LMFDB - Embedded newform 135.5.c.c.26.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,5,Mod(26,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("135.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	135.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$13.9549450163$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 73x^{4} + 1456x^{2} + 4500 $ x^6 + 73x^4 + 1456x^2 + 4500
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.4
Root			$1.94025i$ of defining polynomial
Character	$\chi$	$=$	135.26
Dual form			135.5.c.c.26.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.94025i q^{2} +12.2354 q^{4} -11.1803i q^{5} -50.3522 q^{7} +54.7837i q^{8} +21.6926 q^{10} +202.956i q^{11} +183.564 q^{13} -97.6958i q^{14} +89.4731 q^{16} +174.739i q^{17} +306.208 q^{19} -136.796i q^{20} -393.785 q^{22} +121.013i q^{23} -125.000 q^{25} +356.160i q^{26} -616.082 q^{28} +1431.82i q^{29} +1115.55 q^{31} +1050.14i q^{32} -339.038 q^{34} +562.955i q^{35} -1645.59 q^{37} +594.120i q^{38} +612.501 q^{40} +510.825i q^{41} -544.361 q^{43} +2483.26i q^{44} -234.794 q^{46} -4157.23i q^{47} +134.346 q^{49} -242.531i q^{50} +2245.99 q^{52} +1814.46i q^{53} +2269.12 q^{55} -2758.48i q^{56} -2778.09 q^{58} -4837.65i q^{59} -3605.33 q^{61} +2164.44i q^{62} -605.961 q^{64} -2052.31i q^{65} +1367.51 q^{67} +2138.01i q^{68} -1092.27 q^{70} +2401.14i q^{71} +2473.24 q^{73} -3192.85i q^{74} +3746.59 q^{76} -10219.3i q^{77} +11646.0 q^{79} -1000.34i q^{80} -991.127 q^{82} -10062.9i q^{83} +1953.65 q^{85} -1056.20i q^{86} -11118.7 q^{88} +8373.34i q^{89} -9242.87 q^{91} +1480.64i q^{92} +8066.04 q^{94} -3423.51i q^{95} +1573.80 q^{97} +260.665i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 50 q^{4} + 4 q^{7} + 50 q^{10} + 566 q^{13} - 638 q^{16} + 1124 q^{19} + 1402 q^{22} - 750 q^{25} - 3584 q^{28} - 2582 q^{31} + 1558 q^{34} + 2040 q^{37} + 750 q^{40} + 1922 q^{43} - 5862 q^{46} + 6782 q^{49}+ \cdots - 27792 q^{97}+O(q^{100}) $ 6 * q - 50 * q^4 + 4 * q^7 + 50 * q^10 + 566 * q^13 - 638 * q^16 + 1124 * q^19 + 1402 * q^22 - 750 * q^25 - 3584 * q^28 - 2582 * q^31 + 1558 * q^34 + 2040 * q^37 + 750 * q^40 + 1922 * q^43 - 5862 * q^46 + 6782 * q^49 - 834 * q^52 - 850 * q^55 - 7234 * q^58 - 11656 * q^61 + 19734 * q^64 + 1288 * q^67 + 13200 * q^70 - 25828 * q^73 - 2520 * q^76 + 25690 * q^79 - 11692 * q^82 + 16250 * q^85 - 32994 * q^88 + 40676 * q^91 + 35266 * q^94 - 27792 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.94025i	0.485062i	0.970144	+	0.242531i	$0.0779776\pi$
$2$	1.94025i	0.485062i	−0.970144	+	0.242531i	$0.922022\pi$
$3$	0	0
$3$	0	0
$4$	12.2354	0.764715
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	−50.3522	−1.02760	−0.513798	−	0.857911i	$-0.671762\pi$
$7$	−50.3522	−1.02760	−0.513798	+	0.857911i	$0.671762\pi$
$8$	54.7837i	0.855996i
$9$	0	0
$10$	21.6926	0.216926
$11$	202.956i	1.67732i	0.544653	+	0.838662i	$0.316662\pi$
$11$	202.956i	1.67732i	−0.544653	+	0.838662i	$0.683338\pi$
$12$	0	0
$13$	183.564	1.08618	0.543090	−	0.839675i	$-0.317254\pi$
$13$	183.564	1.08618	0.543090	+	0.839675i	$0.317254\pi$
$14$	− 97.6958i	− 0.498448i
$15$	0	0
$16$	89.4731	0.349504
$17$	174.739i	0.604635i	0.953207	+	0.302317i	$0.0977603\pi$
$17$	174.739i	0.604635i	−0.953207	+	0.302317i	$0.902240\pi$
$18$	0	0
$19$	306.208	0.848223	0.424111	−	0.905610i	$-0.360587\pi$
$19$	306.208	0.848223	0.424111	+	0.905610i	$0.360587\pi$
$20$	− 136.796i	− 0.341991i
$21$	0	0
$22$	−393.785	−0.813606
$23$	121.013i	0.228757i	0.993437	+	0.114379i	$0.0364877\pi$
$23$	121.013i	0.228757i	−0.993437	+	0.114379i	$0.963512\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	356.160i	0.526864i
$27$	0	0
$28$	−616.082	−0.785818
$29$	1431.82i	1.70252i	0.524743	+	0.851261i	$0.324162\pi$
$29$	1431.82i	1.70252i	−0.524743	+	0.851261i	$0.675838\pi$
$30$	0	0
$31$	1115.55	1.16082	0.580411	−	0.814324i	$-0.302892\pi$
$31$	1115.55	1.16082	0.580411	+	0.814324i	$0.302892\pi$
$32$	1050.14i	1.02553i
$33$	0	0
$34$	−339.038	−0.293285
$35$	562.955i	0.459555i
$36$	0	0
$37$	−1645.59	−1.20204	−0.601019	−	0.799235i	$-0.705238\pi$
$37$	−1645.59	−1.20204	−0.601019	+	0.799235i	$0.705238\pi$
$38$	594.120i	0.411440i
$39$	0	0
$40$	612.501	0.382813
$41$	510.825i	0.303882i	0.988390	+	0.151941i	$0.0485523\pi$
$41$	510.825i	0.303882i	−0.988390	+	0.151941i	$0.951448\pi$
$42$	0	0
$43$	−544.361	−0.294408	−0.147204	−	0.989106i	$-0.547027\pi$
$43$	−544.361	−0.294408	−0.147204	+	0.989106i	$0.547027\pi$
$44$	2483.26i	1.28267i
$45$	0	0
$46$	−234.794	−0.110961
$47$	− 4157.23i	− 1.88195i	−0.338478	−	0.940974i	$-0.609912\pi$
$47$	− 4157.23i	− 1.88195i	0.338478	−	0.940974i	$-0.390088\pi$
$48$	0	0
$49$	134.346	0.0559544
$50$	− 242.531i	− 0.0970124i
$51$	0	0
$52$	2245.99	0.830618
$53$	1814.46i	0.645946i	0.946408	+	0.322973i	$0.104682\pi$
$53$	1814.46i	0.645946i	−0.946408	+	0.322973i	$0.895318\pi$
$54$	0	0
$55$	2269.12	0.750122
$56$	− 2758.48i	− 0.879618i
$57$	0	0
$58$	−2778.09	−0.825828
$59$	− 4837.65i	− 1.38973i	−0.719140	−	0.694866i	$-0.755464\pi$
$59$	− 4837.65i	− 1.38973i	0.719140	−	0.694866i	$-0.244536\pi$
$60$	0	0
$61$	−3605.33	−0.968915	−0.484457	−	0.874815i	$-0.660983\pi$
$61$	−3605.33	−0.968915	−0.484457	+	0.874815i	$0.660983\pi$
$62$	2164.44i	0.563070i
$63$	0	0
$64$	−605.961	−0.147940
$65$	− 2052.31i	− 0.485754i
$66$	0	0
$67$	1367.51	0.304635	0.152317	−	0.988332i	$-0.451326\pi$
$67$	1367.51	0.304635	0.152317	+	0.988332i	$0.451326\pi$
$68$	2138.01i	0.462373i
$69$	0	0
$70$	−1092.27	−0.222913
$71$	2401.14i	0.476322i	0.971226	+	0.238161i	$0.0765446\pi$
$71$	2401.14i	0.476322i	−0.971226	+	0.238161i	$0.923455\pi$
$72$	0	0
$73$	2473.24	0.464109	0.232055	−	0.972703i	$-0.425455\pi$
$73$	2473.24	0.464109	0.232055	+	0.972703i	$0.425455\pi$
$74$	− 3192.85i	− 0.583062i
$75$	0	0
$76$	3746.59	0.648649
$77$	− 10219.3i	− 1.72361i
$78$	0	0
$79$	11646.0	1.86604	0.933022	−	0.359820i	$-0.117162\pi$
$79$	11646.0	1.86604	0.933022	+	0.359820i	$0.117162\pi$
$80$	− 1000.34i	− 0.156303i
$81$	0	0
$82$	−991.127	−0.147401
$83$	− 10062.9i	− 1.46071i	−0.683066	−	0.730357i	$-0.739354\pi$
$83$	− 10062.9i	− 1.46071i	0.683066	−	0.730357i	$-0.260646\pi$
$84$	0	0
$85$	1953.65	0.270401
$86$	− 1056.20i	− 0.142806i
$87$	0	0
$88$	−11118.7	−1.43578
$89$	8373.34i	1.05711i	0.848900	+	0.528553i	$0.177265\pi$
$89$	8373.34i	1.05711i	−0.848900	+	0.528553i	$0.822735\pi$
$90$	0	0
$91$	−9242.87	−1.11615
$92$	1480.64i	0.174934i
$93$	0	0
$94$	8066.04	0.912862
$95$	− 3423.51i	− 0.379337i
$96$	0	0
$97$	1573.80	0.167266	0.0836328	−	0.996497i	$-0.473348\pi$
$97$	1573.80	0.167266	0.0836328	+	0.996497i	$0.473348\pi$
$98$	260.665i	0.0271413i
$99$	0	0
$100$	−1529.43	−0.152943
$101$	− 11677.7i	− 1.14476i	−0.819988	−	0.572380i	$-0.806020\pi$
$101$	− 11677.7i	− 1.14476i	0.819988	−	0.572380i	$-0.193980\pi$
$102$	0	0
$103$	−13464.4	−1.26915	−0.634576	−	0.772861i	$-0.718825\pi$
$103$	−13464.4	−1.26915	−0.634576	+	0.772861i	$0.718825\pi$
$104$	10056.3i	0.929765i
$105$	0	0
$106$	−3520.50	−0.313324
$107$	− 4631.07i	− 0.404495i	−0.979334	−	0.202248i	$-0.935175\pi$
$107$	− 4631.07i	− 0.404495i	0.979334	−	0.202248i	$-0.0648246\pi$
$108$	0	0
$109$	18127.8	1.52578	0.762890	−	0.646528i	$-0.223780\pi$
$109$	18127.8	1.52578	0.762890	+	0.646528i	$0.223780\pi$
$110$	4402.65i	0.363855i
$111$	0	0
$112$	−4505.17	−0.359149
$113$	− 12304.8i	− 0.963647i	−0.876268	−	0.481823i	$-0.839975\pi$
$113$	− 12304.8i	− 0.963647i	0.876268	−	0.481823i	$-0.160025\pi$
$114$	0	0
$115$	1352.96	0.102303
$116$	17519.0i	1.30194i
$117$	0	0
$118$	9386.25	0.674106
$119$	− 8798.52i	− 0.621321i
$120$	0	0
$121$	−26550.2	−1.81341
$122$	− 6995.23i	− 0.469984i
$123$	0	0
$124$	13649.2	0.887697
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	−9955.91	−0.617267	−0.308634	−	0.951181i	$-0.599872\pi$
$127$	−9955.91	−0.617267	−0.308634	+	0.951181i	$0.599872\pi$
$128$	15626.5i	0.953767i
$129$	0	0
$130$	3981.99	0.235621
$131$	− 155.782i	− 0.00907767i	−0.999990	−	0.00453884i	$-0.998555\pi$
$131$	− 155.782i	− 0.00907767i	0.999990	−	0.00453884i	$-0.00144476\pi$
$132$	0	0
$133$	−15418.3	−0.871630
$134$	2653.30i	0.147767i
$135$	0	0
$136$	−9572.88	−0.517565
$137$	23957.9i	1.27646i	0.769844	+	0.638232i	$0.220334\pi$
$137$	23957.9i	1.27646i	−0.769844	+	0.638232i	$0.779666\pi$
$138$	0	0
$139$	8399.21	0.434719	0.217360	−	0.976092i	$-0.430256\pi$
$139$	8399.21	0.434719	0.217360	+	0.976092i	$0.430256\pi$
$140$	6888.00i	0.351429i
$141$	0	0
$142$	−4658.81	−0.231046
$143$	37255.5i	1.82187i
$144$	0	0
$145$	16008.2	0.761391
$146$	4798.69i	0.225122i
$147$	0	0
$148$	−20134.5	−0.919216
$149$	388.709i	0.0175086i	0.999962	+	0.00875432i	$0.00278662\pi$
$149$	388.709i	0.0175086i	−0.999962	+	0.00875432i	$0.997213\pi$
$150$	0	0
$151$	27587.9	1.20994	0.604972	−	0.796247i	$-0.293184\pi$
$151$	27587.9	1.20994	0.604972	+	0.796247i	$0.293184\pi$
$152$	16775.2i	0.726075i
$153$	0	0
$154$	19828.0	0.836058
$155$	− 12472.2i	− 0.519135i
$156$	0	0
$157$	34802.8	1.41194	0.705968	−	0.708244i	$-0.250512\pi$
$157$	34802.8	1.41194	0.705968	+	0.708244i	$0.250512\pi$
$158$	22596.1i	0.905146i
$159$	0	0
$160$	11740.9	0.458630
$161$	− 6093.25i	− 0.235070i
$162$	0	0
$163$	−37147.8	−1.39816	−0.699081	−	0.715043i	$-0.746407\pi$
$163$	−37147.8	−1.39816	−0.699081	+	0.715043i	$0.746407\pi$
$164$	6250.17i	0.232383i
$165$	0	0
$166$	19524.4	0.708536
$167$	17254.1i	0.618672i	0.950953	+	0.309336i	$0.100107\pi$
$167$	17254.1i	0.618672i	−0.950953	+	0.309336i	$0.899893\pi$
$168$	0	0
$169$	5134.86	0.179786
$170$	3790.56i	0.131161i
$171$	0	0
$172$	−6660.50	−0.225139
$173$	− 16749.1i	− 0.559629i	−0.960054	−	0.279815i	$-0.909727\pi$
$173$	− 16749.1i	− 0.559629i	0.960054	−	0.279815i	$-0.0902730\pi$
$174$	0	0
$175$	6294.03	0.205519
$176$	18159.1i	0.586232i
$177$	0	0
$178$	−16246.3	−0.512762
$179$	− 37065.5i	− 1.15681i	−0.815748	−	0.578407i	$-0.803675\pi$
$179$	− 37065.5i	− 1.15681i	0.815748	−	0.578407i	$-0.196325\pi$
$180$	0	0
$181$	−25651.5	−0.782990	−0.391495	−	0.920180i	$-0.628042\pi$
$181$	−25651.5	−0.782990	−0.391495	+	0.920180i	$0.628042\pi$
$182$	− 17933.5i	− 0.541404i
$183$	0	0
$184$	−6629.52	−0.195815
$185$	18398.2i	0.537567i
$186$	0	0
$187$	−35464.5	−1.01417
$188$	− 50865.5i	− 1.43915i
$189$	0	0
$190$	6642.46	0.184002
$191$	− 11801.5i	− 0.323498i	−0.986832	−	0.161749i	$-0.948286\pi$
$191$	− 11801.5i	− 0.323498i	0.986832	−	0.161749i	$-0.0517135\pi$
$192$	0	0
$193$	−30954.0	−0.831002	−0.415501	−	0.909593i	$-0.636394\pi$
$193$	−30954.0	−0.831002	−0.415501	+	0.909593i	$0.636394\pi$
$194$	3053.56i	0.0811341i
$195$	0	0
$196$	1643.79	0.0427892
$197$	− 12462.4i	− 0.321120i	−0.987026	−	0.160560i	$-0.948670\pi$
$197$	− 12462.4i	− 0.321120i	0.987026	−	0.160560i	$-0.0513301\pi$
$198$	0	0
$199$	68252.0	1.72349	0.861746	−	0.507340i	$-0.169371\pi$
$199$	68252.0	1.72349	0.861746	+	0.507340i	$0.169371\pi$
$200$	− 6847.97i	− 0.171199i
$201$	0	0
$202$	22657.6	0.555280
$203$	− 72095.4i	− 1.74951i
$204$	0	0
$205$	5711.20	0.135900
$206$	− 26124.3i	− 0.615617i
$207$	0	0
$208$	16424.1	0.379624
$209$	62146.9i	1.42274i
$210$	0	0
$211$	−21887.4	−0.491621	−0.245810	−	0.969318i	$-0.579054\pi$
$211$	−21887.4	−0.491621	−0.245810	+	0.969318i	$0.579054\pi$
$212$	22200.7i	0.493964i
$213$	0	0
$214$	8985.42	0.196205
$215$	6086.14i	0.131663i
$216$	0	0
$217$	−56170.4	−1.19286
$218$	35172.4i	0.740098i
$219$	0	0
$220$	27763.7	0.573629
$221$	32075.9i	0.656742i
$222$	0	0
$223$	5919.40	0.119033	0.0595166	−	0.998227i	$-0.481044\pi$
$223$	5919.40	0.119033	0.0595166	+	0.998227i	$0.481044\pi$
$224$	− 52876.9i	− 1.05383i
$225$	0	0
$226$	23874.4	0.467428
$227$	55687.9i	1.08071i	0.841437	+	0.540355i	$0.181710\pi$
$227$	55687.9i	1.08071i	−0.841437	+	0.540355i	$0.818290\pi$
$228$	0	0
$229$	6312.99	0.120383	0.0601914	−	0.998187i	$-0.480829\pi$
$229$	6312.99	0.120383	0.0601914	+	0.998187i	$0.480829\pi$
$230$	2625.08i	0.0496234i
$231$	0	0
$232$	−78440.5	−1.45735
$233$	− 84453.4i	− 1.55563i	−0.628495	−	0.777814i	$-0.716329\pi$
$233$	− 84453.4i	− 1.55563i	0.628495	−	0.777814i	$-0.283671\pi$
$234$	0	0
$235$	−46479.2	−0.841633
$236$	− 59190.8i	− 1.06275i
$237$	0	0
$238$	17071.3	0.301379
$239$	− 106001.i	− 1.85572i	−0.372930	−	0.927860i	$-0.621647\pi$
$239$	− 106001.i	− 1.85572i	0.372930	−	0.927860i	$-0.378353\pi$
$240$	0	0
$241$	−22592.3	−0.388979	−0.194490	−	0.980905i	$-0.562305\pi$
$241$	−22592.3	−0.388979	−0.194490	+	0.980905i	$0.562305\pi$
$242$	− 51513.9i	− 0.879618i
$243$	0	0
$244$	−44112.8	−0.740944
$245$	− 1502.04i	− 0.0250236i
$246$	0	0
$247$	56208.9	0.921322
$248$	61113.9i	0.993658i
$249$	0	0
$250$	−2711.58	−0.0433852
$251$	− 6641.52i	− 0.105419i	−0.998610	−	0.0527096i	$-0.983214\pi$
$251$	− 6641.52i	− 0.105419i	0.998610	−	0.0527096i	$-0.0167858\pi$
$252$	0	0
$253$	−24560.2	−0.383700
$254$	− 19316.9i	− 0.299413i
$255$	0	0
$256$	−40014.7	−0.610576
$257$	− 63640.5i	− 0.963534i	−0.876299	−	0.481767i	$-0.839995\pi$
$257$	− 63640.5i	− 0.963534i	0.876299	−	0.481767i	$-0.160005\pi$
$258$	0	0
$259$	82859.0	1.23521
$260$	− 25110.9i	− 0.371464i
$261$	0	0
$262$	302.256	0.00440323
$263$	63237.1i	0.914241i	0.889405	+	0.457120i	$0.151119\pi$
$263$	63237.1i	0.914241i	−0.889405	+	0.457120i	$0.848881\pi$
$264$	0	0
$265$	20286.3	0.288876
$266$	− 29915.3i	− 0.422795i
$267$	0	0
$268$	16732.0	0.232959
$269$	3204.84i	0.0442896i	0.999755	+	0.0221448i	$0.00704949\pi$
$269$	3204.84i	0.0442896i	−0.999755	+	0.0221448i	$0.992951\pi$
$270$	0	0
$271$	38026.4	0.517781	0.258891	−	0.965907i	$-0.416643\pi$
$271$	38026.4	0.517781	0.258891	+	0.965907i	$0.416643\pi$
$272$	15634.5i	0.211322i
$273$	0	0
$274$	−46484.3	−0.619164
$275$	− 25369.5i	− 0.335465i
$276$	0	0
$277$	118697.	1.54696	0.773481	−	0.633819i	$-0.218514\pi$
$277$	118697.	1.54696	0.773481	+	0.633819i	$0.218514\pi$
$278$	16296.5i	0.210866i
$279$	0	0
$280$	−30840.8	−0.393377
$281$	79375.0i	1.00524i	0.864506	+	0.502622i	$0.167631\pi$
$281$	79375.0i	1.00524i	−0.864506	+	0.502622i	$0.832369\pi$
$282$	0	0
$283$	−10423.0	−0.130143	−0.0650716	−	0.997881i	$-0.520728\pi$
$283$	−10423.0	−0.130143	−0.0650716	+	0.997881i	$0.520728\pi$
$284$	29379.0i	0.364251i
$285$	0	0
$286$	−72284.9	−0.883722
$287$	− 25721.2i	− 0.312268i
$288$	0	0
$289$	52987.1	0.634417
$290$	31060.0i	0.369322i
$291$	0	0
$292$	30261.2	0.354911
$293$	108970.i	1.26932i	0.772792	+	0.634659i	$0.218859\pi$
$293$	108970.i	1.26932i	−0.772792	+	0.634659i	$0.781141\pi$
$294$	0	0
$295$	−54086.6	−0.621507
$296$	− 90151.5i	− 1.02894i
$297$	0	0
$298$	−754.192	−0.00849277
$299$	22213.6i	0.248471i
$300$	0	0
$301$	27409.8	0.302533
$302$	53527.4i	0.586897i
$303$	0	0
$304$	27397.4	0.296457
$305$	40308.8i	0.433312i
$306$	0	0
$307$	29250.9	0.310357	0.155179	−	0.987886i	$-0.450405\pi$
$307$	29250.9	0.310357	0.155179	+	0.987886i	$0.450405\pi$
$308$	− 125038.i	− 1.31807i
$309$	0	0
$310$	24199.2	0.251813
$311$	93436.3i	0.966039i	0.875609	+	0.483020i	$0.160460\pi$
$311$	93436.3i	0.966039i	−0.875609	+	0.483020i	$0.839540\pi$
$312$	0	0
$313$	135052.	1.37852	0.689258	−	0.724516i	$-0.257937\pi$
$313$	135052.	1.37852	0.689258	+	0.724516i	$0.257937\pi$
$314$	67526.0i	0.684876i
$315$	0	0
$316$	142494.	1.42699
$317$	167909.i	1.67092i	0.549549	+	0.835462i	$0.314800\pi$
$317$	167909.i	1.67092i	−0.549549	+	0.835462i	$0.685200\pi$
$318$	0	0
$319$	−290597.	−2.85568
$320$	6774.85i	0.0661607i
$321$	0	0
$322$	11822.4	0.114024
$323$	53506.7i	0.512865i
$324$	0	0
$325$	−22945.5	−0.217236
$326$	− 72075.8i	− 0.678195i
$327$	0	0
$328$	−27984.9	−0.260122
$329$	209326.i	1.93388i
$330$	0	0
$331$	−159004.	−1.45129	−0.725644	−	0.688071i	$-0.758458\pi$
$331$	−159004.	−1.45129	−0.725644	+	0.688071i	$0.758458\pi$
$332$	− 123123.i	− 1.11703i
$333$	0	0
$334$	−33477.3	−0.300094
$335$	− 15289.2i	− 0.136237i
$336$	0	0
$337$	195916.	1.72508	0.862541	−	0.505987i	$-0.168872\pi$
$337$	195916.	1.72508	0.862541	+	0.505987i	$0.168872\pi$
$338$	9962.90i	0.0872072i
$339$	0	0
$340$	23903.7	0.206780
$341$	226408.i	1.94707i
$342$	0	0
$343$	114131.	0.970098
$344$	− 29822.1i	− 0.252012i
$345$	0	0
$346$	32497.5	0.271455
$347$	− 145963.i	− 1.21223i	−0.795378	−	0.606113i	$-0.792728\pi$
$347$	− 145963.i	− 1.21223i	0.795378	−	0.606113i	$-0.207272\pi$
$348$	0	0
$349$	−217782.	−1.78801	−0.894006	−	0.448055i	$-0.852117\pi$
$349$	−217782.	−1.78801	−0.894006	+	0.448055i	$0.852117\pi$
$350$	12212.0i	0.0996896i
$351$	0	0
$352$	−213132.	−1.72014
$353$	410.870i	0.00329727i	0.999999	+	0.00164864i	$0.000524777\pi$
$353$	410.870i	0.00329727i	−0.999999	+	0.00164864i	$0.999475\pi$
$354$	0	0
$355$	26845.6	0.213018
$356$	102451.i	0.808385i
$357$	0	0
$358$	71916.2	0.561126
$359$	147600.i	1.14524i	0.819821	+	0.572620i	$0.194073\pi$
$359$	147600.i	1.14524i	−0.819821	+	0.572620i	$0.805927\pi$
$360$	0	0
$361$	−36557.4	−0.280518
$362$	− 49770.3i	− 0.379799i
$363$	0	0
$364$	−113091.	−0.853540
$365$	− 27651.6i	− 0.207556i
$366$	0	0
$367$	−103326.	−0.767145	−0.383572	−	0.923511i	$-0.625306\pi$
$367$	−103326.	−0.767145	−0.383572	+	0.923511i	$0.625306\pi$
$368$	10827.4i	0.0799516i
$369$	0	0
$370$	−35697.1	−0.260753
$371$	− 91362.2i	− 0.663771i
$372$	0	0
$373$	48216.2	0.346558	0.173279	−	0.984873i	$-0.444564\pi$
$373$	48216.2	0.346558	0.173279	+	0.984873i	$0.444564\pi$
$374$	− 68809.8i	− 0.491934i
$375$	0	0
$376$	227748.	1.61094
$377$	262831.i	1.84924i
$378$	0	0
$379$	173643.	1.20887	0.604433	−	0.796656i	$-0.293400\pi$
$379$	173643.	1.20887	0.604433	+	0.796656i	$0.293400\pi$
$380$	− 41888.2i	− 0.290084i
$381$	0	0
$382$	22897.9	0.156917
$383$	− 195296.i	− 1.33136i	−0.746238	−	0.665679i	$-0.768142\pi$
$383$	− 195296.i	− 1.33136i	0.746238	−	0.665679i	$-0.231858\pi$
$384$	0	0
$385$	−114255.	−0.770823
$386$	− 60058.4i	− 0.403087i
$387$	0	0
$388$	19256.2	0.127910
$389$	39786.7i	0.262929i	0.991321	+	0.131465i	$0.0419679\pi$
$389$	39786.7i	0.262929i	−0.991321	+	0.131465i	$0.958032\pi$
$390$	0	0
$391$	−21145.7	−0.138315
$392$	7360.00i	0.0478967i
$393$	0	0
$394$	24180.1	0.155763
$395$	− 130206.i	− 0.834520i
$396$	0	0
$397$	−59732.1	−0.378989	−0.189495	−	0.981882i	$-0.560685\pi$
$397$	−59732.1	−0.378989	−0.189495	+	0.981882i	$0.560685\pi$
$398$	132426.i	0.836000i
$399$	0	0
$400$	−11184.1	−0.0699008
$401$	− 233569.i	− 1.45253i	−0.687414	−	0.726266i	$-0.741254\pi$
$401$	− 233569.i	− 1.45253i	0.687414	−	0.726266i	$-0.258746\pi$
$402$	0	0
$403$	204775.	1.26086
$404$	− 142882.i	− 0.875416i
$405$	0	0
$406$	139883.	0.848618
$407$	− 333982.i	− 2.01620i
$408$	0	0
$409$	−331803.	−1.98350	−0.991752	−	0.128168i	$-0.959090\pi$
$409$	−331803.	−1.98350	−0.991752	+	0.128168i	$0.959090\pi$
$410$	11081.1i	0.0659199i
$411$	0	0
$412$	−164743.	−0.970539
$413$	243587.i	1.42808i
$414$	0	0
$415$	−112506.	−0.653251
$416$	192768.i	1.11391i
$417$	0	0
$418$	−120580.	−0.690119
$419$	− 207488.i	− 1.18186i	−0.806723	−	0.590930i	$-0.798761\pi$
$419$	− 207488.i	− 1.18186i	0.806723	−	0.590930i	$-0.201239\pi$
$420$	0	0
$421$	190580.	1.07526	0.537631	−	0.843181i	$-0.319320\pi$
$421$	190580.	1.07526	0.537631	+	0.843181i	$0.319320\pi$
$422$	− 42467.1i	− 0.238466i
$423$	0	0
$424$	−99403.0	−0.552927
$425$	− 21842.4i	− 0.120927i
$426$	0	0
$427$	181536.	0.995653
$428$	− 56663.2i	− 0.309324i
$429$	0	0
$430$	−11808.6	−0.0638649
$431$	43553.5i	0.234460i	0.993105	+	0.117230i	$0.0374015\pi$
$431$	43553.5i	0.234460i	−0.993105	+	0.117230i	$0.962599\pi$
$432$	0	0
$433$	123115.	0.656650	0.328325	−	0.944565i	$-0.393516\pi$
$433$	123115.	0.656650	0.328325	+	0.944565i	$0.393516\pi$
$434$	− 108984.i	− 0.578609i
$435$	0	0
$436$	221802.	1.16679
$437$	37055.1i	0.194037i
$438$	0	0
$439$	125357.	0.650459	0.325229	−	0.945635i	$-0.394558\pi$
$439$	125357.	0.650459	0.325229	+	0.945635i	$0.394558\pi$
$440$	124311.i	0.642101i
$441$	0	0
$442$	−62235.2	−0.318560
$443$	− 67781.5i	− 0.345385i	−0.984976	−	0.172693i	$-0.944753\pi$
$443$	− 67781.5i	− 0.345385i	0.984976	−	0.172693i	$-0.0552468\pi$
$444$	0	0
$445$	93616.8	0.472752
$446$	11485.1i	0.0577385i
$447$	0	0
$448$	30511.5	0.152022
$449$	− 11399.8i	− 0.0565462i	−0.999600	−	0.0282731i	$-0.990999\pi$
$449$	− 11399.8i	− 0.0565462i	0.999600	−	0.0282731i	$-0.00900081\pi$
$450$	0	0
$451$	−103675.	−0.509708
$452$	− 150555.i	− 0.736915i
$453$	0	0
$454$	−108048.	−0.524211
$455$	103338.i	0.499159i
$456$	0	0
$457$	−173541.	−0.830939	−0.415470	−	0.909607i	$-0.636383\pi$
$457$	−173541.	−0.830939	−0.415470	+	0.909607i	$0.636383\pi$
$458$	12248.8i	0.0583931i
$459$	0	0
$460$	16554.1	0.0782329
$461$	− 207824.i	− 0.977900i	−0.872312	−	0.488950i	$-0.837380\pi$
$461$	− 207824.i	− 0.977900i	0.872312	−	0.488950i	$-0.162620\pi$
$462$	0	0
$463$	295939.	1.38051	0.690256	−	0.723565i	$-0.257498\pi$
$463$	295939.	1.38051	0.690256	+	0.723565i	$0.257498\pi$
$464$	128109.i	0.595038i
$465$	0	0
$466$	163861.	0.754575
$467$	38962.8i	0.178655i	0.996002	+	0.0893277i	$0.0284718\pi$
$467$	38962.8i	0.178655i	−0.996002	+	0.0893277i	$0.971528\pi$
$468$	0	0
$469$	−68856.9	−0.313042
$470$	− 90181.1i	− 0.408244i
$471$	0	0
$472$	265025.	1.18960
$473$	− 110481.i	− 0.493818i
$474$	0	0
$475$	−38276.0	−0.169645
$476$	− 107654.i	− 0.475133i
$477$	0	0
$478$	205667.	0.900138
$479$	42930.4i	0.187109i	0.995614	+	0.0935543i	$0.0298229\pi$
$479$	42930.4i	0.187109i	−0.995614	+	0.0935543i	$0.970177\pi$
$480$	0	0
$481$	−302071.	−1.30563
$482$	− 43834.7i	− 0.188679i
$483$	0	0
$484$	−324853.	−1.38675
$485$	− 17595.6i	− 0.0748034i
$486$	0	0
$487$	−242477.	−1.02238	−0.511191	−	0.859467i	$-0.670796\pi$
$487$	−242477.	−1.02238	−0.511191	+	0.859467i	$0.670796\pi$
$488$	− 197514.i	− 0.829387i
$489$	0	0
$490$	2914.33	0.0121380
$491$	− 423092.i	− 1.75498i	−0.479598	−	0.877488i	$-0.659217\pi$
$491$	− 423092.i	− 1.75498i	0.479598	−	0.877488i	$-0.340783\pi$
$492$	0	0
$493$	−250196.	−1.02940
$494$	109059.i	0.446898i
$495$	0	0
$496$	99811.6	0.405712
$497$	− 120903.i	− 0.489467i
$498$	0	0
$499$	−152654.	−0.613065	−0.306533	−	0.951860i	$-0.599169\pi$
$499$	−152654.	−0.613065	−0.306533	+	0.951860i	$0.599169\pi$
$500$	17099.5i	0.0683982i
$501$	0	0
$502$	12886.2	0.0511348
$503$	− 334692.i	− 1.32285i	−0.750013	−	0.661423i	$-0.769953\pi$
$503$	− 334692.i	− 1.32285i	0.750013	−	0.661423i	$-0.230047\pi$
$504$	0	0
$505$	−130561.	−0.511953
$506$	− 47652.9i	− 0.186118i
$507$	0	0
$508$	−121815.	−0.472034
$509$	− 63575.3i	− 0.245388i	−0.992445	−	0.122694i	$-0.960847\pi$
$509$	− 63575.3i	− 0.245388i	0.992445	−	0.122694i	$-0.0391533\pi$
$510$	0	0
$511$	−124533.	−0.476917
$512$	172386.i	0.657600i
$513$	0	0
$514$	123478.	0.467374
$515$	150537.i	0.567582i
$516$	0	0
$517$	843734.	3.15664
$518$	160767.i	0.599153i
$519$	0	0
$520$	112433.	0.415804
$521$	412565.i	1.51991i	0.649977	+	0.759954i	$0.274778\pi$
$521$	412565.i	1.51991i	−0.649977	+	0.759954i	$0.725222\pi$
$522$	0	0
$523$	436558.	1.59602	0.798011	−	0.602643i	$-0.205886\pi$
$523$	436558.	1.59602	0.798011	+	0.602643i	$0.205886\pi$
$524$	− 1906.06i	− 0.00694183i
$525$	0	0
$526$	−122696.	−0.443463
$527$	194930.i	0.701873i
$528$	0	0
$529$	265197.	0.947670
$530$	39360.4i	0.140123i
$531$	0	0
$532$	−188649.	−0.666549
$533$	93769.3i	0.330070i
$534$	0	0
$535$	−51776.9	−0.180896
$536$	74917.1i	0.260766i
$537$	0	0
$538$	−6218.18	−0.0214832
$539$	27266.4i	0.0938536i
$540$	0	0
$541$	−386165.	−1.31941	−0.659703	−	0.751526i	$-0.729318\pi$
$541$	−386165.	−1.31941	−0.659703	+	0.751526i	$0.729318\pi$
$542$	73780.6i	0.251156i
$543$	0	0
$544$	−183501.	−0.620069
$545$	− 202675.i	− 0.682350i
$546$	0	0
$547$	99445.7	0.332362	0.166181	−	0.986095i	$-0.446856\pi$
$547$	99445.7	0.332362	0.166181	+	0.986095i	$0.446856\pi$
$548$	293136.i	0.976131i
$549$	0	0
$550$	49223.1	0.162721
$551$	438436.i	1.44412i
$552$	0	0
$553$	−586401.	−1.91754
$554$	230301.i	0.750372i
$555$	0	0
$556$	102768.	0.332436
$557$	220057.i	0.709292i	0.935001	+	0.354646i	$0.115399\pi$
$557$	220057.i	0.709292i	−0.935001	+	0.354646i	$0.884601\pi$
$558$	0	0
$559$	−99925.3	−0.319780
$560$	50369.3i	0.160616i
$561$	0	0
$562$	−154007.	−0.487605
$563$	− 73858.7i	− 0.233016i	−0.993190	−	0.116508i	$-0.962830\pi$
$563$	− 73858.7i	− 0.233016i	0.993190	−	0.116508i	$-0.0371700\pi$
$564$	0	0
$565$	−137572.	−0.430956
$566$	− 20223.3i	− 0.0631275i
$567$	0	0
$568$	−131543.	−0.407730
$569$	− 517855.i	− 1.59950i	−0.600335	−	0.799749i	$-0.704966\pi$
$569$	− 517855.i	− 1.59950i	0.600335	−	0.799749i	$-0.295034\pi$
$570$	0	0
$571$	−30998.5	−0.0950754	−0.0475377	−	0.998869i	$-0.515137\pi$
$571$	−30998.5	−0.0950754	−0.0475377	+	0.998869i	$0.515137\pi$
$572$	455838.i	1.39321i
$573$	0	0
$574$	49905.5	0.151469
$575$	− 15126.6i	− 0.0457514i
$576$	0	0
$577$	42846.7	0.128696	0.0643481	−	0.997928i	$-0.479503\pi$
$577$	42846.7	0.128696	0.0643481	+	0.997928i	$0.479503\pi$
$578$	102808.i	0.307731i
$579$	0	0
$580$	195868.	0.582247
$581$	506687.i	1.50102i
$582$	0	0
$583$	−368256.	−1.08346
$584$	135493.i	0.397276i
$585$	0	0
$586$	−211428.	−0.615698
$587$	329934.i	0.957528i	0.877944	+	0.478764i	$0.158915\pi$
$587$	329934.i	0.957528i	−0.877944	+	0.478764i	$0.841085\pi$
$588$	0	0
$589$	341590.	0.984635
$590$	− 104941.i	− 0.301469i
$591$	0	0
$592$	−147236.	−0.420117
$593$	167618.i	0.476664i	0.971184	+	0.238332i	$0.0766006\pi$
$593$	167618.i	0.476664i	−0.971184	+	0.238332i	$0.923399\pi$
$594$	0	0
$595$	−98370.5	−0.277863
$596$	4756.03i	0.0133891i
$597$	0	0
$598$	−43099.9	−0.120524
$599$	− 227547.i	− 0.634188i	−0.948394	−	0.317094i	$-0.897293\pi$
$599$	− 227547.i	− 0.634188i	0.948394	−	0.317094i	$-0.102707\pi$
$600$	0	0
$601$	46077.2	0.127567	0.0637833	−	0.997964i	$-0.479683\pi$
$601$	46077.2	0.127567	0.0637833	+	0.997964i	$0.479683\pi$
$602$	53181.8i	0.146747i
$603$	0	0
$604$	337550.	0.925262
$605$	296840.i	0.810983i
$606$	0	0
$607$	277108.	0.752095	0.376047	−	0.926600i	$-0.377283\pi$
$607$	277108.	0.752095	0.376047	+	0.926600i	$0.377283\pi$
$608$	321562.i	0.869875i
$609$	0	0
$610$	−78209.1	−0.210183
$611$	− 763118.i	− 2.04413i
$612$	0	0
$613$	315023.	0.838343	0.419171	−	0.907907i	$-0.362321\pi$
$613$	315023.	0.838343	0.419171	+	0.907907i	$0.362321\pi$
$614$	56753.9i	0.150543i
$615$	0	0
$616$	559851.	1.47540
$617$	− 282133.i	− 0.741112i	−0.928810	−	0.370556i	$-0.879167\pi$
$617$	− 282133.i	− 0.741112i	0.928810	−	0.370556i	$-0.120833\pi$
$618$	0	0
$619$	288014.	0.751679	0.375839	−	0.926685i	$-0.377354\pi$
$619$	288014.	0.751679	0.375839	+	0.926685i	$0.377354\pi$
$620$	− 152603.i	− 0.396990i
$621$	0	0
$622$	−181290.	−0.468589
$623$	− 421616.i	− 1.08628i
$624$	0	0
$625$	15625.0	0.0400000
$626$	262034.i	0.668665i
$627$	0	0
$628$	425828.	1.07973
$629$	− 287549.i	− 0.726793i
$630$	0	0
$631$	−77926.3	−0.195716	−0.0978578	−	0.995200i	$-0.531199\pi$
$631$	−77926.3	−0.195716	−0.0978578	+	0.995200i	$0.531199\pi$
$632$	638010.i	1.59733i
$633$	0	0
$634$	−325786.	−0.810501
$635$	111310.i	0.276050i
$636$	0	0
$637$	24661.2	0.0607765
$638$	− 563830.i	− 1.38518i
$639$	0	0
$640$	174710.	0.426538
$641$	339129.i	0.825372i	0.910873	+	0.412686i	$0.135409\pi$
$641$	339129.i	0.825372i	−0.910873	+	0.412686i	$0.864591\pi$
$642$	0	0
$643$	−178034.	−0.430606	−0.215303	−	0.976547i	$-0.569074\pi$
$643$	−178034.	−0.430606	−0.215303	+	0.976547i	$0.569074\pi$
$644$	− 74553.6i	− 0.179762i
$645$	0	0
$646$	−103816.	−0.248771
$647$	− 778198.i	− 1.85901i	−0.368809	−	0.929505i	$-0.620234\pi$
$647$	− 778198.i	− 1.85901i	0.368809	−	0.929505i	$-0.379766\pi$
$648$	0	0
$649$	981832.	2.33103
$650$	− 44520.0i	− 0.105373i
$651$	0	0
$652$	−454519.	−1.06920
$653$	570456.i	1.33782i	0.743346	+	0.668908i	$0.233238\pi$
$653$	570456.i	1.33782i	−0.743346	+	0.668908i	$0.766762\pi$
$654$	0	0
$655$	−1741.70	−0.00405966
$656$	45705.1i	0.106208i
$657$	0	0
$658$	−406143.	−0.938053
$659$	493094.i	1.13543i	0.823227	+	0.567713i	$0.192172\pi$
$659$	493094.i	1.13543i	−0.823227	+	0.567713i	$0.807828\pi$
$660$	0	0
$661$	−654379.	−1.49771	−0.748853	−	0.662736i	$-0.769395\pi$
$661$	−654379.	−1.49771	−0.748853	+	0.662736i	$0.769395\pi$
$662$	− 308508.i	− 0.703964i
$663$	0	0
$664$	551281.	1.25036
$665$	172382.i	0.389805i
$666$	0	0
$667$	−173268.	−0.389464
$668$	211112.i	0.473108i
$669$	0	0
$670$	29664.8	0.0660833
$671$	− 731724.i	− 1.62518i
$672$	0	0
$673$	1427.22	0.00315109	0.00157554	−	0.999999i	$-0.499498\pi$
$673$	1427.22	0.00315109	0.00157554	+	0.999999i	$0.499498\pi$
$674$	380125.i	0.836771i
$675$	0	0
$676$	62827.3	0.137485
$677$	− 397819.i	− 0.867976i	−0.900919	−	0.433988i	$-0.857106\pi$
$677$	− 397819.i	− 0.867976i	0.900919	−	0.433988i	$-0.142894\pi$
$678$	0	0
$679$	−79244.4	−0.171881
$680$	107028.i	0.231462i
$681$	0	0
$682$	−439287.	−0.944450
$683$	− 137477.i	− 0.294707i	−0.989084	−	0.147353i	$-0.952925\pi$
$683$	− 137477.i	− 0.294707i	0.989084	−	0.147353i	$-0.0470755\pi$
$684$	0	0
$685$	267858.	0.570852
$686$	221442.i	0.470557i
$687$	0	0
$688$	−48705.7	−0.102897
$689$	333070.i	0.701613i
$690$	0	0
$691$	−307963.	−0.644975	−0.322488	−	0.946574i	$-0.604519\pi$
$691$	−307963.	−0.644975	−0.322488	+	0.946574i	$0.604519\pi$
$692$	− 204933.i	− 0.427957i
$693$	0	0
$694$	283204.	0.588005
$695$	− 93906.0i	− 0.194412i
$696$	0	0
$697$	−89261.3	−0.183737
$698$	− 422550.i	− 0.867296i
$699$	0	0
$700$	77010.2	0.157164
$701$	− 779311.i	− 1.58590i	−0.609289	−	0.792948i	$-0.708545\pi$
$701$	− 779311.i	− 1.58590i	0.609289	−	0.792948i	$-0.291455\pi$
$702$	0	0
$703$	−503893.	−1.01959
$704$	− 122984.i	− 0.248143i
$705$	0	0
$706$	−797.189	−0.00159938
$707$	587998.i	1.17635i
$708$	0	0
$709$	350809.	0.697876	0.348938	−	0.937146i	$-0.386542\pi$
$709$	350809.	0.697876	0.348938	+	0.937146i	$0.386542\pi$
$710$	52087.0i	0.103327i
$711$	0	0
$712$	−458723.	−0.904878
$713$	134995.i	0.265546i
$714$	0	0
$715$	416529.	0.814767
$716$	− 453512.i	− 0.884633i
$717$	0	0
$718$	−286380.	−0.555512
$719$	− 840579.i	− 1.62600i	−0.582263	−	0.813001i	$-0.697833\pi$
$719$	− 840579.i	− 1.62600i	0.582263	−	0.813001i	$-0.302167\pi$
$720$	0	0
$721$	677964.	1.30418
$722$	− 70930.5i	− 0.136069i
$723$	0	0
$724$	−313858.	−0.598764
$725$	− 178978.i	− 0.340504i
$726$	0	0
$727$	1.01200e6	1.91475	0.957374	−	0.288851i	$-0.0932732\pi$
$727$	1.01200e6	1.91475	0.957374	+	0.288851i	$0.0932732\pi$
$728$	− 506359.i	− 0.955423i
$729$	0	0
$730$	53651.0	0.100677
$731$	− 95121.4i	− 0.178010i
$732$	0	0
$733$	−421087.	−0.783725	−0.391863	−	0.920024i	$-0.628169\pi$
$733$	−421087.	−0.783725	−0.391863	+	0.920024i	$0.628169\pi$
$734$	− 200478.i	− 0.372113i
$735$	0	0
$736$	−127080.	−0.234597
$737$	277544.i	0.510971i
$738$	0	0
$739$	−282437.	−0.517170	−0.258585	−	0.965989i	$-0.583256\pi$
$739$	−282437.	−0.517170	−0.258585	+	0.965989i	$0.583256\pi$
$740$	225111.i	0.411086i
$741$	0	0
$742$	177265.	0.321970
$743$	− 369786.i	− 0.669842i	−0.942246	−	0.334921i	$-0.891290\pi$
$743$	− 369786.i	− 0.669842i	0.942246	−	0.334921i	$-0.108710\pi$
$744$	0	0
$745$	4345.90	0.00783010
$746$	93551.4i	0.168102i
$747$	0	0
$748$	−433923.	−0.775550
$749$	233185.i	0.415658i
$750$	0	0
$751$	86095.1	0.152651	0.0763253	−	0.997083i	$-0.475681\pi$
$751$	86095.1	0.152651	0.0763253	+	0.997083i	$0.475681\pi$
$752$	− 371960.i	− 0.657749i
$753$	0	0
$754$	−509958.	−0.896998
$755$	− 308442.i	− 0.541103i
$756$	0	0
$757$	−189172.	−0.330115	−0.165058	−	0.986284i	$-0.552781\pi$
$757$	−189172.	−0.330115	−0.165058	+	0.986284i	$0.552781\pi$
$758$	336910.i	0.586374i
$759$	0	0
$760$	187553.	0.324711
$761$	473747.i	0.818045i	0.912524	+	0.409022i	$0.134130\pi$
$761$	473747.i	0.818045i	−0.912524	+	0.409022i	$0.865870\pi$
$762$	0	0
$763$	−912775.	−1.56789
$764$	− 144397.i	− 0.247384i
$765$	0	0
$766$	378922.	0.645791
$767$	− 888021.i	− 1.50950i
$768$	0	0
$769$	783721.	1.32528	0.662642	−	0.748936i	$-0.269435\pi$
$769$	783721.	1.32528	0.662642	+	0.748936i	$0.269435\pi$
$770$	− 221683.i	− 0.373897i
$771$	0	0
$772$	−378736.	−0.635480
$773$	393774.i	0.659004i	0.944155	+	0.329502i	$0.106881\pi$
$773$	393774.i	0.659004i	−0.944155	+	0.329502i	$0.893119\pi$
$774$	0	0
$775$	−139444.	−0.232164
$776$	86218.7i	0.143179i
$777$	0	0
$778$	−77196.0	−0.127537
$779$	156419.i	0.257759i
$780$	0	0
$781$	−487326.	−0.798947
$782$	− 41027.8i	− 0.0670911i
$783$	0	0
$784$	12020.4	0.0195563
$785$	− 389107.i	− 0.631437i
$786$	0	0
$787$	−262539.	−0.423881	−0.211940	−	0.977283i	$-0.567978\pi$
$787$	−262539.	−0.423881	−0.211940	+	0.977283i	$0.567978\pi$
$788$	− 152482.i	− 0.245566i
$789$	0	0
$790$	252632.	0.404794
$791$	619574.i	0.990240i
$792$	0	0
$793$	−661810.	−1.05242
$794$	− 115895.i	− 0.183833i
$795$	0	0
$796$	835094.	1.31798
$797$	433451.i	0.682375i	0.939995	+	0.341187i	$0.110829\pi$
$797$	433451.i	0.682375i	−0.939995	+	0.341187i	$0.889171\pi$
$798$	0	0
$799$	726431.	1.13789
$800$	− 131267.i	− 0.205105i
$801$	0	0
$802$	453181.	0.704568
$803$	501959.i	0.778461i
$804$	0	0
$805$	−68124.6	−0.105127
$806$	397314.i	0.611595i
$807$	0	0
$808$	639748.	0.979910
$809$	− 394934.i	− 0.603431i	−0.953398	−	0.301716i	$-0.902441\pi$
$809$	− 394934.i	− 0.603431i	0.953398	−	0.301716i	$-0.0975593\pi$
$810$	0	0
$811$	−795473.	−1.20944	−0.604719	−	0.796439i	$-0.706715\pi$
$811$	−795473.	−1.20944	−0.604719	+	0.796439i	$0.706715\pi$
$812$	− 882119.i	− 1.33787i
$813$	0	0
$814$	648008.	0.977984
$815$	415325.i	0.625277i
$816$	0	0
$817$	−166688.	−0.249724
$818$	− 643779.i	− 0.962122i
$819$	0	0
$820$	69879.0	0.103925
$821$	− 288613.i	− 0.428183i	−0.976814	−	0.214092i	$-0.931321\pi$
$821$	− 288613.i	− 0.428183i	0.976814	−	0.214092i	$-0.0686791\pi$
$822$	0	0
$823$	90831.5	0.134103	0.0670513	−	0.997750i	$-0.478641\pi$
$823$	90831.5	0.134103	0.0670513	+	0.997750i	$0.478641\pi$
$824$	− 737632.i	− 1.08639i
$825$	0	0
$826$	−472618.	−0.692708
$827$	− 442580.i	− 0.647114i	−0.946209	−	0.323557i	$-0.895121\pi$
$827$	− 442580.i	− 0.647114i	0.946209	−	0.323557i	$-0.104879\pi$
$828$	0	0
$829$	−259965.	−0.378273	−0.189137	−	0.981951i	$-0.560569\pi$
$829$	−259965.	−0.378273	−0.189137	+	0.981951i	$0.560569\pi$
$830$	− 218290.i	− 0.316867i
$831$	0	0
$832$	−111233.	−0.160689
$833$	23475.6i	0.0338320i
$834$	0	0
$835$	192907.	0.276679
$836$	760394.i	1.08799i
$837$	0	0
$838$	402579.	0.573275
$839$	413062.i	0.586802i	0.955989	+	0.293401i	$0.0947871\pi$
$839$	413062.i	0.586802i	−0.955989	+	0.293401i	$0.905213\pi$
$840$	0	0
$841$	−1.34283e6	−1.89858
$842$	369773.i	0.521568i
$843$	0	0
$844$	−267803.	−0.375950
$845$	− 57409.5i	− 0.0804026i
$846$	0	0
$847$	1.33686e6	1.86346
$848$	162345.i	0.225761i
$849$	0	0
$850$	42379.7	0.0586571
$851$	− 199137.i	− 0.274975i
$852$	0	0
$853$	48283.6	0.0663593	0.0331796	−	0.999449i	$-0.489437\pi$
$853$	48283.6	0.0663593	0.0331796	+	0.999449i	$0.489437\pi$
$854$	352226.i	0.482953i
$855$	0	0
$856$	253707.	0.346246
$857$	639554.i	0.870794i	0.900238	+	0.435397i	$0.143392\pi$
$857$	639554.i	0.870794i	−0.900238	+	0.435397i	$0.856608\pi$
$858$	0	0
$859$	391575.	0.530675	0.265337	−	0.964156i	$-0.414517\pi$
$859$	391575.	0.530675	0.265337	+	0.964156i	$0.414517\pi$
$860$	74466.7i	0.100685i
$861$	0	0
$862$	−84504.6	−0.113728
$863$	1.20719e6i	1.62089i	0.585816	+	0.810444i	$0.300774\pi$
$863$	1.20719e6i	1.62089i	−0.585816	+	0.810444i	$0.699226\pi$
$864$	0	0
$865$	−187261.	−0.250274
$866$	238873.i	0.318516i
$867$	0	0
$868$	−687269.	−0.912195
$869$	2.36362e6i	3.12996i
$870$	0	0
$871$	251025.	0.330888
$872$	993108.i	1.30606i
$873$	0	0
$874$	−71896.0	−0.0941200
$875$	− 70369.4i	− 0.0919110i
$876$	0	0
$877$	−413297.	−0.537357	−0.268678	−	0.963230i	$-0.586587\pi$
$877$	−413297.	−0.537357	−0.268678	+	0.963230i	$0.586587\pi$
$878$	243224.i	0.315513i
$879$	0	0
$880$	203025.	0.262171
$881$	− 789955.i	− 1.01777i	−0.860834	−	0.508886i	$-0.830057\pi$
$881$	− 789955.i	− 1.01777i	0.860834	−	0.508886i	$-0.169943\pi$
$882$	0	0
$883$	558493.	0.716302	0.358151	−	0.933664i	$-0.383407\pi$
$883$	558493.	0.716302	0.358151	+	0.933664i	$0.383407\pi$
$884$	392463.i	0.502220i
$885$	0	0
$886$	131513.	0.167533
$887$	− 124383.i	− 0.158093i	−0.996871	−	0.0790465i	$-0.974812\pi$
$887$	− 124383.i	− 0.158093i	0.996871	−	0.0790465i	$-0.0251876\pi$
$888$	0	0
$889$	501302.	0.634302
$890$	181640.i	0.229314i
$891$	0	0
$892$	72426.5	0.0910265
$893$	− 1.27298e6i	− 1.59631i
$894$	0	0
$895$	−414405.	−0.517343
$896$	− 786830.i	− 0.980088i
$897$	0	0
$898$	22118.4	0.0274284
$899$	1.59727e6i	1.97632i
$900$	0	0
$901$	−317058.	−0.390561
$902$	− 201155.i	− 0.247240i
$903$	0	0
$904$	674103.	0.824877
$905$	286793.i	0.350164i
$906$	0	0
$907$	−683001.	−0.830246	−0.415123	−	0.909765i	$-0.636261\pi$
$907$	−683001.	−0.830246	−0.415123	+	0.909765i	$0.636261\pi$
$908$	681366.i	0.826435i
$909$	0	0
$910$	−200502.	−0.242123
$911$	683114.i	0.823107i	0.911386	+	0.411553i	$0.135014\pi$
$911$	683114.i	0.823107i	−0.911386	+	0.411553i	$0.864986\pi$
$912$	0	0
$913$	2.04232e6	2.45009
$914$	− 336712.i	− 0.403057i
$915$	0	0
$916$	77242.2	0.0920585
$917$	7843.97i	0.00932819i
$918$	0	0
$919$	247081.	0.292556	0.146278	−	0.989244i	$-0.453271\pi$
$919$	247081.	0.292556	0.146278	+	0.989244i	$0.453271\pi$
$920$	74120.3i	0.0875712i
$921$	0	0
$922$	403230.	0.474342
$923$	440764.i	0.517372i
$924$	0	0
$925$	205699.	0.240407
$926$	574195.i	0.669633i
$927$	0	0
$928$	−1.50361e6	−1.74598
$929$	− 377632.i	− 0.437560i	−0.975774	−	0.218780i	$-0.929792\pi$
$929$	− 377632.i	− 0.437560i	0.975774	−	0.218780i	$-0.0702077\pi$
$930$	0	0
$931$	41138.0	0.0474618
$932$	− 1.03333e6i	− 1.18961i
$933$	0	0
$934$	−75597.4	−0.0866589
$935$	396505.i	0.453550i
$936$	0	0
$937$	−987404.	−1.12465	−0.562323	−	0.826918i	$-0.690092\pi$
$937$	−987404.	−1.12465	−0.562323	+	0.826918i	$0.690092\pi$
$938$	− 133599.i	− 0.151845i
$939$	0	0
$940$	−568693.	−0.643610
$941$	795104.i	0.897934i	0.893548	+	0.448967i	$0.148208\pi$
$941$	795104.i	0.897934i	−0.893548	+	0.448967i	$0.851792\pi$
$942$	0	0
$943$	−61816.3	−0.0695151
$944$	− 432840.i	− 0.485717i
$945$	0	0
$946$	214361.	0.239532
$947$	1.54212e6i	1.71956i	0.510665	+	0.859780i	$0.329399\pi$
$947$	1.54212e6i	1.71956i	−0.510665	+	0.859780i	$0.670601\pi$
$948$	0	0
$949$	453998.	0.504106
$950$	− 74265.0i	− 0.0822881i
$951$	0	0
$952$	482016.	0.531848
$953$	706894.i	0.778339i	0.921166	+	0.389169i	$0.127238\pi$
$953$	706894.i	0.778339i	−0.921166	+	0.389169i	$0.872762\pi$
$954$	0	0
$955$	−131945.	−0.144673
$956$	− 1.29696e6i	− 1.41910i
$957$	0	0
$958$	−83295.6	−0.0907593
$959$	− 1.20634e6i	− 1.31169i
$960$	0	0
$961$	320929.	0.347506
$962$	− 586093.i	− 0.633310i
$963$	0	0
$964$	−276427.	−0.297458
$965$	346076.i	0.371636i
$966$	0	0
$967$	−619740.	−0.662761	−0.331380	−	0.943497i	$-0.607514\pi$
$967$	−619740.	−0.662761	−0.331380	+	0.943497i	$0.607514\pi$
$968$	− 1.45452e6i	− 1.55227i
$969$	0	0
$970$	34139.9	0.0362843
$971$	777633.i	0.824776i	0.911008	+	0.412388i	$0.135305\pi$
$971$	777633.i	0.824776i	−0.911008	+	0.412388i	$0.864695\pi$
$972$	0	0
$973$	−422919.	−0.446716
$974$	− 470466.i	− 0.495919i
$975$	0	0
$976$	−322580.	−0.338640
$977$	− 1.35355e6i	− 1.41802i	−0.705196	−	0.709012i	$-0.749141\pi$
$977$	− 1.35355e6i	− 1.41802i	0.705196	−	0.709012i	$-0.250859\pi$
$978$	0	0
$979$	−1.69942e6	−1.77311
$980$	− 18378.1i	− 0.0191359i
$981$	0	0
$982$	820902.	0.851272
$983$	1.41672e6i	1.46614i	0.680151	+	0.733072i	$0.261914\pi$
$983$	1.41672e6i	1.46614i	−0.680151	+	0.733072i	$0.738086\pi$
$984$	0	0
$985$	−139333.	−0.143609
$986$	− 485441.i	− 0.499325i
$987$	0	0
$988$	687741.	0.704549
$989$	− 65874.6i	− 0.0673481i
$990$	0	0
$991$	−697784.	−0.710516	−0.355258	−	0.934768i	$-0.615607\pi$
$991$	−697784.	−0.710516	−0.355258	+	0.934768i	$0.615607\pi$
$992$	1.17148e6i	1.19045i
$993$	0	0
$994$	234581.	0.237422
$995$	− 763081.i	− 0.770769i
$996$	0	0
$997$	537542.	0.540782	0.270391	−	0.962751i	$-0.412847\pi$
$997$	537542.	0.540782	0.270391	+	0.962751i	$0.412847\pi$
$998$	− 296186.i	− 0.297375i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.5.c.c.26.4	yes	6
3.2	odd	2	inner	135.5.c.c.26.3	✓	6
5.2	odd	4		675.5.d.l.674.5		12
5.3	odd	4		675.5.d.l.674.8		12
5.4	even	2		675.5.c.p.26.3		6
15.2	even	4		675.5.d.l.674.7		12
15.8	even	4		675.5.d.l.674.6		12
15.14	odd	2		675.5.c.p.26.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.5.c.c.26.3	✓	6	3.2	odd	2	inner
135.5.c.c.26.4	yes	6	1.1	even	1	trivial
675.5.c.p.26.3		6	5.4	even	2
675.5.c.p.26.4		6	15.14	odd	2
675.5.d.l.674.5		12	5.2	odd	4
675.5.d.l.674.6		12	15.8	even	4
675.5.d.l.674.7		12	15.2	even	4
675.5.d.l.674.8		12	5.3	odd	4

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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	135.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.94025i q^{2} +12.2354 q^{4} -11.1803i q^{5} -50.3522 q^{7} +54.7837i q^{8} +21.6926 q^{10} +202.956i q^{11} +183.564 q^{13} -97.6958i q^{14} +89.4731 q^{16} +174.739i q^{17} +306.208 q^{19} -136.796i q^{20} -393.785 q^{22} +121.013i q^{23} -125.000 q^{25} +356.160i q^{26} -616.082 q^{28} +1431.82i q^{29} +1115.55 q^{31} +1050.14i q^{32} -339.038 q^{34} +562.955i q^{35} -1645.59 q^{37} +594.120i q^{38} +612.501 q^{40} +510.825i q^{41} -544.361 q^{43} +2483.26i q^{44} -234.794 q^{46} -4157.23i q^{47} +134.346 q^{49} -242.531i q^{50} +2245.99 q^{52} +1814.46i q^{53} +2269.12 q^{55} -2758.48i q^{56} -2778.09 q^{58} -4837.65i q^{59} -3605.33 q^{61} +2164.44i q^{62} -605.961 q^{64} -2052.31i q^{65} +1367.51 q^{67} +2138.01i q^{68} -1092.27 q^{70} +2401.14i q^{71} +2473.24 q^{73} -3192.85i q^{74} +3746.59 q^{76} -10219.3i q^{77} +11646.0 q^{79} -1000.34i q^{80} -991.127 q^{82} -10062.9i q^{83} +1953.65 q^{85} -1056.20i q^{86} -11118.7 q^{88} +8373.34i q^{89} -9242.87 q^{91} +1480.64i q^{92} +8066.04 q^{94} -3423.51i q^{95} +1573.80 q^{97} +260.665i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 50 q^{4} + 4 q^{7} + 50 q^{10} + 566 q^{13} - 638 q^{16} + 1124 q^{19} + 1402 q^{22} - 750 q^{25} - 3584 q^{28} - 2582 q^{31} + 1558 q^{34} + 2040 q^{37} + 750 q^{40} + 1922 q^{43} - 5862 q^{46} + 6782 q^{49}+ \cdots - 27792 q^{97}+O(q^{100}) \) 6 * q - 50 * q^4 + 4 * q^7 + 50 * q^10 + 566 * q^13 - 638 * q^16 + 1124 * q^19 + 1402 * q^22 - 750 * q^25 - 3584 * q^28 - 2582 * q^31 + 1558 * q^34 + 2040 * q^37 + 750 * q^40 + 1922 * q^43 - 5862 * q^46 + 6782 * q^49 - 834 * q^52 - 850 * q^55 - 7234 * q^58 - 11656 * q^61 + 19734 * q^64 + 1288 * q^67 + 13200 * q^70 - 25828 * q^73 - 2520 * q^76 + 25690 * q^79 - 11692 * q^82 + 16250 * q^85 - 32994 * q^88 + 40676 * q^91 + 35266 * q^94 - 27792 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more