LMFDB - Embedded newform 2205.4.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
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:	no (minimal twist has level 735)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.92771$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.92771 q^{2} +0.571487 q^{4} +5.00000 q^{5} +21.7485 q^{8} +O(q^{10})$	$q-2.92771 q^{2} +0.571487 q^{4} +5.00000 q^{5} +21.7485 q^{8} -14.6386 q^{10} -45.3855 q^{11} -24.1478 q^{13} -68.2453 q^{16} +7.89162 q^{17} +151.949 q^{19} +2.85744 q^{20} +132.876 q^{22} -39.3342 q^{23} +25.0000 q^{25} +70.6977 q^{26} -200.572 q^{29} +15.0636 q^{31} +25.8142 q^{32} -23.1044 q^{34} +119.753 q^{37} -444.864 q^{38} +108.743 q^{40} +32.1480 q^{41} +358.400 q^{43} -25.9372 q^{44} +115.159 q^{46} +171.051 q^{47} -73.1928 q^{50} -13.8002 q^{52} -387.811 q^{53} -226.927 q^{55} +587.218 q^{58} +504.229 q^{59} -328.392 q^{61} -44.1017 q^{62} +470.386 q^{64} -120.739 q^{65} +519.007 q^{67} +4.50996 q^{68} -1173.62 q^{71} +113.604 q^{73} -350.603 q^{74} +86.8371 q^{76} -805.369 q^{79} -341.227 q^{80} -94.1201 q^{82} -1350.22 q^{83} +39.4581 q^{85} -1049.29 q^{86} -987.068 q^{88} +658.726 q^{89} -22.4790 q^{92} -500.788 q^{94} +759.747 q^{95} -886.401 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 39 q^{4} + 20 q^{5} - 33 q^{8}+O(q^{10}) $ 4 * q - q^2 + 39 * q^4 + 20 * q^5 - 33 * q^8	$ 4 q - q^{2} + 39 q^{4} + 20 q^{5} - 33 q^{8} - 5 q^{10} - 60 q^{11} - 14 q^{13} + 283 q^{16} + 46 q^{17} - 40 q^{19} + 195 q^{20} - 14 q^{22} - 490 q^{23} + 100 q^{25} - 546 q^{26} + 6 q^{29} - 122 q^{31} - 525 q^{32} - 748 q^{34} + 142 q^{37} - 152 q^{38} - 165 q^{40} + 168 q^{41} + 286 q^{43} - 1214 q^{44} - 148 q^{46} - 306 q^{47} - 25 q^{50} + 1934 q^{52} + 276 q^{53} - 300 q^{55} + 652 q^{58} - 328 q^{59} - 174 q^{61} - 1742 q^{62} + 3615 q^{64} - 70 q^{65} - 994 q^{67} + 132 q^{68} - 2442 q^{71} - 14 q^{73} + 2424 q^{74} - 5160 q^{76} - 252 q^{79} + 1415 q^{80} - 2014 q^{82} - 1556 q^{83} + 230 q^{85} - 734 q^{86} - 4486 q^{88} + 600 q^{89} - 5844 q^{92} - 1210 q^{94} - 200 q^{95} + 238 q^{97}+O(q^{100}) $ 4 * q - q^2 + 39 * q^4 + 20 * q^5 - 33 * q^8 - 5 * q^10 - 60 * q^11 - 14 * q^13 + 283 * q^16 + 46 * q^17 - 40 * q^19 + 195 * q^20 - 14 * q^22 - 490 * q^23 + 100 * q^25 - 546 * q^26 + 6 * q^29 - 122 * q^31 - 525 * q^32 - 748 * q^34 + 142 * q^37 - 152 * q^38 - 165 * q^40 + 168 * q^41 + 286 * q^43 - 1214 * q^44 - 148 * q^46 - 306 * q^47 - 25 * q^50 + 1934 * q^52 + 276 * q^53 - 300 * q^55 + 652 * q^58 - 328 * q^59 - 174 * q^61 - 1742 * q^62 + 3615 * q^64 - 70 * q^65 - 994 * q^67 + 132 * q^68 - 2442 * q^71 - 14 * q^73 + 2424 * q^74 - 5160 * q^76 - 252 * q^79 + 1415 * q^80 - 2014 * q^82 - 1556 * q^83 + 230 * q^85 - 734 * q^86 - 4486 * q^88 + 600 * q^89 - 5844 * q^92 - 1210 * q^94 - 200 * q^95 + 238 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.92771	−1.03510	−0.517551	−	0.855652i	$-0.673156\pi$
$2$	−2.92771	−1.03510	−0.517551	+	0.855652i	$0.673156\pi$
$3$	0	0
$3$	0	0
$4$	0.571487	0.0714359
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.7485	0.961158
$9$	0	0
$10$	−14.6386	−0.462912
$11$	−45.3855	−1.24402	−0.622011	−	0.783009i	$-0.713684\pi$
$11$	−45.3855	−1.24402	−0.622011	+	0.783009i	$0.713684\pi$
$12$	0	0
$13$	−24.1478	−0.515184	−0.257592	−	0.966254i	$-0.582929\pi$
$13$	−24.1478	−0.515184	−0.257592	+	0.966254i	$0.582929\pi$
$14$	0	0
$15$	0	0
$16$	−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	$0.482072\pi$
$18$	0	0
$19$	151.949	1.83471	0.917357	−	0.398066i	$-0.130318\pi$
$19$	151.949	1.83471	0.917357	+	0.398066i	$0.130318\pi$
$20$	2.85744	0.0319471
$21$	0	0
$22$	132.876	1.28769
$23$	−39.3342	−0.356598	−0.178299	−	0.983976i	$-0.557059\pi$
$23$	−39.3342	−0.356598	−0.178299	+	0.983976i	$0.557059\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	70.6977	0.533268
$27$	0	0
$28$	0	0
$29$	−200.572	−1.28432	−0.642162	−	0.766569i	$-0.721962\pi$
$29$	−200.572	−1.28432	−0.642162	+	0.766569i	$0.721962\pi$
$30$	0	0
$31$	15.0636	0.0872740	0.0436370	−	0.999047i	$-0.486105\pi$
$31$	15.0636	0.0872740	0.0436370	+	0.999047i	$0.486105\pi$
$32$	25.8142	0.142605
$33$	0	0
$34$	−23.1044	−0.116540
$35$	0	0
$36$	0	0
$37$	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
$39$	0	0
$40$	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
$45$	0	0
$46$	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	$0.414486\pi$
$48$	0	0
$49$	0	0
$50$	−73.1928	−0.207020
$51$	0	0
$52$	−13.8002	−0.0368026
$53$	−387.811	−1.00509	−0.502547	−	0.864550i	$-0.667603\pi$
$53$	−387.811	−1.00509	−0.502547	+	0.864550i	$0.667603\pi$
$54$	0	0
$55$	−226.927	−0.556343
$56$	0	0
$57$	0	0
$58$	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	$0.312215\pi$
$60$	0	0
$61$	−328.392	−0.689284	−0.344642	−	0.938734i	$-0.612000\pi$
$61$	−328.392	−0.689284	−0.344642	+	0.938734i	$0.612000\pi$
$62$	−44.1017	−0.0903375
$63$	0	0
$64$	470.386	0.918722
$65$	−120.739	−0.230397
$66$	0	0
$67$	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
$69$	0	0
$70$	0	0
$71$	−1173.62	−1.96173	−0.980863	−	0.194697i	$-0.937628\pi$
$71$	−1173.62	−1.96173	−0.980863	+	0.194697i	$0.937628\pi$
$72$	0	0
$73$	113.604	0.182142	0.0910708	−	0.995844i	$-0.470971\pi$
$73$	113.604	0.182142	0.0910708	+	0.995844i	$0.470971\pi$
$74$	−350.603	−0.550767
$75$	0	0
$76$	86.8371	0.131064
$77$	0	0
$78$	0	0
$79$	−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
$81$	0	0
$82$	−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
$87$	0	0
$88$	−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	$0.371688\pi$
$90$	0	0
$91$	0	0
$92$	−22.4790	−0.0254739
$93$	0	0
$94$	−500.788	−0.549493
$95$	759.747	0.820509
$96$	0	0
$97$	−886.401	−0.927839	−0.463920	−	0.885877i	$-0.653557\pi$
$97$	−886.401	−0.927839	−0.463920	+	0.885877i	$0.653557\pi$
$98$	0	0
$99$	0	0
$100$	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	$0.133935\pi$
$102$	0	0
$103$	1673.73	1.60114	0.800572	−	0.599237i	$-0.204529\pi$
$103$	1673.73	1.60114	0.800572	+	0.599237i	$0.204529\pi$
$104$	−525.179	−0.495173
$105$	0	0
$106$	1135.40	1.04037
$107$	3.50571	0.00316738	0.00158369	−	0.999999i	$-0.499496\pi$
$107$	3.50571	0.00316738	0.00158369	+	0.999999i	$0.499496\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	0	0
$113$	−211.621	−0.176173	−0.0880867	−	0.996113i	$-0.528075\pi$
$113$	−211.621	−0.176173	−0.0880867	+	0.996113i	$0.528075\pi$
$114$	0	0
$115$	−196.671	−0.159475
$116$	−114.625	−0.0917468
$117$	0	0
$118$	−1476.24	−1.15168
$119$	0	0
$120$	0	0
$121$	728.842	0.547590
$122$	961.437	0.713479
$123$	0	0
$124$	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
$129$	0	0
$130$	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	$0.476055\pi$
$132$	0	0
$133$	0	0
$134$	−1519.50	−0.979590
$135$	0	0
$136$	171.631	0.108215
$137$	1827.22	1.13949	0.569743	−	0.821823i	$-0.307043\pi$
$137$	1827.22	1.13949	0.569743	+	0.821823i	$0.307043\pi$
$138$	0	0
$139$	1656.39	1.01074	0.505370	−	0.862903i	$-0.331356\pi$
$139$	1656.39	1.01074	0.505370	+	0.862903i	$0.331356\pi$
$140$	0	0
$141$	0	0
$142$	3436.01	2.03059
$143$	1095.96	0.640900
$144$	0	0
$145$	−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	$-0.599335\pi$
$149$	−1116.84	−0.614060	−0.307030	+	0.951700i	$0.599335\pi$
$150$	0	0
$151$	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
$153$	0	0
$154$	0	0
$155$	75.3178	0.0390301
$156$	0	0
$157$	1136.54	0.577743	0.288872	−	0.957368i	$-0.406720\pi$
$157$	1136.54	0.577743	0.288872	+	0.957368i	$0.406720\pi$
$158$	2357.89	1.18724
$159$	0	0
$160$	129.071	0.0637748
$161$	0	0
$162$	0	0
$163$	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
$165$	0	0
$166$	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
$171$	0	0
$172$	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	$0.463602\pi$
$174$	0	0
$175$	0	0
$176$	3097.35	1.32654
$177$	0	0
$178$	−1928.56	−0.812088
$179$	432.844	0.180739	0.0903696	−	0.995908i	$-0.471195\pi$
$179$	432.844	0.180739	0.0903696	+	0.995908i	$0.471195\pi$
$180$	0	0
$181$	−2791.71	−1.14644	−0.573221	−	0.819401i	$-0.694306\pi$
$181$	−2791.71	−1.14644	−0.573221	+	0.819401i	$0.694306\pi$
$182$	0	0
$183$	0	0
$184$	−855.461	−0.342747
$185$	598.766	0.237958
$186$	0	0
$187$	−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	$-0.291100\pi$
$191$	3221.32	1.22035	0.610173	+	0.792268i	$0.291100\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	0	0
$197$	−3178.85	−1.14966	−0.574832	−	0.818271i	$-0.694933\pi$
$197$	−3178.85	−1.14966	−0.574832	+	0.818271i	$0.694933\pi$
$198$	0	0
$199$	−3869.55	−1.37842	−0.689208	−	0.724563i	$-0.742041\pi$
$199$	−3869.55	−1.37842	−0.689208	+	0.724563i	$0.742041\pi$
$200$	543.713	0.192232
$201$	0	0
$202$	−5425.05	−1.88963
$203$	0	0
$204$	0	0
$205$	160.740	0.0547638
$206$	−4900.21	−1.65735
$207$	0	0
$208$	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
$213$	0	0
$214$	−10.2637	−0.00327856
$215$	1792.00	0.568435
$216$	0	0
$217$	0	0
$218$	4401.86	1.36758
$219$	0	0
$220$	−129.686	−0.0397429
$221$	−190.565	−0.0580036
$222$	0	0
$223$	305.552	0.0917546	0.0458773	−	0.998947i	$-0.485392\pi$
$223$	305.552	0.0917546	0.0458773	+	0.998947i	$0.485392\pi$
$224$	0	0
$225$	0	0
$226$	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	$0.919376\pi$
$228$	0	0
$229$	5240.39	1.51220	0.756102	−	0.654454i	$-0.227101\pi$
$229$	5240.39	1.51220	0.756102	+	0.654454i	$0.227101\pi$
$230$	575.796	0.165073
$231$	0	0
$232$	−4362.16	−1.23444
$233$	2614.11	0.735005	0.367502	−	0.930023i	$-0.380213\pi$
$233$	2614.11	0.735005	0.367502	+	0.930023i	$0.380213\pi$
$234$	0	0
$235$	855.255	0.237407
$236$	288.161	0.0794816
$237$	0	0
$238$	0	0
$239$	319.944	0.0865919	0.0432960	−	0.999062i	$-0.486214\pi$
$239$	319.944	0.0865919	0.0432960	+	0.999062i	$0.486214\pi$
$240$	0	0
$241$	−3429.97	−0.916778	−0.458389	−	0.888752i	$-0.651573\pi$
$241$	−3429.97	−0.916778	−0.458389	+	0.888752i	$0.651573\pi$
$242$	−2133.84	−0.566811
$243$	0	0
$244$	−187.672	−0.0492396
$245$	0	0
$246$	0	0
$247$	−3669.24	−0.945215
$248$	327.610	0.0838842
$249$	0	0
$250$	−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
$255$	0	0
$256$	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	$0.424449\pi$
$258$	0	0
$259$	0	0
$260$	−69.0008	−0.0164586
$261$	0	0
$262$	−659.809	−0.155584
$263$	−2817.53	−0.660594	−0.330297	−	0.943877i	$-0.607149\pi$
$263$	−2817.53	−0.660594	−0.330297	+	0.943877i	$0.607149\pi$
$264$	0	0
$265$	−1939.05	−0.449491
$266$	0	0
$267$	0	0
$268$	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	$0.170952\pi$
$270$	0	0
$271$	2007.56	0.450002	0.225001	−	0.974359i	$-0.427761\pi$
$271$	2007.56	0.450002	0.225001	+	0.974359i	$0.427761\pi$
$272$	−538.566	−0.120057
$273$	0	0
$274$	−5349.56	−1.17948
$275$	−1134.64	−0.248804
$276$	0	0
$277$	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
$279$	0	0
$280$	0	0
$281$	−2528.04	−0.536691	−0.268345	−	0.963323i	$-0.586477\pi$
$281$	−2528.04	−0.536691	−0.268345	+	0.963323i	$0.586477\pi$
$282$	0	0
$283$	2608.02	0.547812	0.273906	−	0.961756i	$-0.411684\pi$
$283$	2608.02	0.547812	0.273906	+	0.961756i	$0.411684\pi$
$284$	−670.707	−0.140138
$285$	0	0
$286$	−3208.65	−0.663397
$287$	0	0
$288$	0	0
$289$	−4850.72	−0.987324
$290$	2936.09	0.594528
$291$	0	0
$292$	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
$297$	0	0
$298$	3269.78	0.635614
$299$	949.834	0.183713
$300$	0	0
$301$	0	0
$302$	−1283.21	−0.244505
$303$	0	0
$304$	−10369.8	−1.95642
$305$	−1641.96	−0.308257
$306$	0	0
$307$	−8752.67	−1.62717	−0.813585	−	0.581446i	$-0.802487\pi$
$307$	−8752.67	−1.62717	−0.813585	+	0.581446i	$0.802487\pi$
$308$	0	0
$309$	0	0
$310$	−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	$0.626495\pi$
$312$	0	0
$313$	5888.63	1.06340	0.531702	−	0.846932i	$-0.321553\pi$
$313$	5888.63	1.06340	0.531702	+	0.846932i	$0.321553\pi$
$314$	−3327.46	−0.598023
$315$	0	0
$316$	−460.258	−0.0819353
$317$	7074.82	1.25351	0.626753	−	0.779218i	$-0.284384\pi$
$317$	7074.82	1.25351	0.626753	+	0.779218i	$0.284384\pi$
$318$	0	0
$319$	9103.08	1.59773
$320$	2351.93	0.410865
$321$	0	0
$322$	0	0
$323$	1199.13	0.206567
$324$	0	0
$325$	−603.695	−0.103037
$326$	−2562.40	−0.435332
$327$	0	0
$328$	699.172	0.117699
$329$	0	0
$330$	0	0
$331$	−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
$333$	0	0
$334$	−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
$339$	0	0
$340$	22.5498	0.00359687
$341$	−683.667	−0.108571
$342$	0	0
$343$	0	0
$344$	7794.68	1.22169
$345$	0	0
$346$	−1520.24	−0.236210
$347$	−9526.56	−1.47381	−0.736906	−	0.675995i	$-0.763714\pi$
$347$	−9526.56	−1.47381	−0.736906	+	0.675995i	$0.763714\pi$
$348$	0	0
$349$	−198.263	−0.0304092	−0.0152046	−	0.999884i	$-0.504840\pi$
$349$	−198.263	−0.0304092	−0.0152046	+	0.999884i	$0.504840\pi$
$350$	0	0
$351$	0	0
$352$	−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	$0.746659\pi$
$354$	0	0
$355$	−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	$-0.339048\pi$
$359$	6589.47	0.968743	0.484372	+	0.874862i	$0.339048\pi$
$360$	0	0
$361$	16229.6	2.36617
$362$	8173.31	1.18668
$363$	0	0
$364$	0	0
$365$	568.020	0.0814562
$366$	0	0
$367$	−10984.5	−1.56236	−0.781180	−	0.624305i	$-0.785382\pi$
$367$	−10984.5	−1.56236	−0.781180	+	0.624305i	$0.785382\pi$
$368$	2684.37	0.380252
$369$	0	0
$370$	−1753.01	−0.246310
$371$	0	0
$372$	0	0
$373$	−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
$375$	0	0
$376$	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
$381$	0	0
$382$	−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	$0.749457\pi$
$384$	0	0
$385$	0	0
$386$	10658.0	1.40539
$387$	0	0
$388$	−506.567	−0.0662811
$389$	−7301.27	−0.951643	−0.475822	−	0.879542i	$-0.657849\pi$
$389$	−7301.27	−0.951643	−0.475822	+	0.879542i	$0.657849\pi$
$390$	0	0
$391$	−310.411	−0.0401487
$392$	0	0
$393$	0	0
$394$	9306.76	1.19002
$395$	−4026.84	−0.512943
$396$	0	0
$397$	−7650.00	−0.967110	−0.483555	−	0.875314i	$-0.660655\pi$
$397$	−7650.00	−0.967110	−0.483555	+	0.875314i	$0.660655\pi$
$398$	11328.9	1.42680
$399$	0	0
$400$	−1706.13	−0.213267
$401$	−7383.41	−0.919476	−0.459738	−	0.888055i	$-0.652057\pi$
$401$	−7383.41	−0.919476	−0.459738	+	0.888055i	$0.652057\pi$
$402$	0	0
$403$	−363.752	−0.0449622
$404$	1058.97	0.130410
$405$	0	0
$406$	0	0
$407$	−5435.06	−0.661931
$408$	0	0
$409$	−13061.0	−1.57903	−0.789517	−	0.613729i	$-0.789669\pi$
$409$	−13061.0	−1.57903	−0.789517	+	0.613729i	$0.789669\pi$
$410$	−470.600	−0.0566861
$411$	0	0
$412$	956.517	0.114379
$413$	0	0
$414$	0	0
$415$	−6751.08	−0.798549
$416$	−623.356	−0.0734677
$417$	0	0
$418$	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
$423$	0	0
$424$	−8434.32	−0.966054
$425$	197.291	0.0225176
$426$	0	0
$427$	0	0
$428$	2.00347	0.000226265 0
$429$	0	0
$430$	−5246.46	−0.588388
$431$	12983.2	1.45099	0.725495	−	0.688227i	$-0.241611\pi$
$431$	12983.2	1.45099	0.725495	+	0.688227i	$0.241611\pi$
$432$	0	0
$433$	−10976.3	−1.21822	−0.609109	−	0.793087i	$-0.708473\pi$
$433$	−10976.3	−1.21822	−0.609109	+	0.793087i	$0.708473\pi$
$434$	0	0
$435$	0	0
$436$	−859.241	−0.0943812
$437$	−5976.81	−0.654255
$438$	0	0
$439$	−17477.2	−1.90009	−0.950045	−	0.312112i	$-0.898964\pi$
$439$	−17477.2	−1.90009	−0.950045	+	0.312112i	$0.898964\pi$
$440$	−4935.34	−0.534734
$441$	0	0
$442$	557.920	0.0600397
$443$	−2083.78	−0.223484	−0.111742	−	0.993737i	$-0.535643\pi$
$443$	−2083.78	−0.223484	−0.111742	+	0.993737i	$0.535643\pi$
$444$	0	0
$445$	3293.63	0.350861
$446$	−894.568	−0.0949754
$447$	0	0
$448$	0	0
$449$	1874.17	0.196988	0.0984938	−	0.995138i	$-0.468598\pi$
$449$	1874.17	0.196988	0.0984938	+	0.995138i	$0.468598\pi$
$450$	0	0
$451$	−1459.05	−0.152337
$452$	−120.939	−0.0125851
$453$	0	0
$454$	19387.2	2.00415
$455$	0	0
$456$	0	0
$457$	−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
$459$	0	0
$460$	−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
$465$	0	0
$466$	−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	$0.828062\pi$
$468$	0	0
$469$	0	0
$470$	−2503.94	−0.245741
$471$	0	0
$472$	10966.2	1.06941
$473$	−16266.2	−1.58122
$474$	0	0
$475$	3798.73	0.366943
$476$	0	0
$477$	0	0
$478$	−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	$0.464045\pi$
$480$	0	0
$481$	−2891.78	−0.274124
$482$	10041.9	0.948959
$483$	0	0
$484$	416.524	0.0391176
$485$	−4432.01	−0.414942
$486$	0	0
$487$	−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
$489$	0	0
$490$	0	0
$491$	−1968.10	−0.180894	−0.0904469	−	0.995901i	$-0.528830\pi$
$491$	−1968.10	−0.180894	−0.0904469	+	0.995901i	$0.528830\pi$
$492$	0	0
$493$	−1582.84	−0.144600
$494$	10742.5	0.978394
$495$	0	0
$496$	−1028.02	−0.0930632
$497$	0	0
$498$	0	0
$499$	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
$501$	0	0
$502$	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
$507$	0	0
$508$	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	$0.427932\pi$
$510$	0	0
$511$	0	0
$512$	10112.2	0.872851
$513$	0	0
$514$	−5672.32	−0.486761
$515$	8368.67	0.716053
$516$	0	0
$517$	−7763.23	−0.660400
$518$	0	0
$519$	0	0
$520$	−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	$0.522409\pi$
$522$	0	0
$523$	−612.609	−0.0512190	−0.0256095	−	0.999672i	$-0.508153\pi$
$523$	−612.609	−0.0512190	−0.0256095	+	0.999672i	$0.508153\pi$
$524$	128.794	0.0107374
$525$	0	0
$526$	8248.90	0.683782
$527$	118.876	0.00982603
$528$	0	0
$529$	−10619.8	−0.872838
$530$	5676.99	0.465269
$531$	0	0
$532$	0	0
$533$	−776.304	−0.0630871
$534$	0	0
$535$	17.5285	0.00141649
$536$	11287.6	0.909612
$537$	0	0
$538$	−22196.7	−1.77875
$539$	0	0
$540$	0	0
$541$	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
$543$	0	0
$544$	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
$549$	0	0
$550$	3321.89	0.257538
$551$	−30476.9	−2.35637
$552$	0	0
$553$	0	0
$554$	−1529.83	−0.117322
$555$	0	0
$556$	946.605	0.0722032
$557$	6765.42	0.514650	0.257325	−	0.966325i	$-0.417159\pi$
$557$	6765.42	0.514650	0.257325	+	0.966325i	$0.417159\pi$
$558$	0	0
$559$	−8654.58	−0.654829
$560$	0	0
$561$	0	0
$562$	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	$0.842316\pi$
$564$	0	0
$565$	−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	$-0.505601\pi$
$569$	−477.650	−0.0351918	−0.0175959	+	0.999845i	$0.505601\pi$
$570$	0	0
$571$	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
$573$	0	0
$574$	0	0
$575$	−983.355	−0.0713196
$576$	0	0
$577$	−10405.0	−0.750719	−0.375359	−	0.926879i	$-0.622481\pi$
$577$	−10405.0	−0.750719	−0.375359	+	0.926879i	$0.622481\pi$
$578$	14201.5	1.02198
$579$	0	0
$580$	−573.123	−0.0410304
$581$	0	0
$582$	0	0
$583$	17601.0	1.25036
$584$	2470.72	0.175067
$585$	0	0
$586$	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
$591$	0	0
$592$	−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	$0.632136\pi$
$594$	0	0
$595$	0	0
$596$	−638.258	−0.0438659
$597$	0	0
$598$	−2780.84	−0.190162
$599$	−6289.16	−0.428995	−0.214498	−	0.976725i	$-0.568811\pi$
$599$	−6289.16	−0.428995	−0.214498	+	0.976725i	$0.568811\pi$
$600$	0	0
$601$	−10340.0	−0.701796	−0.350898	−	0.936414i	$-0.614124\pi$
$601$	−10340.0	−0.701796	−0.350898	+	0.936414i	$0.614124\pi$
$602$	0	0
$603$	0	0
$604$	250.482	0.0168741
$605$	3644.21	0.244890
$606$	0	0
$607$	−15606.3	−1.04356	−0.521781	−	0.853080i	$-0.674732\pi$
$607$	−15606.3	−1.04356	−0.521781	+	0.853080i	$0.674732\pi$
$608$	3922.45	0.261639
$609$	0	0
$610$	4807.19	0.319078
$611$	−4130.51	−0.273490
$612$	0	0
$613$	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
$615$	0	0
$616$	0	0
$617$	−21516.8	−1.40394	−0.701971	−	0.712205i	$-0.747696\pi$
$617$	−21516.8	−1.40394	−0.701971	+	0.712205i	$0.747696\pi$
$618$	0	0
$619$	−7302.79	−0.474191	−0.237095	−	0.971486i	$-0.576195\pi$
$619$	−7302.79	−0.474191	−0.237095	+	0.971486i	$0.576195\pi$
$620$	43.0432	0.00278815
$621$	0	0
$622$	12428.8	0.801204
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−17240.2	−1.10073
$627$	0	0
$628$	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
$633$	0	0
$634$	−20713.0	−1.29751
$635$	3734.31	0.233373
$636$	0	0
$637$	0	0
$638$	−26651.2	−1.65381
$639$	0	0
$640$	−7918.34	−0.489062
$641$	7479.14	0.460855	0.230428	−	0.973089i	$-0.425987\pi$
$641$	7479.14	0.460855	0.230428	+	0.973089i	$0.425987\pi$
$642$	0	0
$643$	25832.0	1.58432	0.792158	−	0.610316i	$-0.208958\pi$
$643$	25832.0	1.58432	0.792158	+	0.610316i	$0.208958\pi$
$644$	0	0
$645$	0	0
$646$	−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	$0.349686\pi$
$648$	0	0
$649$	−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	$-0.571400\pi$
$653$	−7423.31	−0.444865	−0.222432	+	0.974948i	$0.571400\pi$
$654$	0	0
$655$	1126.83	0.0672199
$656$	−2193.95	−0.130578
$657$	0	0
$658$	0	0
$659$	−3126.66	−0.184822	−0.0924109	−	0.995721i	$-0.529457\pi$
$659$	−3126.66	−0.184822	−0.0924109	+	0.995721i	$0.529457\pi$
$660$	0	0
$661$	7725.29	0.454582	0.227291	−	0.973827i	$-0.427013\pi$
$661$	7725.29	0.454582	0.227291	+	0.973827i	$0.427013\pi$
$662$	29969.3	1.75950
$663$	0	0
$664$	−29365.2	−1.71625
$665$	0	0
$666$	0	0
$667$	7889.36	0.457987
$668$	1212.59	0.0702343
$669$	0	0
$670$	−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
$675$	0	0
$676$	−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	$0.707528\pi$
$678$	0	0
$679$	0	0
$680$	858.156	0.0483953
$681$	0	0
$682$	2001.58	0.112382
$683$	−19700.2	−1.10367	−0.551834	−	0.833954i	$-0.686072\pi$
$683$	−19700.2	−1.10367	−0.551834	+	0.833954i	$0.686072\pi$
$684$	0	0
$685$	9136.08	0.509594
$686$	0	0
$687$	0	0
$688$	−24459.1	−1.35537
$689$	9364.78	0.517808
$690$	0	0
$691$	4568.18	0.251493	0.125747	−	0.992062i	$-0.459867\pi$
$691$	4568.18	0.251493	0.125747	+	0.992062i	$0.459867\pi$
$692$	296.750	0.0163017
$693$	0	0
$694$	27891.0	1.52555
$695$	8281.94	0.452017
$696$	0	0
$697$	253.700	0.0137871
$698$	580.458	0.0314766
$699$	0	0
$700$	0	0
$701$	22203.6	1.19632	0.598159	−	0.801377i	$-0.295899\pi$
$701$	22203.6	1.19632	0.598159	+	0.801377i	$0.295899\pi$
$702$	0	0
$703$	18196.4	0.976232
$704$	−21348.7	−1.14291
$705$	0	0
$706$	27170.6	1.44841
$707$	0	0
$708$	0	0
$709$	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
$711$	0	0
$712$	14326.3	0.754076
$713$	−592.513	−0.0311217
$714$	0	0
$715$	5479.79	0.286619
$716$	247.365	0.0129113
$717$	0	0
$718$	−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	$0.361724\pi$
$720$	0	0
$721$	0	0
$722$	−47515.5	−2.44923
$723$	0	0
$724$	−1595.43	−0.0818971
$725$	−5014.31	−0.256865
$726$	0	0
$727$	34994.2	1.78523	0.892615	−	0.450820i	$-0.148868\pi$
$727$	34994.2	1.78523	0.892615	+	0.450820i	$0.148868\pi$
$728$	0	0
$729$	0	0
$730$	−1663.00	−0.0843155
$731$	2828.36	0.143106
$732$	0	0
$733$	−24430.8	−1.23107	−0.615533	−	0.788111i	$-0.711059\pi$
$733$	−24430.8	−1.23107	−0.615533	+	0.788111i	$0.711059\pi$
$734$	32159.4	1.61720
$735$	0	0
$736$	−1015.38	−0.0508525
$737$	−23555.4	−1.17731
$738$	0	0
$739$	−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
$741$	0	0
$742$	0	0
$743$	−14.3426	−0.000708183 0	−0.000354092	−	1.00000i	$-0.500113\pi$
$743$	−14.3426	−0.000708183 0	−0.000354092		1.00000i	$0.500113\pi$
$744$	0	0
$745$	−5584.19	−0.274616
$746$	6411.80	0.314682
$747$	0	0
$748$	−204.687	−0.0100055
$749$	0	0
$750$	0	0
$751$	−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
$753$	0	0
$754$	−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
$759$	0	0
$760$	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	$0.421480\pi$
$762$	0	0
$763$	0	0
$764$	1840.94	0.0871766
$765$	0	0
$766$	30981.2	1.46135
$767$	−12176.0	−0.573208
$768$	0	0
$769$	37870.0	1.77585	0.887925	−	0.459989i	$-0.152147\pi$
$769$	37870.0	1.77585	0.887925	+	0.459989i	$0.152147\pi$
$770$	0	0
$771$	0	0
$772$	−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	$0.372631\pi$
$774$	0	0
$775$	376.589	0.0174548
$776$	−19277.9	−0.891801
$777$	0	0
$778$	21376.0	0.985047
$779$	4884.87	0.224671
$780$	0	0
$781$	53265.1	2.44043
$782$	908.793	0.0415580
$783$	0	0
$784$	0	0
$785$	5682.70	0.258375
$786$	0	0
$787$	17080.1	0.773622	0.386811	−	0.922159i	$-0.373577\pi$
$787$	17080.1	0.773622	0.386811	+	0.922159i	$0.373577\pi$
$788$	−1816.68	−0.0821274
$789$	0	0
$790$	11789.4	0.530948
$791$	0	0
$792$	0	0
$793$	7929.95	0.355108
$794$	22397.0	1.00106
$795$	0	0
$796$	−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
$801$	0	0
$802$	21616.5	0.951752
$803$	−5155.97	−0.226588
$804$	0	0
$805$	0	0
$806$	1064.96	0.0465404
$807$	0	0
$808$	40300.0	1.75464
$809$	−36969.6	−1.60665	−0.803325	−	0.595541i	$-0.796938\pi$
$809$	−36969.6	−1.60665	−0.803325	+	0.595541i	$0.796938\pi$
$810$	0	0
$811$	−35231.2	−1.52544	−0.762722	−	0.646727i	$-0.776137\pi$
$811$	−35231.2	−1.52544	−0.762722	+	0.646727i	$0.776137\pi$
$812$	0	0
$813$	0	0
$814$	15912.3	0.685166
$815$	4376.12	0.188084
$816$	0	0
$817$	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	$-0.473902\pi$
$821$	3853.10	0.163793	0.0818965	+	0.996641i	$0.473902\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	0	0
$827$	−42985.0	−1.80742	−0.903710	−	0.428145i	$-0.859167\pi$
$827$	−42985.0	−1.80742	−0.903710	+	0.428145i	$0.859167\pi$
$828$	0	0
$829$	−7707.12	−0.322894	−0.161447	−	0.986881i	$-0.551616\pi$
$829$	−7707.12	−0.322894	−0.161447	+	0.986881i	$0.551616\pi$
$830$	19765.2	0.826579
$831$	0	0
$832$	−11358.8	−0.473311
$833$	0	0
$834$	0	0
$835$	10609.1	0.439691
$836$	−3941.14	−0.163047
$837$	0	0
$838$	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
$843$	0	0
$844$	2558.95	0.104363
$845$	−8069.42	−0.328517
$846$	0	0
$847$	0	0
$848$	26466.3	1.07176
$849$	0	0
$850$	−577.610	−0.0233081
$851$	−4710.40	−0.189742
$852$	0	0
$853$	−21332.0	−0.856266	−0.428133	−	0.903716i	$-0.640828\pi$
$853$	−21332.0	−0.856266	−0.428133	+	0.903716i	$0.640828\pi$
$854$	0	0
$855$	0	0
$856$	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	$0.273792\pi$
$858$	0	0
$859$	−35989.8	−1.42952	−0.714759	−	0.699371i	$-0.753464\pi$
$859$	−35989.8	−1.42952	−0.714759	+	0.699371i	$0.753464\pi$
$860$	1024.11	0.0406067
$861$	0	0
$862$	−38010.9	−1.50192
$863$	40400.4	1.59356	0.796781	−	0.604268i	$-0.206534\pi$
$863$	40400.4	1.59356	0.796781	+	0.604268i	$0.206534\pi$
$864$	0	0
$865$	2596.30	0.102054
$866$	32135.5	1.26098
$867$	0	0
$868$	0	0
$869$	36552.1	1.42686
$870$	0	0
$871$	−12532.9	−0.487555
$872$	−32699.3	−1.26988
$873$	0	0
$874$	17498.4	0.677221
$875$	0	0
$876$	0	0
$877$	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
$879$	0	0
$880$	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
$885$	0	0
$886$	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	$0.529668\pi$
$888$	0	0
$889$	0	0
$890$	−9642.80	−0.363177
$891$	0	0
$892$	174.619	0.00655458
$893$	25991.1	0.973974
$894$	0	0
$895$	2164.22	0.0808290
$896$	0	0
$897$	0	0
$898$	−5487.02	−0.203902
$899$	−3021.34	−0.112088
$900$	0	0
$901$	−3060.46	−0.113162
$902$	4271.69	0.157685
$903$	0	0
$904$	−4602.44	−0.169331
$905$	−13958.5	−0.512704
$906$	0	0
$907$	−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
$909$	0	0
$910$	0	0
$911$	−5336.79	−0.194090	−0.0970449	−	0.995280i	$-0.530939\pi$
$911$	−5336.79	−0.194090	−0.0970449	+	0.995280i	$0.530939\pi$
$912$	0	0
$913$	61280.2	2.22134
$914$	19465.5	0.704444
$915$	0	0
$916$	2994.82	0.108026
$917$	0	0
$918$	0	0
$919$	−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
$921$	0	0
$922$	−45464.3	−1.62396
$923$	28340.2	1.01065
$924$	0	0
$925$	2993.83	0.106418
$926$	−6926.64	−0.245814
$927$	0	0
$928$	−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	$0.591012\pi$
$930$	0	0
$931$	0	0
$932$	1493.93	0.0525057
$933$	0	0
$934$	50679.4	1.77546
$935$	−1790.83	−0.0626377
$936$	0	0
$937$	47349.8	1.65085	0.825426	−	0.564510i	$-0.190935\pi$
$937$	47349.8	1.65085	0.825426	+	0.564510i	$0.190935\pi$
$938$	0	0
$939$	0	0
$940$	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	$0.322689\pi$
$942$	0	0
$943$	−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	$-0.394591\pi$
$947$	18950.3	0.650267	0.325134	+	0.945668i	$0.394591\pi$
$948$	0	0
$949$	−2743.29	−0.0938365
$950$	−11121.6	−0.379823
$951$	0	0
$952$	0	0
$953$	34526.2	1.17357	0.586785	−	0.809743i	$-0.300393\pi$
$953$	34526.2	1.17357	0.586785	+	0.809743i	$0.300393\pi$
$954$	0	0
$955$	16106.6	0.545756
$956$	182.844	0.00618577
$957$	0	0
$958$	−6919.14	−0.233348
$959$	0	0
$960$	0	0
$961$	−29564.1	−0.992383
$962$	8466.28	0.283746
$963$	0	0
$964$	−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
$969$	0	0
$970$	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	$0.572858\pi$
$972$	0	0
$973$	0	0
$974$	52158.7	1.71589
$975$	0	0
$976$	22411.2	0.735006
$977$	15039.7	0.492490	0.246245	−	0.969208i	$-0.420803\pi$
$977$	15039.7	0.492490	0.246245	+	0.969208i	$0.420803\pi$
$978$	0	0
$979$	−29896.6	−0.975996
$980$	0	0
$981$	0	0
$982$	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	$0.649223\pi$
$984$	0	0
$985$	−15894.3	−0.514146
$986$	4634.10	0.149675
$987$	0	0
$988$	−2096.92	−0.0675223
$989$	−14097.4	−0.453257
$990$	0	0
$991$	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
$993$	0	0
$994$	0	0
$995$	−19347.7	−0.616446
$996$	0	0
$997$	−37173.5	−1.18084	−0.590419	−	0.807097i	$-0.701038\pi$
$997$	−37173.5	−1.18084	−0.590419	+	0.807097i	$0.701038\pi$
$998$	−50117.9	−1.58963
$999$	0	0

Twists

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

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

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-2.92771 q^{2} +0.571487 q^{4} +5.00000 q^{5} +21.7485 q^{8} +O(q^{10})\)	\(q-2.92771 q^{2} +0.571487 q^{4} +5.00000 q^{5} +21.7485 q^{8} -14.6386 q^{10} -45.3855 q^{11} -24.1478 q^{13} -68.2453 q^{16} +7.89162 q^{17} +151.949 q^{19} +2.85744 q^{20} +132.876 q^{22} -39.3342 q^{23} +25.0000 q^{25} +70.6977 q^{26} -200.572 q^{29} +15.0636 q^{31} +25.8142 q^{32} -23.1044 q^{34} +119.753 q^{37} -444.864 q^{38} +108.743 q^{40} +32.1480 q^{41} +358.400 q^{43} -25.9372 q^{44} +115.159 q^{46} +171.051 q^{47} -73.1928 q^{50} -13.8002 q^{52} -387.811 q^{53} -226.927 q^{55} +587.218 q^{58} +504.229 q^{59} -328.392 q^{61} -44.1017 q^{62} +470.386 q^{64} -120.739 q^{65} +519.007 q^{67} +4.50996 q^{68} -1173.62 q^{71} +113.604 q^{73} -350.603 q^{74} +86.8371 q^{76} -805.369 q^{79} -341.227 q^{80} -94.1201 q^{82} -1350.22 q^{83} +39.4581 q^{85} -1049.29 q^{86} -987.068 q^{88} +658.726 q^{89} -22.4790 q^{92} -500.788 q^{94} +759.747 q^{95} -886.401 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 39 q^{4} + 20 q^{5} - 33 q^{8}+O(q^{10}) \) 4 * q - q^2 + 39 * q^4 + 20 * q^5 - 33 * q^8	\( 4 q - q^{2} + 39 q^{4} + 20 q^{5} - 33 q^{8} - 5 q^{10} - 60 q^{11} - 14 q^{13} + 283 q^{16} + 46 q^{17} - 40 q^{19} + 195 q^{20} - 14 q^{22} - 490 q^{23} + 100 q^{25} - 546 q^{26} + 6 q^{29} - 122 q^{31} - 525 q^{32} - 748 q^{34} + 142 q^{37} - 152 q^{38} - 165 q^{40} + 168 q^{41} + 286 q^{43} - 1214 q^{44} - 148 q^{46} - 306 q^{47} - 25 q^{50} + 1934 q^{52} + 276 q^{53} - 300 q^{55} + 652 q^{58} - 328 q^{59} - 174 q^{61} - 1742 q^{62} + 3615 q^{64} - 70 q^{65} - 994 q^{67} + 132 q^{68} - 2442 q^{71} - 14 q^{73} + 2424 q^{74} - 5160 q^{76} - 252 q^{79} + 1415 q^{80} - 2014 q^{82} - 1556 q^{83} + 230 q^{85} - 734 q^{86} - 4486 q^{88} + 600 q^{89} - 5844 q^{92} - 1210 q^{94} - 200 q^{95} + 238 q^{97}+O(q^{100}) \) 4 * q - q^2 + 39 * q^4 + 20 * q^5 - 33 * q^8 - 5 * q^10 - 60 * q^11 - 14 * q^13 + 283 * q^16 + 46 * q^17 - 40 * q^19 + 195 * q^20 - 14 * q^22 - 490 * q^23 + 100 * q^25 - 546 * q^26 + 6 * q^29 - 122 * q^31 - 525 * q^32 - 748 * q^34 + 142 * q^37 - 152 * q^38 - 165 * q^40 + 168 * q^41 + 286 * q^43 - 1214 * q^44 - 148 * q^46 - 306 * q^47 - 25 * q^50 + 1934 * q^52 + 276 * q^53 - 300 * q^55 + 652 * q^58 - 328 * q^59 - 174 * q^61 - 1742 * q^62 + 3615 * q^64 - 70 * q^65 - 994 * q^67 + 132 * q^68 - 2442 * q^71 - 14 * q^73 + 2424 * q^74 - 5160 * q^76 - 252 * q^79 + 1415 * q^80 - 2014 * q^82 - 1556 * q^83 + 230 * q^85 - 734 * q^86 - 4486 * q^88 + 600 * q^89 - 5844 * q^92 - 1210 * q^94 - 200 * q^95 + 238 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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