23143.0 q^{48} -2852.93 q^{49} -39874.0 q^{51} +35668.0 q^{52} -9998.59 q^{53} +81.0795 q^{54} -1020.12 q^{56} -48052.8 q^{57} -393.605 q^{58} +21134.3 q^{59} +11087.2 q^{61} +1217.61 q^{62} +31839.3 q^{63} -32656.1 q^{64} -369.736 q^{66} -39791.8 q^{67} +56328.8 q^{68} +82718.7 q^{69} +38238.6 q^{71} -2327.64 q^{72} -15983.0 q^{73} +107.979 q^{74} +67882.8 q^{76} -14293.4 q^{77} -3407.87 q^{78} -44791.2 q^{79} -51897.2 q^{81} -876.028 q^{82} -54122.1 q^{83} -85529.2 q^{84} -1707.73 q^{86} -66020.2 q^{87} +1044.93 q^{88} -23144.7 q^{89} -131743. q^{91} -116854. q^{92} +204232. q^{93} -3638.57 q^{94} +9379.90 q^{96} -10056.4 q^{97} -385.067 q^{98} -32613.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9}+O(q^{10}) \) 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9	\( 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9} - 605 q^{11} + 1605 q^{12} - 1498 q^{13} + 2113 q^{14} + 4883 q^{16} - 3874 q^{17} - 5838 q^{18} + 882 q^{19} + 1092 q^{21} + 1089 q^{22} + 5344 q^{23} - 13119 q^{24} - 4478 q^{26} + 2160 q^{27} - 12565 q^{28} + 5318 q^{29} - 7916 q^{31} - 21385 q^{32} - 18605 q^{34} + 5628 q^{36} + 1788 q^{37} + 34421 q^{38} - 29760 q^{39} + 5854 q^{41} + 46725 q^{42} + 4364 q^{43} - 13915 q^{44} - 33834 q^{46} - 46452 q^{47} + 127545 q^{48} - 34217 q^{49} + 19842 q^{51} - 3222 q^{52} - 4412 q^{53} - 86535 q^{54} + 115575 q^{56} - 137160 q^{57} + 58221 q^{58} + 17896 q^{59} - 35930 q^{61} + 19627 q^{62} - 100980 q^{63} + 14779 q^{64} + 28677 q^{66} - 73136 q^{67} + 83409 q^{68} + 34296 q^{69} + 43612 q^{71} - 372276 q^{72} - 142306 q^{73} - 95609 q^{74} - 6617 q^{76} + 8470 q^{77} - 15750 q^{78} - 46504 q^{79} + 79101 q^{81} - 175798 q^{82} - 81604 q^{83} - 532533 q^{84} - 101788 q^{86} - 219750 q^{87} + 91113 q^{88} + 8664 q^{89} - 203380 q^{91} - 251174 q^{92} + 46470 q^{93} - 71458 q^{94} - 925479 q^{96} + 22230 q^{97} - 59962 q^{98} - 128139 q^{99}+O(q^{100}) \) 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9 - 605 * q^11 + 1605 * q^12 - 1498 * q^13 + 2113 * q^14 + 4883 * q^16 - 3874 * q^17 - 5838 * q^18 + 882 * q^19 + 1092 * q^21 + 1089 * q^22 + 5344 * q^23 - 13119 * q^24 - 4478 * q^26 + 2160 * q^27 - 12565 * q^28 + 5318 * q^29 - 7916 * q^31 - 21385 * q^32 - 18605 * q^34 + 5628 * q^36 + 1788 * q^37 + 34421 * q^38 - 29760 * q^39 + 5854 * q^41 + 46725 * q^42 + 4364 * q^43 - 13915 * q^44 - 33834 * q^46 - 46452 * q^47 + 127545 * q^48 - 34217 * q^49 + 19842 * q^51 - 3222 * q^52 - 4412 * q^53 - 86535 * q^54 + 115575 * q^56 - 137160 * q^57 + 58221 * q^58 + 17896 * q^59 - 35930 * q^61 + 19627 * q^62 - 100980 * q^63 + 14779 * q^64 + 28677 * q^66 - 73136 * q^67 + 83409 * q^68 + 34296 * q^69 + 43612 * q^71 - 372276 * q^72 - 142306 * q^73 - 95609 * q^74 - 6617 * q^76 + 8470 * q^77 - 15750 * q^78 - 46504 * q^79 + 79101 * q^81 - 175798 * q^82 - 81604 * q^83 - 532533 * q^84 - 101788 * q^86 - 219750 * q^87 + 91113 * q^88 + 8664 * q^89 - 203380 * q^91 - 251174 * q^92 + 46470 * q^93 - 71458 * q^94 - 925479 * q^96 + 22230 * q^97 - 59962 * q^98 - 128139 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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