LMFDB - Embedded newform 735.4.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,4,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	735.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:	$43.3664038542$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 35x^{2} + 19x + 196 $ x^4 - x^3 - 35x^2 + 19x + 196
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.92771$ of defining polynomial
Character	$\chi$	$=$	735.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.92771 q^{2} -3.00000 q^{3} +0.571487 q^{4} +5.00000 q^{5} -8.78313 q^{6} -21.7485 q^{8} +9.00000 q^{9} +14.6386 q^{10} +45.3855 q^{11} -1.71446 q^{12} +24.1478 q^{13} -15.0000 q^{15} -68.2453 q^{16} +7.89162 q^{17} +26.3494 q^{18} -151.949 q^{19} +2.85744 q^{20} +132.876 q^{22} +39.3342 q^{23} +65.2456 q^{24} +25.0000 q^{25} +70.6977 q^{26} -27.0000 q^{27} +200.572 q^{29} -43.9157 q^{30} -15.0636 q^{31} -25.8142 q^{32} -136.156 q^{33} +23.1044 q^{34} +5.14339 q^{36} +119.753 q^{37} -444.864 q^{38} -72.4434 q^{39} -108.743 q^{40} +32.1480 q^{41} +358.400 q^{43} +25.9372 q^{44} +45.0000 q^{45} +115.159 q^{46} +171.051 q^{47} +204.736 q^{48} +73.1928 q^{50} -23.6749 q^{51} +13.8002 q^{52} +387.811 q^{53} -79.0482 q^{54} +226.927 q^{55} +455.848 q^{57} +587.218 q^{58} +504.229 q^{59} -8.57231 q^{60} +328.392 q^{61} -44.1017 q^{62} +470.386 q^{64} +120.739 q^{65} -398.627 q^{66} +519.007 q^{67} +4.50996 q^{68} -118.003 q^{69} +1173.62 q^{71} -195.737 q^{72} -113.604 q^{73} +350.603 q^{74} -75.0000 q^{75} -86.8371 q^{76} -212.093 q^{78} -805.369 q^{79} -341.227 q^{80} +81.0000 q^{81} +94.1201 q^{82} -1350.22 q^{83} +39.4581 q^{85} +1049.29 q^{86} -601.717 q^{87} -987.068 q^{88} +658.726 q^{89} +131.747 q^{90} +22.4790 q^{92} +45.1907 q^{93} +500.788 q^{94} -759.747 q^{95} +77.4426 q^{96} +886.401 q^{97} +408.469 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 12 q^{3} + 39 q^{4} + 20 q^{5} - 3 q^{6} + 33 q^{8} + 36 q^{9} + 5 q^{10} + 60 q^{11} - 117 q^{12} + 14 q^{13} - 60 q^{15} + 283 q^{16} + 46 q^{17} + 9 q^{18} + 40 q^{19} + 195 q^{20} - 14 q^{22}+ \cdots + 540 q^{99}+O(q^{100}) $ 4 * q + q^2 - 12 * q^3 + 39 * q^4 + 20 * q^5 - 3 * q^6 + 33 * q^8 + 36 * q^9 + 5 * q^10 + 60 * q^11 - 117 * q^12 + 14 * q^13 - 60 * q^15 + 283 * q^16 + 46 * q^17 + 9 * q^18 + 40 * q^19 + 195 * q^20 - 14 * q^22 + 490 * q^23 - 99 * q^24 + 100 * q^25 - 546 * q^26 - 108 * q^27 - 6 * q^29 - 15 * q^30 + 122 * q^31 + 525 * q^32 - 180 * q^33 + 748 * q^34 + 351 * q^36 + 142 * q^37 - 152 * q^38 - 42 * q^39 + 165 * q^40 + 168 * q^41 + 286 * q^43 + 1214 * q^44 + 180 * q^45 - 148 * q^46 - 306 * q^47 - 849 * q^48 + 25 * q^50 - 138 * q^51 - 1934 * q^52 - 276 * q^53 - 27 * q^54 + 300 * q^55 - 120 * q^57 + 652 * q^58 - 328 * q^59 - 585 * q^60 + 174 * q^61 - 1742 * q^62 + 3615 * q^64 + 70 * q^65 + 42 * q^66 - 994 * q^67 + 132 * q^68 - 1470 * q^69 + 2442 * q^71 + 297 * q^72 + 14 * q^73 - 2424 * q^74 - 300 * q^75 + 5160 * q^76 + 1638 * q^78 - 252 * q^79 + 1415 * q^80 + 324 * q^81 + 2014 * q^82 - 1556 * q^83 + 230 * q^85 + 734 * q^86 + 18 * q^87 - 4486 * q^88 + 600 * q^89 + 45 * q^90 + 5844 * q^92 - 366 * q^93 + 1210 * q^94 + 200 * q^95 - 1575 * q^96 - 238 * q^97 + 540 * q^99

Coefficient data

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

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

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

$2$	2.92771	1.03510	0.517551	−	0.855652i	$-0.326844\pi$
$2$	2.92771	1.03510	0.517551	+	0.855652i	$0.326844\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0.571487	0.0714359
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−8.78313	−0.597616
$7$	0	0
$7$	0	0
$8$	−21.7485	−0.961158
$9$	9.00000	0.333333
$10$	14.6386	0.462912
$11$	45.3855	1.24402	0.622011	−	0.783009i	$-0.286316\pi$
$11$	45.3855	1.24402	0.622011	+	0.783009i	$0.286316\pi$
$12$	−1.71446	−0.0412436
$13$	24.1478	0.515184	0.257592	−	0.966254i	$-0.417071\pi$
$13$	24.1478	0.515184	0.257592	+	0.966254i	$0.417071\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	−68.2453	−1.06633
$17$	7.89162	0.112588	0.0562941	−	0.998414i	$-0.482072\pi$
$17$	7.89162	0.112588	0.0562941	+	0.998414i	$0.482072\pi$
$18$	26.3494	0.345034
$19$	−151.949	−1.83471	−0.917357	−	0.398066i	$-0.869682\pi$
$19$	−151.949	−1.83471	−0.917357	+	0.398066i	$0.869682\pi$
$20$	2.85744	0.0319471
$21$	0	0
$22$	132.876	1.28769
$23$	39.3342	0.356598	0.178299	−	0.983976i	$-0.442941\pi$
$23$	39.3342	0.356598	0.178299	+	0.983976i	$0.442941\pi$
$24$	65.2456	0.554925
$25$	25.0000	0.200000
$26$	70.6977	0.533268
$27$	−27.0000	−0.192450
$28$	0	0
$29$	200.572	1.28432	0.642162	−	0.766569i	$-0.278038\pi$
$29$	200.572	1.28432	0.642162	+	0.766569i	$0.278038\pi$
$30$	−43.9157	−0.267262
$31$	−15.0636	−0.0872740	−0.0436370	−	0.999047i	$-0.513895\pi$
$31$	−15.0636	−0.0872740	−0.0436370	+	0.999047i	$0.513895\pi$
$32$	−25.8142	−0.142605
$33$	−136.156	−0.718236
$34$	23.1044	0.116540
$35$	0	0
$36$	5.14339	0.0238120
$37$	119.753	0.532089	0.266045	−	0.963961i	$-0.414283\pi$
$37$	119.753	0.532089	0.266045	+	0.963961i	$0.414283\pi$
$38$	−444.864	−1.89912
$39$	−72.4434	−0.297442
$40$	−108.743	−0.429843
$41$	32.1480	0.122456	0.0612278	−	0.998124i	$-0.480498\pi$
$41$	32.1480	0.122456	0.0612278	+	0.998124i	$0.480498\pi$
$42$	0	0
$43$	358.400	1.27106	0.635530	−	0.772077i	$-0.280782\pi$
$43$	358.400	1.27106	0.635530	+	0.772077i	$0.280782\pi$
$44$	25.9372	0.0888678
$45$	45.0000	0.149071
$46$	115.159	0.369115
$47$	171.051	0.530859	0.265429	−	0.964130i	$-0.414486\pi$
$47$	171.051	0.530859	0.265429	+	0.964130i	$0.414486\pi$
$48$	204.736	0.615648
$49$	0	0
$50$	73.1928	0.207020
$51$	−23.6749	−0.0650028
$52$	13.8002	0.0368026
$53$	387.811	1.00509	0.502547	−	0.864550i	$-0.332397\pi$
$53$	387.811	1.00509	0.502547	+	0.864550i	$0.332397\pi$
$54$	−79.0482	−0.199205
$55$	226.927	0.556343
$56$	0	0
$57$	455.848	1.05927
$58$	587.218	1.32941
$59$	504.229	1.11263	0.556314	−	0.830972i	$-0.312215\pi$
$59$	504.229	1.11263	0.556314	+	0.830972i	$0.312215\pi$
$60$	−8.57231	−0.0184447
$61$	328.392	0.689284	0.344642	−	0.938734i	$-0.388000\pi$
$61$	328.392	0.689284	0.344642	+	0.938734i	$0.388000\pi$
$62$	−44.1017	−0.0903375
$63$	0	0
$64$	470.386	0.918722
$65$	120.739	0.230397
$66$	−398.627	−0.743448
$67$	519.007	0.946370	0.473185	−	0.880963i	$-0.343104\pi$
$67$	519.007	0.946370	0.473185	+	0.880963i	$0.343104\pi$
$68$	4.50996	0.00804284
$69$	−118.003	−0.205882
$70$	0	0
$71$	1173.62	1.96173	0.980863	−	0.194697i	$-0.0623723\pi$
$71$	1173.62	1.96173	0.980863	+	0.194697i	$0.0623723\pi$
$72$	−195.737	−0.320386
$73$	−113.604	−0.182142	−0.0910708	−	0.995844i	$-0.529029\pi$
$73$	−113.604	−0.182142	−0.0910708	+	0.995844i	$0.529029\pi$
$74$	350.603	0.550767
$75$	−75.0000	−0.115470
$76$	−86.8371	−0.131064
$77$	0	0
$78$	−212.093	−0.307882
$79$	−805.369	−1.14698	−0.573488	−	0.819214i	$-0.694410\pi$
$79$	−805.369	−1.14698	−0.573488	+	0.819214i	$0.694410\pi$
$80$	−341.227	−0.476879
$81$	81.0000	0.111111
$82$	94.1201	0.126754
$83$	−1350.22	−1.78561	−0.892805	−	0.450444i	$-0.851266\pi$
$83$	−1350.22	−1.78561	−0.892805	+	0.450444i	$0.851266\pi$
$84$	0	0
$85$	39.4581	0.0503510
$86$	1049.29	1.31568
$87$	−601.717	−0.741504
$88$	−987.068	−1.19570
$89$	658.726	0.784549	0.392274	−	0.919848i	$-0.371688\pi$
$89$	658.726	0.784549	0.392274	+	0.919848i	$0.371688\pi$
$90$	131.747	0.154304
$91$	0	0
$92$	22.4790	0.0254739
$93$	45.1907	0.0503877
$94$	500.788	0.549493
$95$	−759.747	−0.820509
$96$	77.4426	0.0823329
$97$	886.401	0.927839	0.463920	−	0.885877i	$-0.346443\pi$
$97$	886.401	0.927839	0.463920	+	0.885877i	$0.346443\pi$
$98$	0	0
$99$	408.469	0.414674
$100$	14.2872	0.0142872
$101$	1853.00	1.82555	0.912775	−	0.408464i	$-0.133935\pi$
$101$	1853.00	1.82555	0.912775	+	0.408464i	$0.133935\pi$
$102$	−69.3132	−0.0672846
$103$	−1673.73	−1.60114	−0.800572	−	0.599237i	$-0.795471\pi$
$103$	−1673.73	−1.60114	−0.800572	+	0.599237i	$0.795471\pi$
$104$	−525.179	−0.495173
$105$	0	0
$106$	1135.40	1.04037
$107$	−3.50571	−0.00316738	−0.00158369	−	0.999999i	$-0.500504\pi$
$107$	−3.50571	−0.00316738	−0.00158369	+	0.999999i	$0.500504\pi$
$108$	−15.4302	−0.0137479
$109$	−1503.52	−1.32120	−0.660600	−	0.750738i	$-0.729698\pi$
$109$	−1503.52	−1.32120	−0.660600	+	0.750738i	$0.729698\pi$
$110$	664.378	0.575872
$111$	−359.260	−0.307202
$112$	0	0
$113$	211.621	0.176173	0.0880867	−	0.996113i	$-0.471925\pi$
$113$	211.621	0.176173	0.0880867	+	0.996113i	$0.471925\pi$
$114$	1334.59	1.09646
$115$	196.671	0.159475
$116$	114.625	0.0917468
$117$	217.330	0.171728
$118$	1476.24	1.15168
$119$	0	0
$120$	326.228	0.248170
$121$	728.842	0.547590
$122$	961.437	0.713479
$123$	−96.4441	−0.0706997
$124$	−8.60863	−0.00623450
$125$	125.000	0.0894427
$126$	0	0
$127$	746.863	0.521837	0.260919	−	0.965361i	$-0.415975\pi$
$127$	746.863	0.521837	0.260919	+	0.965361i	$0.415975\pi$
$128$	1583.67	1.09358
$129$	−1075.20	−0.733846
$130$	353.489	0.238485
$131$	225.367	0.150308	0.0751541	−	0.997172i	$-0.476055\pi$
$131$	225.367	0.150308	0.0751541	+	0.997172i	$0.476055\pi$
$132$	−77.8117	−0.0513079
$133$	0	0
$134$	1519.50	0.979590
$135$	−135.000	−0.0860663
$136$	−171.631	−0.108215
$137$	−1827.22	−1.13949	−0.569743	−	0.821823i	$-0.692957\pi$
$137$	−1827.22	−1.13949	−0.569743	+	0.821823i	$0.692957\pi$
$138$	−345.478	−0.213109
$139$	−1656.39	−1.01074	−0.505370	−	0.862903i	$-0.668644\pi$
$139$	−1656.39	−1.01074	−0.505370	+	0.862903i	$0.668644\pi$
$140$	0	0
$141$	−513.153	−0.306491
$142$	3436.01	2.03059
$143$	1095.96	0.640900
$144$	−614.208	−0.355444
$145$	1002.86	0.574367
$146$	−332.600	−0.188535
$147$	0	0
$148$	68.4375	0.0380103
$149$	1116.84	0.614060	0.307030	−	0.951700i	$-0.400665\pi$
$149$	1116.84	0.614060	0.307030	+	0.951700i	$0.400665\pi$
$150$	−219.578	−0.119523
$151$	438.299	0.236214	0.118107	−	0.993001i	$-0.462317\pi$
$151$	438.299	0.236214	0.118107	+	0.993001i	$0.462317\pi$
$152$	3304.67	1.76345
$153$	71.0246	0.0375294
$154$	0	0
$155$	−75.3178	−0.0390301
$156$	−41.4005	−0.0212480
$157$	−1136.54	−0.577743	−0.288872	−	0.957368i	$-0.593280\pi$
$157$	−1136.54	−0.577743	−0.288872	+	0.957368i	$0.593280\pi$
$158$	−2357.89	−1.18724
$159$	−1163.43	−0.580291
$160$	−129.071	−0.0637748
$161$	0	0
$162$	237.145	0.115011
$163$	875.223	0.420569	0.210285	−	0.977640i	$-0.432561\pi$
$163$	875.223	0.420569	0.210285	+	0.977640i	$0.432561\pi$
$164$	18.3722	0.00874773
$165$	−680.782	−0.321205
$166$	−3953.04	−1.84829
$167$	2121.81	0.983178	0.491589	−	0.870827i	$-0.336416\pi$
$167$	2121.81	0.983178	0.491589	+	0.870827i	$0.336416\pi$
$168$	0	0
$169$	−1613.88	−0.734585
$170$	115.522	0.0521184
$171$	−1367.54	−0.611571
$172$	204.821	0.0907993
$173$	519.259	0.228200	0.114100	−	0.993469i	$-0.463602\pi$
$173$	519.259	0.228200	0.114100	+	0.993469i	$0.463602\pi$
$174$	−1761.65	−0.767533
$175$	0	0
$176$	−3097.35	−1.32654
$177$	−1512.69	−0.642376
$178$	1928.56	0.812088
$179$	−432.844	−0.180739	−0.0903696	−	0.995908i	$-0.528805\pi$
$179$	−432.844	−0.180739	−0.0903696	+	0.995908i	$0.528805\pi$
$180$	25.7169	0.0106490
$181$	2791.71	1.14644	0.573221	−	0.819401i	$-0.305694\pi$
$181$	2791.71	1.14644	0.573221	+	0.819401i	$0.305694\pi$
$182$	0	0
$183$	−985.177	−0.397958
$184$	−855.461	−0.342747
$185$	598.766	0.237958
$186$	132.305	0.0521564
$187$	358.165	0.140062
$188$	97.7535	0.0379224
$189$	0	0
$190$	−2224.32	−0.849310
$191$	−3221.32	−1.22035	−0.610173	−	0.792268i	$-0.708900\pi$
$191$	−3221.32	−1.22035	−0.610173	+	0.792268i	$0.708900\pi$
$192$	−1411.16	−0.530425
$193$	−3640.40	−1.35773	−0.678864	−	0.734264i	$-0.737527\pi$
$193$	−3640.40	−1.35773	−0.678864	+	0.734264i	$0.737527\pi$
$194$	2595.13	0.960408
$195$	−362.217	−0.133020
$196$	0	0
$197$	3178.85	1.14966	0.574832	−	0.818271i	$-0.305067\pi$
$197$	3178.85	1.14966	0.574832	+	0.818271i	$0.305067\pi$
$198$	1195.88	0.429230
$199$	3869.55	1.37842	0.689208	−	0.724563i	$-0.257959\pi$
$199$	3869.55	1.37842	0.689208	+	0.724563i	$0.257959\pi$
$200$	−543.713	−0.192232
$201$	−1557.02	−0.546387
$202$	5425.05	1.88963
$203$	0	0
$204$	−13.5299	−0.00464354
$205$	160.740	0.0547638
$206$	−4900.21	−1.65735
$207$	354.008	0.118866
$208$	−1647.97	−0.549358
$209$	−6896.29	−2.28242
$210$	0	0
$211$	4477.70	1.46094	0.730469	−	0.682946i	$-0.239302\pi$
$211$	4477.70	1.46094	0.730469	+	0.682946i	$0.239302\pi$
$212$	221.629	0.0717998
$213$	−3520.85	−1.13260
$214$	−10.2637	−0.00327856
$215$	1792.00	0.568435
$216$	587.210	0.184975
$217$	0	0
$218$	−4401.86	−1.36758
$219$	340.812	0.105160
$220$	129.686	0.0397429
$221$	190.565	0.0580036
$222$	−1051.81	−0.317985
$223$	−305.552	−0.0917546	−0.0458773	−	0.998947i	$-0.514608\pi$
$223$	−305.552	−0.0917546	−0.0458773	+	0.998947i	$0.514608\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	619.564	0.182357
$227$	−6621.96	−1.93619	−0.968094	−	0.250587i	$-0.919376\pi$
$227$	−6621.96	−1.93619	−0.968094	+	0.250587i	$0.919376\pi$
$228$	260.511	0.0756701
$229$	−5240.39	−1.51220	−0.756102	−	0.654454i	$-0.772899\pi$
$229$	−5240.39	−1.51220	−0.756102	+	0.654454i	$0.772899\pi$
$230$	575.796	0.165073
$231$	0	0
$232$	−4362.16	−1.23444
$233$	−2614.11	−0.735005	−0.367502	−	0.930023i	$-0.619787\pi$
$233$	−2614.11	−0.735005	−0.367502	+	0.930023i	$0.619787\pi$
$234$	636.280	0.177756
$235$	855.255	0.237407
$236$	288.161	0.0794816
$237$	2416.11	0.662207
$238$	0	0
$239$	−319.944	−0.0865919	−0.0432960	−	0.999062i	$-0.513786\pi$
$239$	−319.944	−0.0865919	−0.0432960	+	0.999062i	$0.513786\pi$
$240$	1023.68	0.275326
$241$	3429.97	0.916778	0.458389	−	0.888752i	$-0.348427\pi$
$241$	3429.97	0.916778	0.458389	+	0.888752i	$0.348427\pi$
$242$	2133.84	0.566811
$243$	−243.000	−0.0641500
$244$	187.672	0.0492396
$245$	0	0
$246$	−282.360	−0.0731814
$247$	−3669.24	−0.945215
$248$	327.610	0.0838842
$249$	4050.65	1.03092
$250$	365.964	0.0925823
$251$	−5152.46	−1.29570	−0.647849	−	0.761768i	$-0.724331\pi$
$251$	−5152.46	−1.29570	−0.647849	+	0.761768i	$0.724331\pi$
$252$	0	0
$253$	1785.20	0.443615
$254$	2186.60	0.540155
$255$	−118.374	−0.0290702
$256$	873.432	0.213240
$257$	1937.46	0.470254	0.235127	−	0.971965i	$-0.424449\pi$
$257$	1937.46	0.470254	0.235127	+	0.971965i	$0.424449\pi$
$258$	−3147.88	−0.759606
$259$	0	0
$260$	69.0008	0.0164586
$261$	1805.15	0.428108
$262$	659.809	0.155584
$263$	2817.53	0.660594	0.330297	−	0.943877i	$-0.392851\pi$
$263$	2817.53	0.660594	0.330297	+	0.943877i	$0.392851\pi$
$264$	2961.20	0.690339
$265$	1939.05	0.449491
$266$	0	0
$267$	−1976.18	−0.452959
$268$	296.606	0.0676048
$269$	7581.60	1.71843	0.859216	−	0.511612i	$-0.170952\pi$
$269$	7581.60	1.71843	0.859216	+	0.511612i	$0.170952\pi$
$270$	−395.241	−0.0890874
$271$	−2007.56	−0.450002	−0.225001	−	0.974359i	$-0.572239\pi$
$271$	−2007.56	−0.450002	−0.225001	+	0.974359i	$0.572239\pi$
$272$	−538.566	−0.120057
$273$	0	0
$274$	−5349.56	−1.17948
$275$	1134.64	0.248804
$276$	−67.4370	−0.0147074
$277$	522.535	0.113343	0.0566716	−	0.998393i	$-0.481951\pi$
$277$	522.535	0.113343	0.0566716	+	0.998393i	$0.481951\pi$
$278$	−4849.42	−1.04622
$279$	−135.572	−0.0290913
$280$	0	0
$281$	2528.04	0.536691	0.268345	−	0.963323i	$-0.413523\pi$
$281$	2528.04	0.536691	0.268345	+	0.963323i	$0.413523\pi$
$282$	−1502.36	−0.317250
$283$	−2608.02	−0.547812	−0.273906	−	0.961756i	$-0.588316\pi$
$283$	−2608.02	−0.547812	−0.273906	+	0.961756i	$0.588316\pi$
$284$	670.707	0.140138
$285$	2279.24	0.473721
$286$	3208.65	0.663397
$287$	0	0
$288$	−232.328	−0.0475349
$289$	−4850.72	−0.987324
$290$	2936.09	0.594528
$291$	−2659.20	−0.535688
$292$	−64.9233	−0.0130115
$293$	−2830.10	−0.564287	−0.282143	−	0.959372i	$-0.591045\pi$
$293$	−2830.10	−0.564287	−0.282143	+	0.959372i	$0.591045\pi$
$294$	0	0
$295$	2521.15	0.497582
$296$	−2604.46	−0.511422
$297$	−1225.41	−0.239412
$298$	3269.78	0.635614
$299$	949.834	0.183713
$300$	−42.8616	−0.00824871
$301$	0	0
$302$	1283.21	0.244505
$303$	−5559.00	−1.05398
$304$	10369.8	1.95642
$305$	1641.96	0.308257
$306$	207.939	0.0388468
$307$	8752.67	1.62717	0.813585	−	0.581446i	$-0.197513\pi$
$307$	8752.67	1.62717	0.813585	+	0.581446i	$0.197513\pi$
$308$	0	0
$309$	5021.20	0.924421
$310$	−220.509	−0.0404002
$311$	−4245.23	−0.774034	−0.387017	−	0.922073i	$-0.626495\pi$
$311$	−4245.23	−0.774034	−0.387017	+	0.922073i	$0.626495\pi$
$312$	1575.54	0.285889
$313$	−5888.63	−1.06340	−0.531702	−	0.846932i	$-0.678447\pi$
$313$	−5888.63	−1.06340	−0.531702	+	0.846932i	$0.678447\pi$
$314$	−3327.46	−0.598023
$315$	0	0
$316$	−460.258	−0.0819353
$317$	−7074.82	−1.25351	−0.626753	−	0.779218i	$-0.715616\pi$
$317$	−7074.82	−1.25351	−0.626753	+	0.779218i	$0.715616\pi$
$318$	−3406.19	−0.600660
$319$	9103.08	1.59773
$320$	2351.93	0.410865
$321$	10.5171	0.00182869
$322$	0	0
$323$	−1199.13	−0.206567
$324$	46.2905	0.00793733
$325$	603.695	0.103037
$326$	2562.40	0.435332
$327$	4510.55	0.762795
$328$	−699.172	−0.117699
$329$	0	0
$330$	−1993.13	−0.332480
$331$	−10236.4	−1.69983	−0.849916	−	0.526918i	$-0.823347\pi$
$331$	−10236.4	−1.69983	−0.849916	+	0.526918i	$0.823347\pi$
$332$	−771.632	−0.127557
$333$	1077.78	0.177363
$334$	6212.05	1.01769
$335$	2595.04	0.423230
$336$	0	0
$337$	−5579.42	−0.901871	−0.450935	−	0.892557i	$-0.648909\pi$
$337$	−5579.42	−0.901871	−0.450935	+	0.892557i	$0.648909\pi$
$338$	−4724.99	−0.760371
$339$	−634.862	−0.101714
$340$	22.5498	0.00359687
$341$	−683.667	−0.108571
$342$	−4003.77	−0.633039
$343$	0	0
$344$	−7794.68	−1.22169
$345$	−590.013	−0.0920732
$346$	1520.24	0.236210
$347$	9526.56	1.47381	0.736906	−	0.675995i	$-0.236286\pi$
$347$	9526.56	1.47381	0.736906	+	0.675995i	$0.236286\pi$
$348$	−343.874	−0.0529701
$349$	198.263	0.0304092	0.0152046	−	0.999884i	$-0.495160\pi$
$349$	198.263	0.0304092	0.0152046	+	0.999884i	$0.495160\pi$
$350$	0	0
$351$	−651.990	−0.0991472
$352$	−1171.59	−0.177403
$353$	−9280.48	−1.39929	−0.699646	−	0.714489i	$-0.746659\pi$
$353$	−9280.48	−1.39929	−0.699646	+	0.714489i	$0.746659\pi$
$354$	−4428.71	−0.664925
$355$	5868.08	0.877311
$356$	376.454	0.0560450
$357$	0	0
$358$	−1267.24	−0.187083
$359$	−6589.47	−0.968743	−0.484372	−	0.874862i	$-0.660952\pi$
$359$	−6589.47	−0.968743	−0.484372	+	0.874862i	$0.660952\pi$
$360$	−978.684	−0.143281
$361$	16229.6	2.36617
$362$	8173.31	1.18668
$363$	−2186.53	−0.316151
$364$	0	0
$365$	−568.020	−0.0814562
$366$	−2884.31	−0.411927
$367$	10984.5	1.56236	0.781180	−	0.624305i	$-0.214618\pi$
$367$	10984.5	1.56236	0.781180	+	0.624305i	$0.214618\pi$
$368$	−2684.37	−0.380252
$369$	289.332	0.0408185
$370$	1753.01	0.246310
$371$	0	0
$372$	25.8259	0.00359949
$373$	−2190.04	−0.304010	−0.152005	−	0.988380i	$-0.548573\pi$
$373$	−2190.04	−0.304010	−0.152005	+	0.988380i	$0.548573\pi$
$374$	1048.60	0.144979
$375$	−375.000	−0.0516398
$376$	−3720.11	−0.510239
$377$	4843.38	0.661663
$378$	0	0
$379$	2063.29	0.279642	0.139821	−	0.990177i	$-0.455347\pi$
$379$	2063.29	0.279642	0.139821	+	0.990177i	$0.455347\pi$
$380$	−434.186	−0.0586138
$381$	−2240.59	−0.301283
$382$	−9431.08	−1.26318
$383$	−10582.1	−1.41180	−0.705898	−	0.708313i	$-0.749457\pi$
$383$	−10582.1	−1.41180	−0.705898	+	0.708313i	$0.749457\pi$
$384$	−4751.00	−0.631376
$385$	0	0
$386$	−10658.0	−1.40539
$387$	3225.60	0.423686
$388$	506.567	0.0662811
$389$	7301.27	0.951643	0.475822	−	0.879542i	$-0.342151\pi$
$389$	7301.27	0.951643	0.475822	+	0.879542i	$0.342151\pi$
$390$	−1060.47	−0.137689
$391$	310.411	0.0401487
$392$	0	0
$393$	−676.100	−0.0867805
$394$	9306.76	1.19002
$395$	−4026.84	−0.512943
$396$	233.435	0.0296226
$397$	7650.00	0.967110	0.483555	−	0.875314i	$-0.339345\pi$
$397$	7650.00	0.967110	0.483555	+	0.875314i	$0.339345\pi$
$398$	11328.9	1.42680
$399$	0	0
$400$	−1706.13	−0.213267
$401$	7383.41	0.919476	0.459738	−	0.888055i	$-0.347943\pi$
$401$	7383.41	0.919476	0.459738	+	0.888055i	$0.347943\pi$
$402$	−4558.51	−0.565566
$403$	−363.752	−0.0449622
$404$	1058.97	0.130410
$405$	405.000	0.0496904
$406$	0	0
$407$	5435.06	0.661931
$408$	514.894	0.0624780
$409$	13061.0	1.57903	0.789517	−	0.613729i	$-0.210331\pi$
$409$	13061.0	1.57903	0.789517	+	0.613729i	$0.210331\pi$
$410$	470.600	0.0566861
$411$	5481.65	0.657883
$412$	−956.517	−0.114379
$413$	0	0
$414$	1036.43	0.123038
$415$	−6751.08	−0.798549
$416$	−623.356	−0.0734677
$417$	4969.16	0.583551
$418$	−20190.3	−2.36254
$419$	−12329.9	−1.43760	−0.718800	−	0.695217i	$-0.755308\pi$
$419$	−12329.9	−1.43760	−0.718800	+	0.695217i	$0.755308\pi$
$420$	0	0
$421$	−4011.19	−0.464354	−0.232177	−	0.972673i	$-0.574585\pi$
$421$	−4011.19	−0.464354	−0.232177	+	0.972673i	$0.574585\pi$
$422$	13109.4	1.51222
$423$	1539.46	0.176953
$424$	−8434.32	−0.966054
$425$	197.291	0.0225176
$426$	−10308.0	−1.17236
$427$	0	0
$428$	−2.00347	−0.000226265 0
$429$	−3287.88	−0.370024
$430$	5246.46	0.588388
$431$	−12983.2	−1.45099	−0.725495	−	0.688227i	$-0.758389\pi$
$431$	−12983.2	−1.45099	−0.725495	+	0.688227i	$0.758389\pi$
$432$	1842.62	0.205216
$433$	10976.3	1.21822	0.609109	−	0.793087i	$-0.291527\pi$
$433$	10976.3	1.21822	0.609109	+	0.793087i	$0.291527\pi$
$434$	0	0
$435$	−3008.59	−0.331611
$436$	−859.241	−0.0943812
$437$	−5976.81	−0.654255
$438$	997.799	0.108851
$439$	17477.2	1.90009	0.950045	−	0.312112i	$-0.101036\pi$
$439$	17477.2	1.90009	0.950045	+	0.312112i	$0.101036\pi$
$440$	−4935.34	−0.534734
$441$	0	0
$442$	557.920	0.0600397
$443$	2083.78	0.223484	0.111742	−	0.993737i	$-0.464357\pi$
$443$	2083.78	0.223484	0.111742	+	0.993737i	$0.464357\pi$
$444$	−205.312	−0.0219453
$445$	3293.63	0.350861
$446$	−894.568	−0.0949754
$447$	−3350.51	−0.354527
$448$	0	0
$449$	−1874.17	−0.196988	−0.0984938	−	0.995138i	$-0.531402\pi$
$449$	−1874.17	−0.196988	−0.0984938	+	0.995138i	$0.531402\pi$
$450$	658.735	0.0690068
$451$	1459.05	0.152337
$452$	120.939	0.0125851
$453$	−1314.90	−0.136378
$454$	−19387.2	−2.00415
$455$	0	0
$456$	−9914.02	−1.01813
$457$	−6648.72	−0.680556	−0.340278	−	0.940325i	$-0.610521\pi$
$457$	−6648.72	−0.680556	−0.340278	+	0.940325i	$0.610521\pi$
$458$	−15342.3	−1.56529
$459$	−213.074	−0.0216676
$460$	112.395	0.0113923
$461$	15529.0	1.56888	0.784442	−	0.620202i	$-0.212949\pi$
$461$	15529.0	1.56888	0.784442	+	0.620202i	$0.212949\pi$
$462$	0	0
$463$	2365.89	0.237478	0.118739	−	0.992926i	$-0.462115\pi$
$463$	2365.89	0.237478	0.118739	+	0.992926i	$0.462115\pi$
$464$	−13688.1	−1.36952
$465$	225.953	0.0225341
$466$	−7653.36	−0.760805
$467$	−17310.3	−1.71525	−0.857626	−	0.514274i	$-0.828062\pi$
$467$	−17310.3	−1.71525	−0.857626	+	0.514274i	$0.828062\pi$
$468$	124.201	0.0122675
$469$	0	0
$470$	2503.94	0.245741
$471$	3409.62	0.333560
$472$	−10966.2	−1.06941
$473$	16266.2	1.58122
$474$	7073.66	0.685452
$475$	−3798.73	−0.366943
$476$	0	0
$477$	3490.30	0.335031
$478$	−936.704	−0.0896314
$479$	2363.33	0.225435	0.112717	−	0.993627i	$-0.464045\pi$
$479$	2363.33	0.225435	0.112717	+	0.993627i	$0.464045\pi$
$480$	387.213	0.0368204
$481$	2891.78	0.274124
$482$	10041.9	0.948959
$483$	0	0
$484$	416.524	0.0391176
$485$	4432.01	0.414942
$486$	−711.434	−0.0664018
$487$	−17815.5	−1.65770	−0.828849	−	0.559473i	$-0.811004\pi$
$487$	−17815.5	−1.65770	−0.828849	+	0.559473i	$0.811004\pi$
$488$	−7142.05	−0.662511
$489$	−2625.67	−0.242816
$490$	0	0
$491$	1968.10	0.180894	0.0904469	−	0.995901i	$-0.471170\pi$
$491$	1968.10	0.180894	0.0904469	+	0.995901i	$0.471170\pi$
$492$	−55.1166	−0.00505050
$493$	1582.84	0.144600
$494$	−10742.5	−0.978394
$495$	2042.35	0.185448
$496$	1028.02	0.0930632
$497$	0	0
$498$	11859.1	1.06711
$499$	17118.5	1.53573	0.767863	−	0.640614i	$-0.221320\pi$
$499$	17118.5	1.53573	0.767863	+	0.640614i	$0.221320\pi$
$500$	71.4359	0.00638942
$501$	−6365.44	−0.567638
$502$	−15084.9	−1.34118
$503$	−7025.91	−0.622803	−0.311401	−	0.950278i	$-0.600798\pi$
$503$	−7025.91	−0.622803	−0.311401	+	0.950278i	$0.600798\pi$
$504$	0	0
$505$	9265.00	0.816410
$506$	5226.55	0.459187
$507$	4841.65	0.424113
$508$	426.823	0.0372779
$509$	5155.61	0.448955	0.224478	−	0.974479i	$-0.427932\pi$
$509$	5155.61	0.448955	0.224478	+	0.974479i	$0.427932\pi$
$510$	−346.566	−0.0300906
$511$	0	0
$512$	−10112.2	−0.872851
$513$	4102.63	0.353091
$514$	5672.32	0.486761
$515$	−8368.67	−0.716053
$516$	−614.464	−0.0524230
$517$	7763.23	0.660400
$518$	0	0
$519$	−1557.78	−0.131751
$520$	−2625.89	−0.221448
$521$	−1673.01	−0.140683	−0.0703416	−	0.997523i	$-0.522409\pi$
$521$	−1673.01	−0.140683	−0.0703416	+	0.997523i	$0.522409\pi$
$522$	5284.96	0.443135
$523$	612.609	0.0512190	0.0256095	−	0.999672i	$-0.491847\pi$
$523$	612.609	0.0512190	0.0256095	+	0.999672i	$0.491847\pi$
$524$	128.794	0.0107374
$525$	0	0
$526$	8248.90	0.683782
$527$	−118.876	−0.00982603
$528$	9292.04	0.765879
$529$	−10619.8	−0.872838
$530$	5676.99	0.465269
$531$	4538.06	0.370876
$532$	0	0
$533$	776.304	0.0630871
$534$	−5785.68	−0.468859
$535$	−17.5285	−0.00141649
$536$	−11287.6	−0.909612
$537$	1298.53	0.104350
$538$	22196.7	1.77875
$539$	0	0
$540$	−77.1508	−0.00614823
$541$	12856.9	1.02174	0.510872	−	0.859657i	$-0.329323\pi$
$541$	12856.9	1.02174	0.510872	+	0.859657i	$0.329323\pi$
$542$	−5877.55	−0.465798
$543$	−8375.12	−0.661899
$544$	−203.716	−0.0160556
$545$	−7517.59	−0.590859
$546$	0	0
$547$	−15953.2	−1.24700	−0.623500	−	0.781824i	$-0.714290\pi$
$547$	−15953.2	−1.24700	−0.623500	+	0.781824i	$0.714290\pi$
$548$	−1044.23	−0.0814003
$549$	2955.53	0.229761
$550$	3321.89	0.257538
$551$	−30476.9	−2.35637
$552$	2566.38	0.197885
$553$	0	0
$554$	1529.83	0.117322
$555$	−1796.30	−0.137385
$556$	−946.605	−0.0722032
$557$	−6765.42	−0.514650	−0.257325	−	0.966325i	$-0.582841\pi$
$557$	−6765.42	−0.514650	−0.257325	+	0.966325i	$0.582841\pi$
$558$	−396.916	−0.0301125
$559$	8654.58	0.654829
$560$	0	0
$561$	−1074.50	−0.0808649
$562$	7401.36	0.555529
$563$	−23505.6	−1.75958	−0.879789	−	0.475364i	$-0.842316\pi$
$563$	−23505.6	−1.75958	−0.879789	+	0.475364i	$0.842316\pi$
$564$	−293.261	−0.0218945
$565$	1058.10	0.0787872
$566$	−7635.53	−0.567041
$567$	0	0
$568$	−25524.4	−1.88553
$569$	477.650	0.0351918	0.0175959	−	0.999845i	$-0.494399\pi$
$569$	477.650	0.0351918	0.0175959	+	0.999845i	$0.494399\pi$
$570$	6672.95	0.490350
$571$	14081.6	1.03204	0.516022	−	0.856576i	$-0.327412\pi$
$571$	14081.6	1.03204	0.516022	+	0.856576i	$0.327412\pi$
$572$	626.327	0.0457833
$573$	9663.95	0.704568
$574$	0	0
$575$	983.355	0.0713196
$576$	4233.47	0.306241
$577$	10405.0	0.750719	0.375359	−	0.926879i	$-0.377519\pi$
$577$	10405.0	0.750719	0.375359	+	0.926879i	$0.377519\pi$
$578$	−14201.5	−1.02198
$579$	10921.2	0.783885
$580$	573.123	0.0410304
$581$	0	0
$582$	−7785.38	−0.554492
$583$	17601.0	1.25036
$584$	2470.72	0.175067
$585$	1086.65	0.0767991
$586$	−8285.70	−0.584094
$587$	−14406.0	−1.01295	−0.506473	−	0.862256i	$-0.669051\pi$
$587$	−14406.0	−1.01295	−0.506473	+	0.862256i	$0.669051\pi$
$588$	0	0
$589$	2288.90	0.160123
$590$	7381.19	0.515049
$591$	−9536.56	−0.663759
$592$	−8172.60	−0.567384
$593$	−11647.6	−0.806596	−0.403298	−	0.915069i	$-0.632136\pi$
$593$	−11647.6	−0.806596	−0.403298	+	0.915069i	$0.632136\pi$
$594$	−3587.64	−0.247816
$595$	0	0
$596$	638.258	0.0438659
$597$	−11608.6	−0.795829
$598$	2780.84	0.190162
$599$	6289.16	0.428995	0.214498	−	0.976725i	$-0.431189\pi$
$599$	6289.16	0.428995	0.214498	+	0.976725i	$0.431189\pi$
$600$	1631.14	0.110985
$601$	10340.0	0.701796	0.350898	−	0.936414i	$-0.385876\pi$
$601$	10340.0	0.701796	0.350898	+	0.936414i	$0.385876\pi$
$602$	0	0
$603$	4671.06	0.315457
$604$	250.482	0.0168741
$605$	3644.21	0.244890
$606$	−16275.1	−1.09098
$607$	15606.3	1.04356	0.521781	−	0.853080i	$-0.325268\pi$
$607$	15606.3	1.04356	0.521781	+	0.853080i	$0.325268\pi$
$608$	3922.45	0.261639
$609$	0	0
$610$	4807.19	0.319078
$611$	4130.51	0.273490
$612$	40.5897	0.00268095
$613$	23004.4	1.51572	0.757861	−	0.652416i	$-0.226244\pi$
$613$	23004.4	1.51572	0.757861	+	0.652416i	$0.226244\pi$
$614$	25625.3	1.68429
$615$	−482.220	−0.0316179
$616$	0	0
$617$	21516.8	1.40394	0.701971	−	0.712205i	$-0.252304\pi$
$617$	21516.8	1.40394	0.701971	+	0.712205i	$0.252304\pi$
$618$	14700.6	0.956870
$619$	7302.79	0.474191	0.237095	−	0.971486i	$-0.423805\pi$
$619$	7302.79	0.474191	0.237095	+	0.971486i	$0.423805\pi$
$620$	−43.0432	−0.00278815
$621$	−1062.02	−0.0686273
$622$	−12428.8	−0.801204
$623$	0	0
$624$	4943.92	0.317172
$625$	625.000	0.0400000
$626$	−17240.2	−1.10073
$627$	20688.9	1.31776
$628$	−649.518	−0.0412716
$629$	945.047	0.0599070
$630$	0	0
$631$	28805.7	1.81733	0.908666	−	0.417524i	$-0.137102\pi$
$631$	28805.7	1.81733	0.908666	+	0.417524i	$0.137102\pi$
$632$	17515.6	1.10243
$633$	−13433.1	−0.843473
$634$	−20713.0	−1.29751
$635$	3734.31	0.233373
$636$	−664.887	−0.0414536
$637$	0	0
$638$	26651.2	1.65381
$639$	10562.5	0.653909
$640$	7918.34	0.489062
$641$	−7479.14	−0.460855	−0.230428	−	0.973089i	$-0.574013\pi$
$641$	−7479.14	−0.460855	−0.230428	+	0.973089i	$0.574013\pi$
$642$	30.7911	0.00189288
$643$	−25832.0	−1.58432	−0.792158	−	0.610316i	$-0.791042\pi$
$643$	−25832.0	−1.58432	−0.792158	+	0.610316i	$0.791042\pi$
$644$	0	0
$645$	−5376.01	−0.328186
$646$	−3510.70	−0.213818
$647$	14971.7	0.909736	0.454868	−	0.890559i	$-0.349686\pi$
$647$	14971.7	0.909736	0.454868	+	0.890559i	$0.349686\pi$
$648$	−1761.63	−0.106795
$649$	22884.7	1.38413
$650$	1767.44	0.106654
$651$	0	0
$652$	500.179	0.0300438
$653$	7423.31	0.444865	0.222432	−	0.974948i	$-0.428600\pi$
$653$	7423.31	0.444865	0.222432	+	0.974948i	$0.428600\pi$
$654$	13205.6	0.789571
$655$	1126.83	0.0672199
$656$	−2193.95	−0.130578
$657$	−1022.44	−0.0607139
$658$	0	0
$659$	3126.66	0.184822	0.0924109	−	0.995721i	$-0.470543\pi$
$659$	3126.66	0.184822	0.0924109	+	0.995721i	$0.470543\pi$
$660$	−389.058	−0.0229456
$661$	−7725.29	−0.454582	−0.227291	−	0.973827i	$-0.572987\pi$
$661$	−7725.29	−0.454582	−0.227291	+	0.973827i	$0.572987\pi$
$662$	−29969.3	−1.75950
$663$	−571.696	−0.0334884
$664$	29365.2	1.71625
$665$	0	0
$666$	3155.42	0.183589
$667$	7889.36	0.457987
$668$	1212.59	0.0702343
$669$	916.656	0.0529745
$670$	7597.51	0.438086
$671$	14904.2	0.857484
$672$	0	0
$673$	26556.6	1.52107	0.760536	−	0.649296i	$-0.224936\pi$
$673$	26556.6	1.52107	0.760536	+	0.649296i	$0.224936\pi$
$674$	−16334.9	−0.933528
$675$	−675.000	−0.0384900
$676$	−922.315	−0.0524758
$677$	−21375.9	−1.21350	−0.606752	−	0.794891i	$-0.707528\pi$
$677$	−21375.9	−1.21350	−0.606752	+	0.794891i	$0.707528\pi$
$678$	−1858.69	−0.105284
$679$	0	0
$680$	−858.156	−0.0483953
$681$	19865.9	1.11786
$682$	−2001.58	−0.112382
$683$	19700.2	1.10367	0.551834	−	0.833954i	$-0.313928\pi$
$683$	19700.2	1.10367	0.551834	+	0.833954i	$0.313928\pi$
$684$	−781.534	−0.0436882
$685$	−9136.08	−0.509594
$686$	0	0
$687$	15721.2	0.873071
$688$	−24459.1	−1.35537
$689$	9364.78	0.517808
$690$	−1727.39	−0.0953051
$691$	−4568.18	−0.251493	−0.125747	−	0.992062i	$-0.540133\pi$
$691$	−4568.18	−0.251493	−0.125747	+	0.992062i	$0.540133\pi$
$692$	296.750	0.0163017
$693$	0	0
$694$	27891.0	1.52555
$695$	−8281.94	−0.452017
$696$	13086.5	0.712703
$697$	253.700	0.0137871
$698$	580.458	0.0314766
$699$	7842.33	0.424355
$700$	0	0
$701$	−22203.6	−1.19632	−0.598159	−	0.801377i	$-0.704101\pi$
$701$	−22203.6	−1.19632	−0.598159	+	0.801377i	$0.704101\pi$
$702$	−1908.84	−0.102627
$703$	−18196.4	−0.976232
$704$	21348.7	1.14291
$705$	−2565.77	−0.137067
$706$	−27170.6	−1.44841
$707$	0	0
$708$	−864.482	−0.0458887
$709$	25615.6	1.35686	0.678430	−	0.734665i	$-0.262661\pi$
$709$	25615.6	1.35686	0.678430	+	0.734665i	$0.262661\pi$
$710$	17180.0	0.908106
$711$	−7248.32	−0.382325
$712$	−14326.3	−0.754076
$713$	−592.513	−0.0311217
$714$	0	0
$715$	5479.79	0.286619
$716$	−247.365	−0.0129113
$717$	959.832	0.0499939
$718$	−19292.1	−1.00275
$719$	16228.4	0.841747	0.420873	−	0.907119i	$-0.361724\pi$
$719$	16228.4	0.841747	0.420873	+	0.907119i	$0.361724\pi$
$720$	−3071.04	−0.158960
$721$	0	0
$722$	47515.5	2.44923
$723$	−10289.9	−0.529302
$724$	1595.43	0.0818971
$725$	5014.31	0.256865
$726$	−6401.51	−0.327248
$727$	−34994.2	−1.78523	−0.892615	−	0.450820i	$-0.851132\pi$
$727$	−34994.2	−1.78523	−0.892615	+	0.450820i	$0.851132\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−1663.00	−0.0843155
$731$	2828.36	0.143106
$732$	−563.016	−0.0284285
$733$	24430.8	1.23107	0.615533	−	0.788111i	$-0.288941\pi$
$733$	24430.8	1.23107	0.615533	+	0.788111i	$0.288941\pi$
$734$	32159.4	1.61720
$735$	0	0
$736$	−1015.38	−0.0508525
$737$	23555.4	1.17731
$738$	847.081	0.0422513
$739$	−17001.5	−0.846291	−0.423146	−	0.906062i	$-0.639074\pi$
$739$	−17001.5	−0.846291	−0.423146	+	0.906062i	$0.639074\pi$
$740$	342.187	0.0169987
$741$	11007.7	0.545720
$742$	0	0
$743$	14.3426	0.000708183 0	0.000354092	−	1.00000i	$-0.499887\pi$
$743$	14.3426	0.000708183 0	0.000354092		1.00000i	$0.499887\pi$
$744$	−982.831	−0.0484305
$745$	5584.19	0.274616
$746$	−6411.80	−0.314682
$747$	−12152.0	−0.595203
$748$	204.687	0.0100055
$749$	0	0
$750$	−1097.89	−0.0534524
$751$	−30724.4	−1.49287	−0.746437	−	0.665456i	$-0.768237\pi$
$751$	−30724.4	−1.49287	−0.746437	+	0.665456i	$0.768237\pi$
$752$	−11673.4	−0.566072
$753$	15457.4	0.748072
$754$	14180.0	0.684888
$755$	2191.49	0.105638
$756$	0	0
$757$	−27240.7	−1.30790	−0.653950	−	0.756538i	$-0.726889\pi$
$757$	−27240.7	−1.30790	−0.653950	+	0.756538i	$0.726889\pi$
$758$	6040.72	0.289458
$759$	−5355.61	−0.256121
$760$	16523.4	0.788639
$761$	10252.4	0.488370	0.244185	−	0.969729i	$-0.421480\pi$
$761$	10252.4	0.488370	0.244185	+	0.969729i	$0.421480\pi$
$762$	−6559.79	−0.311859
$763$	0	0
$764$	−1840.94	−0.0871766
$765$	355.123	0.0167837
$766$	−30981.2	−1.46135
$767$	12176.0	0.573208
$768$	−2620.30	−0.123114
$769$	−37870.0	−1.77585	−0.887925	−	0.459989i	$-0.847853\pi$
$769$	−37870.0	−1.77585	−0.887925	+	0.459989i	$0.847853\pi$
$770$	0	0
$771$	−5812.37	−0.271501
$772$	−2080.44	−0.0969906
$773$	16744.1	0.779098	0.389549	−	0.921006i	$-0.372631\pi$
$773$	16744.1	0.779098	0.389549	+	0.921006i	$0.372631\pi$
$774$	9443.63	0.438559
$775$	−376.589	−0.0174548
$776$	−19277.9	−0.891801
$777$	0	0
$778$	21376.0	0.985047
$779$	−4884.87	−0.224671
$780$	−207.002	−0.00950240
$781$	53265.1	2.44043
$782$	908.793	0.0415580
$783$	−5415.46	−0.247168
$784$	0	0
$785$	−5682.70	−0.258375
$786$	−1979.43	−0.0898267
$787$	−17080.1	−0.773622	−0.386811	−	0.922159i	$-0.626423\pi$
$787$	−17080.1	−0.773622	−0.386811	+	0.922159i	$0.626423\pi$
$788$	1816.68	0.0821274
$789$	−8452.58	−0.381394
$790$	−11789.4	−0.530948
$791$	0	0
$792$	−8883.61	−0.398567
$793$	7929.95	0.355108
$794$	22397.0	1.00106
$795$	−5817.16	−0.259514
$796$	2211.40	0.0984684
$797$	−30528.1	−1.35679	−0.678395	−	0.734697i	$-0.737324\pi$
$797$	−30528.1	−1.35679	−0.678395	+	0.734697i	$0.737324\pi$
$798$	0	0
$799$	1349.87	0.0597684
$800$	−645.355	−0.0285209
$801$	5928.54	0.261516
$802$	21616.5	0.951752
$803$	−5155.97	−0.226588
$804$	−889.818	−0.0390317
$805$	0	0
$806$	−1064.96	−0.0465404
$807$	−22744.8	−0.992138
$808$	−40300.0	−1.75464
$809$	36969.6	1.60665	0.803325	−	0.595541i	$-0.203062\pi$
$809$	36969.6	1.60665	0.803325	+	0.595541i	$0.203062\pi$
$810$	1185.72	0.0514346
$811$	35231.2	1.52544	0.762722	−	0.646727i	$-0.223863\pi$
$811$	35231.2	1.52544	0.762722	+	0.646727i	$0.223863\pi$
$812$	0	0
$813$	6022.68	0.259809
$814$	15912.3	0.685166
$815$	4376.12	0.188084
$816$	1615.70	0.0693147
$817$	−54458.7	−2.33203
$818$	38238.8	1.63446
$819$	0	0
$820$	91.8609	0.00391210
$821$	−3853.10	−0.163793	−0.0818965	−	0.996641i	$-0.526098\pi$
$821$	−3853.10	−0.163793	−0.0818965	+	0.996641i	$0.526098\pi$
$822$	16048.7	0.680976
$823$	−19323.3	−0.818431	−0.409216	−	0.912438i	$-0.634198\pi$
$823$	−19323.3	−0.818431	−0.409216	+	0.912438i	$0.634198\pi$
$824$	36401.2	1.53895
$825$	−3403.91	−0.143647
$826$	0	0
$827$	42985.0	1.80742	0.903710	−	0.428145i	$-0.140833\pi$
$827$	42985.0	1.80742	0.903710	+	0.428145i	$0.140833\pi$
$828$	202.311	0.00849130
$829$	7707.12	0.322894	0.161447	−	0.986881i	$-0.448384\pi$
$829$	7707.12	0.322894	0.161447	+	0.986881i	$0.448384\pi$
$830$	−19765.2	−0.826579
$831$	−1567.60	−0.0654387
$832$	11358.8	0.473311
$833$	0	0
$834$	14548.3	0.604035
$835$	10609.1	0.439691
$836$	−3941.14	−0.163047
$837$	406.716	0.0167959
$838$	−36098.3	−1.48806
$839$	15324.4	0.630579	0.315290	−	0.948995i	$-0.397898\pi$
$839$	15324.4	0.630579	0.315290	+	0.948995i	$0.397898\pi$
$840$	0	0
$841$	15840.3	0.649486
$842$	−11743.6	−0.480654
$843$	−7584.11	−0.309858
$844$	2558.95	0.104363
$845$	−8069.42	−0.328517
$846$	4507.09	0.183164
$847$	0	0
$848$	−26466.3	−1.07176
$849$	7824.06	0.316279
$850$	577.610	0.0233081
$851$	4710.40	0.189742
$852$	−2012.12	−0.0809086
$853$	21332.0	0.856266	0.428133	−	0.903716i	$-0.359172\pi$
$853$	21332.0	0.856266	0.428133	+	0.903716i	$0.359172\pi$
$854$	0	0
$855$	−6837.72	−0.273503
$856$	76.2440	0.00304435
$857$	32731.6	1.30466	0.652328	−	0.757936i	$-0.273792\pi$
$857$	32731.6	1.30466	0.652328	+	0.757936i	$0.273792\pi$
$858$	−9625.95	−0.383012
$859$	35989.8	1.42952	0.714759	−	0.699371i	$-0.246536\pi$
$859$	35989.8	1.42952	0.714759	+	0.699371i	$0.246536\pi$
$860$	1024.11	0.0406067
$861$	0	0
$862$	−38010.9	−1.50192
$863$	−40400.4	−1.59356	−0.796781	−	0.604268i	$-0.793466\pi$
$863$	−40400.4	−1.59356	−0.796781	+	0.604268i	$0.793466\pi$
$864$	696.984	0.0274443
$865$	2596.30	0.102054
$866$	32135.5	1.26098
$867$	14552.2	0.570032
$868$	0	0
$869$	−36552.1	−1.42686
$870$	−8808.27	−0.343251
$871$	12532.9	0.487555
$872$	32699.3	1.26988
$873$	7977.61	0.309280
$874$	−17498.4	−0.677221
$875$	0	0
$876$	194.770	0.00751217
$877$	15813.9	0.608890	0.304445	−	0.952530i	$-0.401529\pi$
$877$	15813.9	0.608890	0.304445	+	0.952530i	$0.401529\pi$
$878$	51168.1	1.96679
$879$	8490.29	0.325791
$880$	−15486.7	−0.593247
$881$	11465.6	0.438461	0.219231	−	0.975673i	$-0.429645\pi$
$881$	11465.6	0.438461	0.219231	+	0.975673i	$0.429645\pi$
$882$	0	0
$883$	−15044.6	−0.573376	−0.286688	−	0.958024i	$-0.592554\pi$
$883$	−15044.6	−0.573376	−0.286688	+	0.958024i	$0.592554\pi$
$884$	108.906	0.00414354
$885$	−7563.44	−0.287279
$886$	6100.70	0.231329
$887$	−4917.35	−0.186143	−0.0930714	−	0.995659i	$-0.529668\pi$
$887$	−4917.35	−0.186143	−0.0930714	+	0.995659i	$0.529668\pi$
$888$	7813.37	0.295270
$889$	0	0
$890$	9642.80	0.363177
$891$	3676.22	0.138225
$892$	−174.619	−0.00655458
$893$	−25991.1	−0.973974
$894$	−9809.33	−0.366972
$895$	−2164.22	−0.0808290
$896$	0	0
$897$	−2849.50	−0.106067
$898$	−5487.02	−0.203902
$899$	−3021.34	−0.112088
$900$	128.585	0.00476240
$901$	3060.46	0.113162
$902$	4271.69	0.157685
$903$	0	0
$904$	−4602.44	−0.169331
$905$	13958.5	0.512704
$906$	−3849.64	−0.141165
$907$	−37984.5	−1.39058	−0.695289	−	0.718731i	$-0.744723\pi$
$907$	−37984.5	−1.39058	−0.695289	+	0.718731i	$0.744723\pi$
$908$	−3784.37	−0.138313
$909$	16677.0	0.608516
$910$	0	0
$911$	5336.79	0.194090	0.0970449	−	0.995280i	$-0.469061\pi$
$911$	5336.79	0.194090	0.0970449	+	0.995280i	$0.469061\pi$
$912$	−31109.5	−1.12954
$913$	−61280.2	−2.22134
$914$	−19465.5	−0.704444
$915$	−4925.88	−0.177972
$916$	−2994.82	−0.108026
$917$	0	0
$918$	−623.818	−0.0224282
$919$	−37636.1	−1.35093	−0.675463	−	0.737394i	$-0.736056\pi$
$919$	−37636.1	−1.35093	−0.675463	+	0.737394i	$0.736056\pi$
$920$	−4277.31	−0.153281
$921$	−26258.0	−0.939447
$922$	45464.3	1.62396
$923$	28340.2	1.01065
$924$	0	0
$925$	2993.83	0.106418
$926$	6926.64	0.245814
$927$	−15063.6	−0.533715
$928$	−5177.62	−0.183151
$929$	−15972.3	−0.564084	−0.282042	−	0.959402i	$-0.591012\pi$
$929$	−15972.3	−0.564084	−0.282042	+	0.959402i	$0.591012\pi$
$930$	661.526	0.0233250
$931$	0	0
$932$	−1493.93	−0.0525057
$933$	12735.7	0.446889
$934$	−50679.4	−1.77546
$935$	1790.83	0.0626377
$936$	−4726.61	−0.165058
$937$	−47349.8	−1.65085	−0.825426	−	0.564510i	$-0.809065\pi$
$937$	−47349.8	−1.65085	−0.825426	+	0.564510i	$0.809065\pi$
$938$	0	0
$939$	17665.9	0.613956
$940$	488.768	0.0169594
$941$	30521.3	1.05735	0.528674	−	0.848825i	$-0.322689\pi$
$941$	30521.3	1.05735	0.528674	+	0.848825i	$0.322689\pi$
$942$	9982.37	0.345269
$943$	1264.52	0.0436674
$944$	−34411.3	−1.18643
$945$	0	0
$946$	47622.6	1.63673
$947$	−18950.3	−0.650267	−0.325134	−	0.945668i	$-0.605409\pi$
$947$	−18950.3	−0.650267	−0.325134	+	0.945668i	$0.605409\pi$
$948$	1380.77	0.0473054
$949$	−2743.29	−0.0938365
$950$	−11121.6	−0.379823
$951$	21224.5	0.723712
$952$	0	0
$953$	−34526.2	−1.17357	−0.586785	−	0.809743i	$-0.699607\pi$
$953$	−34526.2	−1.17357	−0.586785	+	0.809743i	$0.699607\pi$
$954$	10218.6	0.346791
$955$	−16106.6	−0.545756
$956$	−182.844	−0.00618577
$957$	−27309.2	−0.922447
$958$	6919.14	0.233348
$959$	0	0
$960$	−7055.79	−0.237213
$961$	−29564.1	−0.992383
$962$	8466.28	0.283746
$963$	−31.5514	−0.00105579
$964$	1960.18	0.0654909
$965$	−18202.0	−0.607195
$966$	0	0
$967$	−41107.3	−1.36703	−0.683517	−	0.729934i	$-0.739551\pi$
$967$	−41107.3	−1.36703	−0.683517	+	0.729934i	$0.739551\pi$
$968$	−15851.2	−0.526320
$969$	3597.38	0.119262
$970$	12975.6	0.429508
$971$	−13730.5	−0.453793	−0.226896	−	0.973919i	$-0.572858\pi$
$971$	−13730.5	−0.453793	−0.226896	+	0.973919i	$0.572858\pi$
$972$	−138.871	−0.00458262
$973$	0	0
$974$	−52158.7	−1.71589
$975$	−1811.08	−0.0594883
$976$	−22411.2	−0.735006
$977$	−15039.7	−0.492490	−0.246245	−	0.969208i	$-0.579197\pi$
$977$	−15039.7	−0.492490	−0.246245	+	0.969208i	$0.579197\pi$
$978$	−7687.20	−0.251339
$979$	29896.6	0.975996
$980$	0	0
$981$	−13531.7	−0.440400
$982$	5762.01	0.187244
$983$	−27849.6	−0.903627	−0.451814	−	0.892112i	$-0.649223\pi$
$983$	−27849.6	−0.903627	−0.451814	+	0.892112i	$0.649223\pi$
$984$	2097.52	0.0679537
$985$	15894.3	0.514146
$986$	4634.10	0.149675
$987$	0	0
$988$	−2096.92	−0.0675223
$989$	14097.4	0.453257
$990$	5979.40	0.191957
$991$	4838.08	0.155082	0.0775411	−	0.996989i	$-0.475293\pi$
$991$	4838.08	0.155082	0.0775411	+	0.996989i	$0.475293\pi$
$992$	388.854	0.0124457
$993$	30709.3	0.981399
$994$	0	0
$995$	19347.7	0.616446
$996$	2314.90	0.0736449
$997$	37173.5	1.18084	0.590419	−	0.807097i	$-0.298962\pi$
$997$	37173.5	1.18084	0.590419	+	0.807097i	$0.298962\pi$
$998$	50117.9	1.58963
$999$	−3233.34	−0.102401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.v.1.3	✓	4
3.2	odd	2		2205.4.a.bn.1.2		4
7.6	odd	2		735.4.a.w.1.3	yes	4
21.20	even	2		2205.4.a.bo.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.v.1.3	✓	4	1.1	even	1	trivial
735.4.a.w.1.3	yes	4	7.6	odd	2
2205.4.a.bn.1.2		4	3.2	odd	2
2205.4.a.bo.1.2		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q+2.92771 q^{2} -3.00000 q^{3} +0.571487 q^{4} +5.00000 q^{5} -8.78313 q^{6} -21.7485 q^{8} +9.00000 q^{9} +14.6386 q^{10} +45.3855 q^{11} -1.71446 q^{12} +24.1478 q^{13} -15.0000 q^{15} -68.2453 q^{16} +7.89162 q^{17} +26.3494 q^{18} -151.949 q^{19} +2.85744 q^{20} +132.876 q^{22} +39.3342 q^{23} +65.2456 q^{24} +25.0000 q^{25} +70.6977 q^{26} -27.0000 q^{27} +200.572 q^{29} -43.9157 q^{30} -15.0636 q^{31} -25.8142 q^{32} -136.156 q^{33} +23.1044 q^{34} +5.14339 q^{36} +119.753 q^{37} -444.864 q^{38} -72.4434 q^{39} -108.743 q^{40} +32.1480 q^{41} +358.400 q^{43} +25.9372 q^{44} +45.0000 q^{45} +115.159 q^{46} +171.051 q^{47} +204.736 q^{48} +73.1928 q^{50} -23.6749 q^{51} +13.8002 q^{52} +387.811 q^{53} -79.0482 q^{54} +226.927 q^{55} +455.848 q^{57} +587.218 q^{58} +504.229 q^{59} -8.57231 q^{60} +328.392 q^{61} -44.1017 q^{62} +470.386 q^{64} +120.739 q^{65} -398.627 q^{66} +519.007 q^{67} +4.50996 q^{68} -118.003 q^{69} +1173.62 q^{71} -195.737 q^{72} -113.604 q^{73} +350.603 q^{74} -75.0000 q^{75} -86.8371 q^{76} -212.093 q^{78} -805.369 q^{79} -341.227 q^{80} +81.0000 q^{81} +94.1201 q^{82} -1350.22 q^{83} +39.4581 q^{85} +1049.29 q^{86} -601.717 q^{87} -987.068 q^{88} +658.726 q^{89} +131.747 q^{90} +22.4790 q^{92} +45.1907 q^{93} +500.788 q^{94} -759.747 q^{95} +77.4426 q^{96} +886.401 q^{97} +408.469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 12 q^{3} + 39 q^{4} + 20 q^{5} - 3 q^{6} + 33 q^{8} + 36 q^{9} + 5 q^{10} + 60 q^{11} - 117 q^{12} + 14 q^{13} - 60 q^{15} + 283 q^{16} + 46 q^{17} + 9 q^{18} + 40 q^{19} + 195 q^{20} - 14 q^{22}+ \cdots + 540 q^{99}+O(q^{100}) \) 4 * q + q^2 - 12 * q^3 + 39 * q^4 + 20 * q^5 - 3 * q^6 + 33 * q^8 + 36 * q^9 + 5 * q^10 + 60 * q^11 - 117 * q^12 + 14 * q^13 - 60 * q^15 + 283 * q^16 + 46 * q^17 + 9 * q^18 + 40 * q^19 + 195 * q^20 - 14 * q^22 + 490 * q^23 - 99 * q^24 + 100 * q^25 - 546 * q^26 - 108 * q^27 - 6 * q^29 - 15 * q^30 + 122 * q^31 + 525 * q^32 - 180 * q^33 + 748 * q^34 + 351 * q^36 + 142 * q^37 - 152 * q^38 - 42 * q^39 + 165 * q^40 + 168 * q^41 + 286 * q^43 + 1214 * q^44 + 180 * q^45 - 148 * q^46 - 306 * q^47 - 849 * q^48 + 25 * q^50 - 138 * q^51 - 1934 * q^52 - 276 * q^53 - 27 * q^54 + 300 * q^55 - 120 * q^57 + 652 * q^58 - 328 * q^59 - 585 * q^60 + 174 * q^61 - 1742 * q^62 + 3615 * q^64 + 70 * q^65 + 42 * q^66 - 994 * q^67 + 132 * q^68 - 1470 * q^69 + 2442 * q^71 + 297 * q^72 + 14 * q^73 - 2424 * q^74 - 300 * q^75 + 5160 * q^76 + 1638 * q^78 - 252 * q^79 + 1415 * q^80 + 324 * q^81 + 2014 * q^82 - 1556 * q^83 + 230 * q^85 + 734 * q^86 + 18 * q^87 - 4486 * q^88 + 600 * q^89 + 45 * q^90 + 5844 * q^92 - 366 * q^93 + 1210 * q^94 + 200 * q^95 - 1575 * q^96 - 238 * q^97 + 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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