LMFDB - Embedded newform 1575.4.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +O(q^{10})$	$q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +5.19224 q^{11} -54.5464 q^{13} +24.9309 q^{14} -79.5312 q^{16} +16.1619 q^{17} +87.4470 q^{19} +18.4924 q^{22} +176.477 q^{23} -194.270 q^{26} +32.7926 q^{28} -142.170 q^{29} -94.3002 q^{31} -188.793 q^{32} +57.5616 q^{34} -17.3305 q^{37} +311.447 q^{38} -210.270 q^{41} -521.570 q^{43} +24.3239 q^{44} +628.533 q^{46} -105.417 q^{47} +49.0000 q^{49} -255.531 q^{52} -108.978 q^{53} -82.6543 q^{56} -506.348 q^{58} -210.365 q^{59} -674.304 q^{61} -335.855 q^{62} -36.1449 q^{64} -324.929 q^{67} +75.7131 q^{68} -793.965 q^{71} -315.417 q^{73} -61.7235 q^{74} +409.659 q^{76} +36.3457 q^{77} -425.840 q^{79} -748.887 q^{82} -283.029 q^{83} -1857.60 q^{86} -61.3087 q^{88} +843.131 q^{89} -381.825 q^{91} +826.736 q^{92} -375.447 q^{94} -1537.33 q^{97} +174.516 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 - 3 * q^4 + 14 * q^7 - 3 * q^8	$ 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + 31 q^{11} - 39 q^{13} + 21 q^{14} - 23 q^{16} - 79 q^{17} - 56 q^{19} + 4 q^{22} + 254 q^{23} - 203 q^{26} - 21 q^{28} + 62 q^{29} - 135 q^{31} - 291 q^{32} + 111 q^{34} - 113 q^{37} + 392 q^{38} - 235 q^{41} - 804 q^{43} - 174 q^{44} + 585 q^{46} + 152 q^{47} + 98 q^{49} - 375 q^{52} + 149 q^{53} - 21 q^{56} - 621 q^{58} + 441 q^{59} - 223 q^{61} - 313 q^{62} - 431 q^{64} - 1157 q^{67} + 807 q^{68} - 619 q^{71} - 268 q^{73} - 8 q^{74} + 1512 q^{76} + 217 q^{77} - 427 q^{79} - 735 q^{82} + 1211 q^{83} - 1699 q^{86} + 166 q^{88} - 466 q^{89} - 273 q^{91} + 231 q^{92} - 520 q^{94} - 172 q^{97} + 147 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 - 3 * q^4 + 14 * q^7 - 3 * q^8 + 31 * q^11 - 39 * q^13 + 21 * q^14 - 23 * q^16 - 79 * q^17 - 56 * q^19 + 4 * q^22 + 254 * q^23 - 203 * q^26 - 21 * q^28 + 62 * q^29 - 135 * q^31 - 291 * q^32 + 111 * q^34 - 113 * q^37 + 392 * q^38 - 235 * q^41 - 804 * q^43 - 174 * q^44 + 585 * q^46 + 152 * q^47 + 98 * q^49 - 375 * q^52 + 149 * q^53 - 21 * q^56 - 621 * q^58 + 441 * q^59 - 223 * q^61 - 313 * q^62 - 431 * q^64 - 1157 * q^67 + 807 * q^68 - 619 * q^71 - 268 * q^73 - 8 * q^74 + 1512 * q^76 + 217 * q^77 - 427 * q^79 - 735 * q^82 + 1211 * q^83 - 1699 * q^86 + 166 * q^88 - 466 * q^89 - 273 * q^91 + 231 * q^92 - 520 * q^94 - 172 * q^97 + 147 * q^98

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 $

$2$	3.56155	1.25920	0.629600	−	0.776920i	$-0.283219\pi$
$2$	3.56155	1.25920	0.629600	+	0.776920i	$0.283219\pi$
$3$	0	0
$3$	0	0
$4$	4.68466	0.585582
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−11.8078	−0.521834
$9$	0	0
$10$	0	0
$11$	5.19224	0.142320	0.0711599	−	0.997465i	$-0.477330\pi$
$11$	5.19224	0.142320	0.0711599	+	0.997465i	$0.477330\pi$
$12$	0	0
$13$	−54.5464	−1.16373	−0.581863	−	0.813287i	$-0.697676\pi$
$13$	−54.5464	−1.16373	−0.581863	+	0.813287i	$0.697676\pi$
$14$	24.9309	0.475933
$15$	0	0
$16$	−79.5312	−1.24268
$17$	16.1619	0.230579	0.115289	−	0.993332i	$-0.463220\pi$
$17$	16.1619	0.230579	0.115289	+	0.993332i	$0.463220\pi$
$18$	0	0
$19$	87.4470	1.05588	0.527940	−	0.849282i	$-0.322965\pi$
$19$	87.4470	1.05588	0.527940	+	0.849282i	$0.322965\pi$
$20$	0	0
$21$	0	0
$22$	18.4924	0.179209
$23$	176.477	1.59992	0.799958	−	0.600056i	$-0.204855\pi$
$23$	176.477	1.59992	0.799958	+	0.600056i	$0.204855\pi$
$24$	0	0
$25$	0	0
$26$	−194.270	−1.46536
$27$	0	0
$28$	32.7926	0.221329
$29$	−142.170	−0.910358	−0.455179	−	0.890400i	$-0.650425\pi$
$29$	−142.170	−0.910358	−0.455179	+	0.890400i	$0.650425\pi$
$30$	0	0
$31$	−94.3002	−0.546349	−0.273174	−	0.961965i	$-0.588074\pi$
$31$	−94.3002	−0.546349	−0.273174	+	0.961965i	$0.588074\pi$
$32$	−188.793	−1.04294
$33$	0	0
$34$	57.5616	0.290345
$35$	0	0
$36$	0	0
$37$	−17.3305	−0.0770031	−0.0385016	−	0.999259i	$-0.512258\pi$
$37$	−17.3305	−0.0770031	−0.0385016	+	0.999259i	$0.512258\pi$
$38$	311.447	1.32956
$39$	0	0
$40$	0	0
$41$	−210.270	−0.800942	−0.400471	−	0.916309i	$-0.631154\pi$
$41$	−210.270	−0.800942	−0.400471	+	0.916309i	$0.631154\pi$
$42$	0	0
$43$	−521.570	−1.84974	−0.924868	−	0.380287i	$-0.875825\pi$
$43$	−521.570	−1.84974	−0.924868	+	0.380287i	$0.875825\pi$
$44$	24.3239	0.0833400
$45$	0	0
$46$	628.533	2.01461
$47$	−105.417	−0.327162	−0.163581	−	0.986530i	$-0.552304\pi$
$47$	−105.417	−0.327162	−0.163581	+	0.986530i	$0.552304\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−255.531	−0.681458
$53$	−108.978	−0.282440	−0.141220	−	0.989978i	$-0.545102\pi$
$53$	−108.978	−0.282440	−0.141220	+	0.989978i	$0.545102\pi$
$54$	0	0
$55$	0	0
$56$	−82.6543	−0.197235
$57$	0	0
$58$	−506.348	−1.14632
$59$	−210.365	−0.464189	−0.232094	−	0.972693i	$-0.574558\pi$
$59$	−210.365	−0.464189	−0.232094	+	0.972693i	$0.574558\pi$
$60$	0	0
$61$	−674.304	−1.41534	−0.707670	−	0.706543i	$-0.750254\pi$
$61$	−674.304	−1.41534	−0.707670	+	0.706543i	$0.750254\pi$
$62$	−335.855	−0.687962
$63$	0	0
$64$	−36.1449	−0.0705955
$65$	0	0
$66$	0	0
$67$	−324.929	−0.592484	−0.296242	−	0.955113i	$-0.595733\pi$
$67$	−324.929	−0.592484	−0.296242	+	0.955113i	$0.595733\pi$
$68$	75.7131	0.135023
$69$	0	0
$70$	0	0
$71$	−793.965	−1.32713	−0.663565	−	0.748118i	$-0.730958\pi$
$71$	−793.965	−1.32713	−0.663565	+	0.748118i	$0.730958\pi$
$72$	0	0
$73$	−315.417	−0.505709	−0.252854	−	0.967504i	$-0.581369\pi$
$73$	−315.417	−0.505709	−0.252854	+	0.967504i	$0.581369\pi$
$74$	−61.7235	−0.0969623
$75$	0	0
$76$	409.659	0.618304
$77$	36.3457	0.0537918
$78$	0	0
$79$	−425.840	−0.606465	−0.303233	−	0.952917i	$-0.598066\pi$
$79$	−425.840	−0.606465	−0.303233	+	0.952917i	$0.598066\pi$
$80$	0	0
$81$	0	0
$82$	−748.887	−1.00855
$83$	−283.029	−0.374295	−0.187148	−	0.982332i	$-0.559924\pi$
$83$	−283.029	−0.374295	−0.187148	+	0.982332i	$0.559924\pi$
$84$	0	0
$85$	0	0
$86$	−1857.60	−2.32919
$87$	0	0
$88$	−61.3087	−0.0742674
$89$	843.131	1.00418	0.502088	−	0.864817i	$-0.332565\pi$
$89$	843.131	1.00418	0.502088	+	0.864817i	$0.332565\pi$
$90$	0	0
$91$	−381.825	−0.439847
$92$	826.736	0.936882
$93$	0	0
$94$	−375.447	−0.411962
$95$	0	0
$96$	0	0
$97$	−1537.33	−1.60920	−0.804601	−	0.593816i	$-0.797621\pi$
$97$	−1537.33	−1.60920	−0.804601	+	0.593816i	$0.797621\pi$
$98$	174.516	0.179886
$99$	0	0
$100$	0	0
$101$	1589.99	1.56644	0.783219	−	0.621745i	$-0.213576\pi$
$101$	1589.99	1.56644	0.783219	+	0.621745i	$0.213576\pi$
$102$	0	0
$103$	164.793	0.157646	0.0788228	−	0.996889i	$-0.474884\pi$
$103$	164.793	0.157646	0.0788228	+	0.996889i	$0.474884\pi$
$104$	644.071	0.607273
$105$	0	0
$106$	−388.132	−0.355648
$107$	1184.08	1.06981	0.534904	−	0.844913i	$-0.320348\pi$
$107$	1184.08	1.06981	0.534904	+	0.844913i	$0.320348\pi$
$108$	0	0
$109$	333.247	0.292837	0.146419	−	0.989223i	$-0.453225\pi$
$109$	333.247	0.292837	0.146419	+	0.989223i	$0.453225\pi$
$110$	0	0
$111$	0	0
$112$	−556.719	−0.469687
$113$	1881.49	1.56634	0.783168	−	0.621810i	$-0.213602\pi$
$113$	1881.49	1.56634	0.783168	+	0.621810i	$0.213602\pi$
$114$	0	0
$115$	0	0
$116$	−666.020	−0.533090
$117$	0	0
$118$	−749.224	−0.584506
$119$	113.133	0.0871507
$120$	0	0
$121$	−1304.04	−0.979745
$122$	−2401.57	−1.78220
$123$	0	0
$124$	−441.764	−0.319932
$125$	0	0
$126$	0	0
$127$	−1638.79	−1.14503	−0.572516	−	0.819893i	$-0.694033\pi$
$127$	−1638.79	−1.14503	−0.572516	+	0.819893i	$0.694033\pi$
$128$	1381.61	0.954048
$129$	0	0
$130$	0	0
$131$	598.142	0.398931	0.199465	−	0.979905i	$-0.436079\pi$
$131$	598.142	0.398931	0.199465	+	0.979905i	$0.436079\pi$
$132$	0	0
$133$	612.129	0.399085
$134$	−1157.25	−0.746055
$135$	0	0
$136$	−190.836	−0.120324
$137$	−1005.25	−0.626894	−0.313447	−	0.949606i	$-0.601484\pi$
$137$	−1005.25	−0.626894	−0.313447	+	0.949606i	$0.601484\pi$
$138$	0	0
$139$	−1875.01	−1.14414	−0.572072	−	0.820204i	$-0.693860\pi$
$139$	−1875.01	−1.14414	−0.572072	+	0.820204i	$0.693860\pi$
$140$	0	0
$141$	0	0
$142$	−2827.75	−1.67112
$143$	−283.218	−0.165621
$144$	0	0
$145$	0	0
$146$	−1123.37	−0.636788
$147$	0	0
$148$	−81.1875	−0.0450917
$149$	1051.80	0.578299	0.289150	−	0.957284i	$-0.406627\pi$
$149$	1051.80	0.578299	0.289150	+	0.957284i	$0.406627\pi$
$150$	0	0
$151$	−750.383	−0.404406	−0.202203	−	0.979344i	$-0.564810\pi$
$151$	−750.383	−0.404406	−0.202203	+	0.979344i	$0.564810\pi$
$152$	−1032.55	−0.550994
$153$	0	0
$154$	129.447	0.0677346
$155$	0	0
$156$	0	0
$157$	1453.90	0.739067	0.369533	−	0.929217i	$-0.379518\pi$
$157$	1453.90	0.739067	0.369533	+	0.929217i	$0.379518\pi$
$158$	−1516.65	−0.763660
$159$	0	0
$160$	0	0
$161$	1235.34	0.604711
$162$	0	0
$163$	−1300.49	−0.624921	−0.312461	−	0.949931i	$-0.601153\pi$
$163$	−1300.49	−0.624921	−0.312461	+	0.949931i	$0.601153\pi$
$164$	−985.043	−0.469018
$165$	0	0
$166$	−1008.02	−0.471312
$167$	−2111.46	−0.978381	−0.489191	−	0.872177i	$-0.662708\pi$
$167$	−2111.46	−0.978381	−0.489191	+	0.872177i	$0.662708\pi$
$168$	0	0
$169$	778.310	0.354260
$170$	0	0
$171$	0	0
$172$	−2443.38	−1.08317
$173$	335.292	0.147351	0.0736756	−	0.997282i	$-0.476527\pi$
$173$	335.292	0.147351	0.0736756	+	0.997282i	$0.476527\pi$
$174$	0	0
$175$	0	0
$176$	−412.945	−0.176857
$177$	0	0
$178$	3002.85	1.26446
$179$	−2322.23	−0.969672	−0.484836	−	0.874605i	$-0.661121\pi$
$179$	−2322.23	−0.969672	−0.484836	+	0.874605i	$0.661121\pi$
$180$	0	0
$181$	−1525.59	−0.626500	−0.313250	−	0.949671i	$-0.601418\pi$
$181$	−1525.59	−0.626500	−0.313250	+	0.949671i	$0.601418\pi$
$182$	−1359.89	−0.553855
$183$	0	0
$184$	−2083.80	−0.834891
$185$	0	0
$186$	0	0
$187$	83.9165	0.0328160
$188$	−493.841	−0.191580
$189$	0	0
$190$	0	0
$191$	−293.912	−0.111344	−0.0556721	−	0.998449i	$-0.517730\pi$
$191$	−293.912	−0.111344	−0.0556721	+	0.998449i	$0.517730\pi$
$192$	0	0
$193$	3664.91	1.36687	0.683435	−	0.730012i	$-0.260485\pi$
$193$	3664.91	1.36687	0.683435	+	0.730012i	$0.260485\pi$
$194$	−5475.29	−2.02630
$195$	0	0
$196$	229.548	0.0836546
$197$	−5101.89	−1.84515	−0.922576	−	0.385816i	$-0.873920\pi$
$197$	−5101.89	−1.84515	−0.922576	+	0.385816i	$0.873920\pi$
$198$	0	0
$199$	5025.86	1.79032	0.895161	−	0.445743i	$-0.147061\pi$
$199$	5025.86	1.79032	0.895161	+	0.445743i	$0.147061\pi$
$200$	0	0
$201$	0	0
$202$	5662.85	1.97246
$203$	−995.193	−0.344083
$204$	0	0
$205$	0	0
$206$	586.918	0.198507
$207$	0	0
$208$	4338.14	1.44614
$209$	454.045	0.150273
$210$	0	0
$211$	−3267.98	−1.06624	−0.533122	−	0.846039i	$-0.678981\pi$
$211$	−3267.98	−1.06624	−0.533122	+	0.846039i	$0.678981\pi$
$212$	−510.526	−0.165392
$213$	0	0
$214$	4217.17	1.34710
$215$	0	0
$216$	0	0
$217$	−660.101	−0.206500
$218$	1186.88	0.368741
$219$	0	0
$220$	0	0
$221$	−881.575	−0.268331
$222$	0	0
$223$	−5457.65	−1.63888	−0.819442	−	0.573162i	$-0.805717\pi$
$223$	−5457.65	−1.63888	−0.819442	+	0.573162i	$0.805717\pi$
$224$	−1321.55	−0.394195
$225$	0	0
$226$	6701.04	1.97233
$227$	281.023	0.0821682	0.0410841	−	0.999156i	$-0.486919\pi$
$227$	281.023	0.0821682	0.0410841	+	0.999156i	$0.486919\pi$
$228$	0	0
$229$	2776.64	0.801248	0.400624	−	0.916243i	$-0.368793\pi$
$229$	2776.64	0.801248	0.400624	+	0.916243i	$0.368793\pi$
$230$	0	0
$231$	0	0
$232$	1678.71	0.475056
$233$	5781.09	1.62546	0.812729	−	0.582642i	$-0.197981\pi$
$233$	5781.09	1.62546	0.812729	+	0.582642i	$0.197981\pi$
$234$	0	0
$235$	0	0
$236$	−985.486	−0.271821
$237$	0	0
$238$	402.931	0.109740
$239$	−1588.17	−0.429833	−0.214916	−	0.976632i	$-0.568948\pi$
$239$	−1588.17	−0.429833	−0.214916	+	0.976632i	$0.568948\pi$
$240$	0	0
$241$	−4330.01	−1.15735	−0.578673	−	0.815560i	$-0.696429\pi$
$241$	−4330.01	−1.15735	−0.578673	+	0.815560i	$0.696429\pi$
$242$	−4644.41	−1.23369
$243$	0	0
$244$	−3158.88	−0.828798
$245$	0	0
$246$	0	0
$247$	−4769.92	−1.22876
$248$	1113.47	0.285104
$249$	0	0
$250$	0	0
$251$	−1400.53	−0.352195	−0.176097	−	0.984373i	$-0.556347\pi$
$251$	−1400.53	−0.352195	−0.176097	+	0.984373i	$0.556347\pi$
$252$	0	0
$253$	916.312	0.227700
$254$	−5836.64	−1.44182
$255$	0	0
$256$	5209.83	1.27193
$257$	4304.86	1.04486	0.522431	−	0.852681i	$-0.325025\pi$
$257$	4304.86	1.04486	0.522431	+	0.852681i	$0.325025\pi$
$258$	0	0
$259$	−121.313	−0.0291045
$260$	0	0
$261$	0	0
$262$	2130.31	0.502333
$263$	−1724.69	−0.404369	−0.202184	−	0.979347i	$-0.564804\pi$
$263$	−1724.69	−0.404369	−0.202184	+	0.979347i	$0.564804\pi$
$264$	0	0
$265$	0	0
$266$	2180.13	0.502527
$267$	0	0
$268$	−1522.18	−0.346948
$269$	−8004.82	−1.81436	−0.907180	−	0.420744i	$-0.861769\pi$
$269$	−8004.82	−1.81436	−0.907180	+	0.420744i	$0.861769\pi$
$270$	0	0
$271$	−1963.65	−0.440160	−0.220080	−	0.975482i	$-0.570632\pi$
$271$	−1963.65	−0.440160	−0.220080	+	0.975482i	$0.570632\pi$
$272$	−1285.38	−0.286535
$273$	0	0
$274$	−3580.26	−0.789384
$275$	0	0
$276$	0	0
$277$	3278.33	0.711104	0.355552	−	0.934656i	$-0.384293\pi$
$277$	3278.33	0.711104	0.355552	+	0.934656i	$0.384293\pi$
$278$	−6677.93	−1.44070
$279$	0	0
$280$	0	0
$281$	−2859.04	−0.606961	−0.303480	−	0.952838i	$-0.598149\pi$
$281$	−2859.04	−0.606961	−0.303480	+	0.952838i	$0.598149\pi$
$282$	0	0
$283$	5433.66	1.14134	0.570668	−	0.821181i	$-0.306685\pi$
$283$	5433.66	1.14134	0.570668	+	0.821181i	$0.306685\pi$
$284$	−3719.45	−0.777144
$285$	0	0
$286$	−1008.70	−0.208550
$287$	−1471.89	−0.302728
$288$	0	0
$289$	−4651.79	−0.946833
$290$	0	0
$291$	0	0
$292$	−1477.62	−0.296134
$293$	8583.43	1.71143	0.855715	−	0.517447i	$-0.173117\pi$
$293$	8583.43	1.71143	0.855715	+	0.517447i	$0.173117\pi$
$294$	0	0
$295$	0	0
$296$	204.634	0.0401829
$297$	0	0
$298$	3746.03	0.728194
$299$	−9626.20	−1.86186
$300$	0	0
$301$	−3650.99	−0.699135
$302$	−2672.53	−0.509228
$303$	0	0
$304$	−6954.77	−1.31212
$305$	0	0
$306$	0	0
$307$	5269.83	0.979691	0.489846	−	0.871809i	$-0.337053\pi$
$307$	5269.83	0.979691	0.489846	+	0.871809i	$0.337053\pi$
$308$	170.267	0.0314995
$309$	0	0
$310$	0	0
$311$	4761.43	0.868154	0.434077	−	0.900876i	$-0.357075\pi$
$311$	4761.43	0.868154	0.434077	+	0.900876i	$0.357075\pi$
$312$	0	0
$313$	7602.95	1.37298	0.686492	−	0.727137i	$-0.259150\pi$
$313$	7602.95	1.37298	0.686492	+	0.727137i	$0.259150\pi$
$314$	5178.13	0.930632
$315$	0	0
$316$	−1994.91	−0.355135
$317$	8064.55	1.42886	0.714432	−	0.699704i	$-0.246685\pi$
$317$	8064.55	1.42886	0.714432	+	0.699704i	$0.246685\pi$
$318$	0	0
$319$	−738.182	−0.129562
$320$	0	0
$321$	0	0
$322$	4399.73	0.761452
$323$	1413.31	0.243464
$324$	0	0
$325$	0	0
$326$	−4631.76	−0.786901
$327$	0	0
$328$	2482.82	0.417959
$329$	−737.917	−0.123655
$330$	0	0
$331$	6960.79	1.15589	0.577945	−	0.816076i	$-0.303855\pi$
$331$	6960.79	1.15589	0.577945	+	0.816076i	$0.303855\pi$
$332$	−1325.90	−0.219181
$333$	0	0
$334$	−7520.08	−1.23198
$335$	0	0
$336$	0	0
$337$	−4731.61	−0.764828	−0.382414	−	0.923991i	$-0.624907\pi$
$337$	−4731.61	−0.764828	−0.382414	+	0.923991i	$0.624907\pi$
$338$	2771.99	0.446084
$339$	0	0
$340$	0	0
$341$	−489.629	−0.0777563
$342$	0	0
$343$	343.000	0.0539949
$344$	6158.58	0.965256
$345$	0	0
$346$	1194.16	0.185544
$347$	9796.67	1.51560	0.757800	−	0.652487i	$-0.226274\pi$
$347$	9796.67	1.51560	0.757800	+	0.652487i	$0.226274\pi$
$348$	0	0
$349$	12702.4	1.94827	0.974134	−	0.225971i	$-0.0725554\pi$
$349$	12702.4	1.94827	0.974134	+	0.225971i	$0.0725554\pi$
$350$	0	0
$351$	0	0
$352$	−980.256	−0.148431
$353$	−9970.21	−1.50329	−0.751644	−	0.659569i	$-0.770739\pi$
$353$	−9970.21	−1.50329	−0.751644	+	0.659569i	$0.770739\pi$
$354$	0	0
$355$	0	0
$356$	3949.78	0.588028
$357$	0	0
$358$	−8270.73	−1.22101
$359$	−4388.21	−0.645128	−0.322564	−	0.946548i	$-0.604545\pi$
$359$	−4388.21	−0.645128	−0.322564	+	0.946548i	$0.604545\pi$
$360$	0	0
$361$	787.970	0.114881
$362$	−5433.49	−0.788889
$363$	0	0
$364$	−1788.72	−0.257567
$365$	0	0
$366$	0	0
$367$	9441.30	1.34287	0.671433	−	0.741065i	$-0.265679\pi$
$367$	9441.30	1.34287	0.671433	+	0.741065i	$0.265679\pi$
$368$	−14035.5	−1.98818
$369$	0	0
$370$	0	0
$371$	−762.847	−0.106752
$372$	0	0
$373$	3219.40	0.446901	0.223451	−	0.974715i	$-0.428268\pi$
$373$	3219.40	0.446901	0.223451	+	0.974715i	$0.428268\pi$
$374$	298.873	0.0413218
$375$	0	0
$376$	1244.73	0.170724
$377$	7754.89	1.05941
$378$	0	0
$379$	−14011.4	−1.89899	−0.949495	−	0.313783i	$-0.898403\pi$
$379$	−14011.4	−1.89899	−0.949495	+	0.313783i	$0.898403\pi$
$380$	0	0
$381$	0	0
$382$	−1046.78	−0.140204
$383$	5322.87	0.710147	0.355073	−	0.934838i	$-0.384456\pi$
$383$	5322.87	0.710147	0.355073	+	0.934838i	$0.384456\pi$
$384$	0	0
$385$	0	0
$386$	13052.8	1.72116
$387$	0	0
$388$	−7201.88	−0.942320
$389$	3844.51	0.501091	0.250545	−	0.968105i	$-0.419390\pi$
$389$	3844.51	0.501091	0.250545	+	0.968105i	$0.419390\pi$
$390$	0	0
$391$	2852.21	0.368907
$392$	−578.580	−0.0745478
$393$	0	0
$394$	−18170.7	−2.32341
$395$	0	0
$396$	0	0
$397$	8046.40	1.01722	0.508611	−	0.860996i	$-0.330159\pi$
$397$	8046.40	1.01722	0.508611	+	0.860996i	$0.330159\pi$
$398$	17899.9	2.25437
$399$	0	0
$400$	0	0
$401$	−7741.38	−0.964055	−0.482027	−	0.876156i	$-0.660099\pi$
$401$	−7741.38	−0.964055	−0.482027	+	0.876156i	$0.660099\pi$
$402$	0	0
$403$	5143.74	0.635801
$404$	7448.58	0.917279
$405$	0	0
$406$	−3544.43	−0.433269
$407$	−89.9840	−0.0109591
$408$	0	0
$409$	−8966.94	−1.08407	−0.542037	−	0.840354i	$-0.682347\pi$
$409$	−8966.94	−1.08407	−0.542037	+	0.840354i	$0.682347\pi$
$410$	0	0
$411$	0	0
$412$	771.997	0.0923145
$413$	−1472.55	−0.175447
$414$	0	0
$415$	0	0
$416$	10298.0	1.21370
$417$	0	0
$418$	1617.11	0.189223
$419$	12413.6	1.44736	0.723681	−	0.690135i	$-0.242449\pi$
$419$	12413.6	1.44736	0.723681	+	0.690135i	$0.242449\pi$
$420$	0	0
$421$	−1672.14	−0.193575	−0.0967875	−	0.995305i	$-0.530857\pi$
$421$	−1672.14	−0.193575	−0.0967875	+	0.995305i	$0.530857\pi$
$422$	−11639.1	−1.34261
$423$	0	0
$424$	1286.79	0.147387
$425$	0	0
$426$	0	0
$427$	−4720.13	−0.534948
$428$	5547.02	0.626461
$429$	0	0
$430$	0	0
$431$	−16021.8	−1.79059	−0.895296	−	0.445472i	$-0.853036\pi$
$431$	−16021.8	−1.79059	−0.895296	+	0.445472i	$0.853036\pi$
$432$	0	0
$433$	−10882.7	−1.20782	−0.603912	−	0.797051i	$-0.706392\pi$
$433$	−10882.7	−1.20782	−0.603912	+	0.797051i	$0.706392\pi$
$434$	−2350.99	−0.260025
$435$	0	0
$436$	1561.15	0.171480
$437$	15432.4	1.68932
$438$	0	0
$439$	7738.40	0.841307	0.420653	−	0.907221i	$-0.361801\pi$
$439$	7738.40	0.841307	0.420653	+	0.907221i	$0.361801\pi$
$440$	0	0
$441$	0	0
$442$	−3139.78	−0.337882
$443$	8766.56	0.940207	0.470103	−	0.882611i	$-0.344217\pi$
$443$	8766.56	0.940207	0.470103	+	0.882611i	$0.344217\pi$
$444$	0	0
$445$	0	0
$446$	−19437.7	−2.06368
$447$	0	0
$448$	−253.014	−0.0266826
$449$	−3099.58	−0.325786	−0.162893	−	0.986644i	$-0.552083\pi$
$449$	−3099.58	−0.325786	−0.162893	+	0.986644i	$0.552083\pi$
$450$	0	0
$451$	−1091.77	−0.113990
$452$	8814.15	0.917219
$453$	0	0
$454$	1000.88	0.103466
$455$	0	0
$456$	0	0
$457$	6122.94	0.626737	0.313369	−	0.949632i	$-0.398542\pi$
$457$	6122.94	0.626737	0.313369	+	0.949632i	$0.398542\pi$
$458$	9889.16	1.00893
$459$	0	0
$460$	0	0
$461$	10412.2	1.05194	0.525970	−	0.850503i	$-0.323703\pi$
$461$	10412.2	1.05194	0.525970	+	0.850503i	$0.323703\pi$
$462$	0	0
$463$	11278.5	1.13209	0.566043	−	0.824376i	$-0.308474\pi$
$463$	11278.5	1.13209	0.566043	+	0.824376i	$0.308474\pi$
$464$	11307.0	1.13128
$465$	0	0
$466$	20589.6	2.04677
$467$	14923.2	1.47872	0.739359	−	0.673311i	$-0.235128\pi$
$467$	14923.2	1.47872	0.739359	+	0.673311i	$0.235128\pi$
$468$	0	0
$469$	−2274.50	−0.223938
$470$	0	0
$471$	0	0
$472$	2483.93	0.242230
$473$	−2708.11	−0.263254
$474$	0	0
$475$	0	0
$476$	529.992	0.0510339
$477$	0	0
$478$	−5656.34	−0.541245
$479$	4674.21	0.445867	0.222933	−	0.974834i	$-0.428437\pi$
$479$	4674.21	0.445867	0.222933	+	0.974834i	$0.428437\pi$
$480$	0	0
$481$	945.316	0.0896106
$482$	−15421.6	−1.45733
$483$	0	0
$484$	−6108.99	−0.573721
$485$	0	0
$486$	0	0
$487$	−17081.7	−1.58941	−0.794706	−	0.606994i	$-0.792375\pi$
$487$	−17081.7	−1.58941	−0.794706	+	0.606994i	$0.792375\pi$
$488$	7962.02	0.738573
$489$	0	0
$490$	0	0
$491$	−18203.9	−1.67318	−0.836588	−	0.547832i	$-0.815453\pi$
$491$	−18203.9	−1.67318	−0.836588	+	0.547832i	$0.815453\pi$
$492$	0	0
$493$	−2297.75	−0.209909
$494$	−16988.3	−1.54725
$495$	0	0
$496$	7499.81	0.678934
$497$	−5557.75	−0.501608
$498$	0	0
$499$	7109.47	0.637803	0.318901	−	0.947788i	$-0.396686\pi$
$499$	7109.47	0.637803	0.318901	+	0.947788i	$0.396686\pi$
$500$	0	0
$501$	0	0
$502$	−4988.07	−0.443483
$503$	−15402.0	−1.36529	−0.682647	−	0.730748i	$-0.739171\pi$
$503$	−15402.0	−1.36529	−0.682647	+	0.730748i	$0.739171\pi$
$504$	0	0
$505$	0	0
$506$	3263.49	0.286719
$507$	0	0
$508$	−7677.17	−0.670511
$509$	−6404.72	−0.557730	−0.278865	−	0.960330i	$-0.589958\pi$
$509$	−6404.72	−0.557730	−0.278865	+	0.960330i	$0.589958\pi$
$510$	0	0
$511$	−2207.92	−0.191140
$512$	7502.22	0.647567
$513$	0	0
$514$	15332.0	1.31569
$515$	0	0
$516$	0	0
$517$	−547.348	−0.0465616
$518$	−432.064	−0.0366483
$519$	0	0
$520$	0	0
$521$	8916.72	0.749806	0.374903	−	0.927064i	$-0.377676\pi$
$521$	8916.72	0.749806	0.374903	+	0.927064i	$0.377676\pi$
$522$	0	0
$523$	6929.40	0.579353	0.289677	−	0.957125i	$-0.406452\pi$
$523$	6929.40	0.579353	0.289677	+	0.957125i	$0.406452\pi$
$524$	2802.09	0.233607
$525$	0	0
$526$	−6142.58	−0.509181
$527$	−1524.07	−0.125977
$528$	0	0
$529$	18977.2	1.55973
$530$	0	0
$531$	0	0
$532$	2867.61	0.233697
$533$	11469.5	0.932078
$534$	0	0
$535$	0	0
$536$	3836.69	0.309178
$537$	0	0
$538$	−28509.6	−2.28464
$539$	254.420	0.0203314
$540$	0	0
$541$	−6929.23	−0.550667	−0.275334	−	0.961349i	$-0.588788\pi$
$541$	−6929.23	−0.550667	−0.275334	+	0.961349i	$0.588788\pi$
$542$	−6993.65	−0.554249
$543$	0	0
$544$	−3051.25	−0.240480
$545$	0	0
$546$	0	0
$547$	8509.95	0.665190	0.332595	−	0.943070i	$-0.392076\pi$
$547$	8509.95	0.665190	0.332595	+	0.943070i	$0.392076\pi$
$548$	−4709.26	−0.367098
$549$	0	0
$550$	0	0
$551$	−12432.4	−0.961228
$552$	0	0
$553$	−2980.88	−0.229222
$554$	11676.0	0.895422
$555$	0	0
$556$	−8783.76	−0.669990
$557$	−4043.94	−0.307625	−0.153813	−	0.988100i	$-0.549155\pi$
$557$	−4043.94	−0.307625	−0.153813	+	0.988100i	$0.549155\pi$
$558$	0	0
$559$	28449.8	2.15259
$560$	0	0
$561$	0	0
$562$	−10182.6	−0.764285
$563$	15878.9	1.18866	0.594331	−	0.804221i	$-0.297417\pi$
$563$	15878.9	1.18866	0.594331	+	0.804221i	$0.297417\pi$
$564$	0	0
$565$	0	0
$566$	19352.3	1.43717
$567$	0	0
$568$	9374.95	0.692543
$569$	−11611.6	−0.855510	−0.427755	−	0.903895i	$-0.640695\pi$
$569$	−11611.6	−0.855510	−0.427755	+	0.903895i	$0.640695\pi$
$570$	0	0
$571$	17395.7	1.27493	0.637466	−	0.770478i	$-0.279982\pi$
$571$	17395.7	1.27493	0.637466	+	0.770478i	$0.279982\pi$
$572$	−1326.78	−0.0969850
$573$	0	0
$574$	−5242.21	−0.381195
$575$	0	0
$576$	0	0
$577$	11474.0	0.827848	0.413924	−	0.910311i	$-0.364158\pi$
$577$	11474.0	0.827848	0.413924	+	0.910311i	$0.364158\pi$
$578$	−16567.6	−1.19225
$579$	0	0
$580$	0	0
$581$	−1981.20	−0.141470
$582$	0	0
$583$	−565.841	−0.0401968
$584$	3724.37	0.263896
$585$	0	0
$586$	30570.3	2.15503
$587$	11870.4	0.834659	0.417330	−	0.908755i	$-0.362966\pi$
$587$	11870.4	0.834659	0.417330	+	0.908755i	$0.362966\pi$
$588$	0	0
$589$	−8246.26	−0.576878
$590$	0	0
$591$	0	0
$592$	1378.32	0.0956899
$593$	−5760.65	−0.398923	−0.199462	−	0.979906i	$-0.563919\pi$
$593$	−5760.65	−0.398923	−0.199462	+	0.979906i	$0.563919\pi$
$594$	0	0
$595$	0	0
$596$	4927.31	0.338642
$597$	0	0
$598$	−34284.2	−2.34446
$599$	21696.5	1.47996	0.739978	−	0.672631i	$-0.234836\pi$
$599$	21696.5	1.47996	0.739978	+	0.672631i	$0.234836\pi$
$600$	0	0
$601$	−12403.0	−0.841812	−0.420906	−	0.907104i	$-0.638288\pi$
$601$	−12403.0	−0.841812	−0.420906	+	0.907104i	$0.638288\pi$
$602$	−13003.2	−0.880350
$603$	0	0
$604$	−3515.29	−0.236813
$605$	0	0
$606$	0	0
$607$	−17066.5	−1.14120	−0.570600	−	0.821228i	$-0.693289\pi$
$607$	−17066.5	−1.14120	−0.570600	+	0.821228i	$0.693289\pi$
$608$	−16509.3	−1.10122
$609$	0	0
$610$	0	0
$611$	5750.10	0.380727
$612$	0	0
$613$	−2707.50	−0.178393	−0.0891965	−	0.996014i	$-0.528430\pi$
$613$	−2707.50	−0.178393	−0.0891965	+	0.996014i	$0.528430\pi$
$614$	18768.8	1.23363
$615$	0	0
$616$	−429.161	−0.0280704
$617$	−23226.3	−1.51549	−0.757743	−	0.652553i	$-0.773698\pi$
$617$	−23226.3	−1.51549	−0.757743	+	0.652553i	$0.773698\pi$
$618$	0	0
$619$	−2298.43	−0.149243	−0.0746216	−	0.997212i	$-0.523775\pi$
$619$	−2298.43	−0.149243	−0.0746216	+	0.997212i	$0.523775\pi$
$620$	0	0
$621$	0	0
$622$	16958.1	1.09318
$623$	5901.91	0.379543
$624$	0	0
$625$	0	0
$626$	27078.3	1.72886
$627$	0	0
$628$	6811.01	0.432785
$629$	−280.094	−0.0177553
$630$	0	0
$631$	−663.913	−0.0418858	−0.0209429	−	0.999781i	$-0.506667\pi$
$631$	−663.913	−0.0418858	−0.0209429	+	0.999781i	$0.506667\pi$
$632$	5028.22	0.316474
$633$	0	0
$634$	28722.3	1.79923
$635$	0	0
$636$	0	0
$637$	−2672.77	−0.166247
$638$	−2629.08	−0.163144
$639$	0	0
$640$	0	0
$641$	−15215.6	−0.937566	−0.468783	−	0.883313i	$-0.655307\pi$
$641$	−15215.6	−0.937566	−0.468783	+	0.883313i	$0.655307\pi$
$642$	0	0
$643$	−12904.0	−0.791420	−0.395710	−	0.918375i	$-0.629502\pi$
$643$	−12904.0	−0.791420	−0.395710	+	0.918375i	$0.629502\pi$
$644$	5787.15	0.354108
$645$	0	0
$646$	5033.58	0.306569
$647$	−9425.54	−0.572730	−0.286365	−	0.958121i	$-0.592447\pi$
$647$	−9425.54	−0.572730	−0.286365	+	0.958121i	$0.592447\pi$
$648$	0	0
$649$	−1092.26	−0.0660632
$650$	0	0
$651$	0	0
$652$	−6092.35	−0.365943
$653$	−29894.7	−1.79153	−0.895765	−	0.444528i	$-0.853372\pi$
$653$	−29894.7	−1.79153	−0.895765	+	0.444528i	$0.853372\pi$
$654$	0	0
$655$	0	0
$656$	16723.0	0.995312
$657$	0	0
$658$	−2628.13	−0.155707
$659$	11593.6	0.685313	0.342656	−	0.939461i	$-0.388673\pi$
$659$	11593.6	0.685313	0.342656	+	0.939461i	$0.388673\pi$
$660$	0	0
$661$	−17149.2	−1.00911	−0.504557	−	0.863378i	$-0.668344\pi$
$661$	−17149.2	−1.00911	−0.504557	+	0.863378i	$0.668344\pi$
$662$	24791.2	1.45550
$663$	0	0
$664$	3341.94	0.195320
$665$	0	0
$666$	0	0
$667$	−25089.9	−1.45650
$668$	−9891.47	−0.572923
$669$	0	0
$670$	0	0
$671$	−3501.15	−0.201431
$672$	0	0
$673$	16475.0	0.943633	0.471817	−	0.881697i	$-0.343598\pi$
$673$	16475.0	0.943633	0.471817	+	0.881697i	$0.343598\pi$
$674$	−16851.9	−0.963071
$675$	0	0
$676$	3646.11	0.207448
$677$	−4559.89	−0.258864	−0.129432	−	0.991588i	$-0.541315\pi$
$677$	−4559.89	−0.258864	−0.129432	+	0.991588i	$0.541315\pi$
$678$	0	0
$679$	−10761.3	−0.608221
$680$	0	0
$681$	0	0
$682$	−1743.84	−0.0979106
$683$	−27895.9	−1.56282	−0.781411	−	0.624017i	$-0.785500\pi$
$683$	−27895.9	−1.56282	−0.781411	+	0.624017i	$0.785500\pi$
$684$	0	0
$685$	0	0
$686$	1221.61	0.0679904
$687$	0	0
$688$	41481.1	2.29862
$689$	5944.37	0.328683
$690$	0	0
$691$	−28178.5	−1.55132	−0.775659	−	0.631152i	$-0.782582\pi$
$691$	−28178.5	−1.55132	−0.775659	+	0.631152i	$0.782582\pi$
$692$	1570.73	0.0862862
$693$	0	0
$694$	34891.4	1.90844
$695$	0	0
$696$	0	0
$697$	−3398.37	−0.184680
$698$	45240.4	2.45326
$699$	0	0
$700$	0	0
$701$	−3912.96	−0.210828	−0.105414	−	0.994428i	$-0.533617\pi$
$701$	−3912.96	−0.210828	−0.105414	+	0.994428i	$0.533617\pi$
$702$	0	0
$703$	−1515.50	−0.0813060
$704$	−187.673	−0.0100471
$705$	0	0
$706$	−35509.4	−1.89294
$707$	11130.0	0.592058
$708$	0	0
$709$	−7782.72	−0.412251	−0.206126	−	0.978526i	$-0.566086\pi$
$709$	−7782.72	−0.412251	−0.206126	+	0.978526i	$0.566086\pi$
$710$	0	0
$711$	0	0
$712$	−9955.49	−0.524014
$713$	−16641.8	−0.874112
$714$	0	0
$715$	0	0
$716$	−10878.8	−0.567823
$717$	0	0
$718$	−15628.9	−0.812345
$719$	27868.8	1.44552	0.722762	−	0.691097i	$-0.242872\pi$
$719$	27868.8	1.44552	0.722762	+	0.691097i	$0.242872\pi$
$720$	0	0
$721$	1153.55	0.0595844
$722$	2806.40	0.144658
$723$	0	0
$724$	−7146.89	−0.366868
$725$	0	0
$726$	0	0
$727$	34202.4	1.74484	0.872419	−	0.488759i	$-0.162550\pi$
$727$	34202.4	1.74484	0.872419	+	0.488759i	$0.162550\pi$
$728$	4508.50	0.229527
$729$	0	0
$730$	0	0
$731$	−8429.58	−0.426510
$732$	0	0
$733$	12544.2	0.632101	0.316051	−	0.948742i	$-0.397643\pi$
$733$	12544.2	0.632101	0.316051	+	0.948742i	$0.397643\pi$
$734$	33625.7	1.69094
$735$	0	0
$736$	−33317.6	−1.66862
$737$	−1687.11	−0.0843221
$738$	0	0
$739$	4563.19	0.227144	0.113572	−	0.993530i	$-0.463771\pi$
$739$	4563.19	0.227144	0.113572	+	0.993530i	$0.463771\pi$
$740$	0	0
$741$	0	0
$742$	−2716.92	−0.134422
$743$	10369.7	0.512017	0.256009	−	0.966674i	$-0.417592\pi$
$743$	10369.7	0.512017	0.256009	+	0.966674i	$0.417592\pi$
$744$	0	0
$745$	0	0
$746$	11466.1	0.562738
$747$	0	0
$748$	393.120	0.0192164
$749$	8288.57	0.404349
$750$	0	0
$751$	−36808.0	−1.78847	−0.894237	−	0.447595i	$-0.852281\pi$
$751$	−36808.0	−1.78847	−0.894237	+	0.447595i	$0.852281\pi$
$752$	8383.92	0.406556
$753$	0	0
$754$	27619.4	1.33401
$755$	0	0
$756$	0	0
$757$	12516.6	0.600955	0.300477	−	0.953789i	$-0.402854\pi$
$757$	12516.6	0.600955	0.300477	+	0.953789i	$0.402854\pi$
$758$	−49902.3	−2.39121
$759$	0	0
$760$	0	0
$761$	−11745.1	−0.559473	−0.279736	−	0.960077i	$-0.590247\pi$
$761$	−11745.1	−0.559473	−0.279736	+	0.960077i	$0.590247\pi$
$762$	0	0
$763$	2332.73	0.110682
$764$	−1376.88	−0.0652011
$765$	0	0
$766$	18957.7	0.894216
$767$	11474.6	0.540189
$768$	0	0
$769$	36497.1	1.71147	0.855735	−	0.517414i	$-0.173105\pi$
$769$	36497.1	1.71147	0.855735	+	0.517414i	$0.173105\pi$
$770$	0	0
$771$	0	0
$772$	17168.8	0.800414
$773$	−29858.1	−1.38929	−0.694644	−	0.719353i	$-0.744438\pi$
$773$	−29858.1	−1.38929	−0.694644	+	0.719353i	$0.744438\pi$
$774$	0	0
$775$	0	0
$776$	18152.5	0.839737
$777$	0	0
$778$	13692.4	0.630973
$779$	−18387.5	−0.845699
$780$	0	0
$781$	−4122.45	−0.188877
$782$	10158.3	0.464527
$783$	0	0
$784$	−3897.03	−0.177525
$785$	0	0
$786$	0	0
$787$	−3168.00	−0.143491	−0.0717453	−	0.997423i	$-0.522857\pi$
$787$	−3168.00	−0.143491	−0.0717453	+	0.997423i	$0.522857\pi$
$788$	−23900.6	−1.08049
$789$	0	0
$790$	0	0
$791$	13170.5	0.592019
$792$	0	0
$793$	36780.8	1.64707
$794$	28657.7	1.28089
$795$	0	0
$796$	23544.5	1.04838
$797$	29317.0	1.30296	0.651481	−	0.758665i	$-0.274148\pi$
$797$	29317.0	1.30296	0.651481	+	0.758665i	$0.274148\pi$
$798$	0	0
$799$	−1703.74	−0.0754366
$800$	0	0
$801$	0	0
$802$	−27571.3	−1.21394
$803$	−1637.72	−0.0719724
$804$	0	0
$805$	0	0
$806$	18319.7	0.800600
$807$	0	0
$808$	−18774.3	−0.817422
$809$	−16657.3	−0.723904	−0.361952	−	0.932197i	$-0.617890\pi$
$809$	−16657.3	−0.723904	−0.361952	+	0.932197i	$0.617890\pi$
$810$	0	0
$811$	5144.55	0.222749	0.111375	−	0.993779i	$-0.464475\pi$
$811$	5144.55	0.222749	0.111375	+	0.993779i	$0.464475\pi$
$812$	−4662.14	−0.201489
$813$	0	0
$814$	−320.483	−0.0137997
$815$	0	0
$816$	0	0
$817$	−45609.7	−1.95310
$818$	−31936.2	−1.36507
$819$	0	0
$820$	0	0
$821$	5217.18	0.221779	0.110890	−	0.993833i	$-0.464630\pi$
$821$	5217.18	0.221779	0.110890	+	0.993833i	$0.464630\pi$
$822$	0	0
$823$	−42326.5	−1.79272	−0.896360	−	0.443327i	$-0.853798\pi$
$823$	−42326.5	−1.79272	−0.896360	+	0.443327i	$0.853798\pi$
$824$	−1945.83	−0.0822649
$825$	0	0
$826$	−5244.57	−0.220922
$827$	−31675.8	−1.33189	−0.665946	−	0.746000i	$-0.731972\pi$
$827$	−31675.8	−1.33189	−0.665946	+	0.746000i	$0.731972\pi$
$828$	0	0
$829$	−3471.22	−0.145429	−0.0727144	−	0.997353i	$-0.523166\pi$
$829$	−3471.22	−0.145429	−0.0727144	+	0.997353i	$0.523166\pi$
$830$	0	0
$831$	0	0
$832$	1971.57	0.0821539
$833$	791.934	0.0329399
$834$	0	0
$835$	0	0
$836$	2127.05	0.0879970
$837$	0	0
$838$	44211.7	1.82252
$839$	20964.9	0.862682	0.431341	−	0.902189i	$-0.358041\pi$
$839$	20964.9	0.862682	0.431341	+	0.902189i	$0.358041\pi$
$840$	0	0
$841$	−4176.57	−0.171248
$842$	−5955.41	−0.243749
$843$	0	0
$844$	−15309.4	−0.624373
$845$	0	0
$846$	0	0
$847$	−9128.28	−0.370309
$848$	8667.17	0.350981
$849$	0	0
$850$	0	0
$851$	−3058.44	−0.123199
$852$	0	0
$853$	−1084.77	−0.0435426	−0.0217713	−	0.999763i	$-0.506931\pi$
$853$	−1084.77	−0.0435426	−0.0217713	+	0.999763i	$0.506931\pi$
$854$	−16811.0	−0.673607
$855$	0	0
$856$	−13981.4	−0.558263
$857$	−12661.6	−0.504679	−0.252340	−	0.967639i	$-0.581200\pi$
$857$	−12661.6	−0.504679	−0.252340	+	0.967639i	$0.581200\pi$
$858$	0	0
$859$	−39678.7	−1.57604	−0.788021	−	0.615648i	$-0.788894\pi$
$859$	−39678.7	−1.57604	−0.788021	+	0.615648i	$0.788894\pi$
$860$	0	0
$861$	0	0
$862$	−57062.6	−2.25471
$863$	−41614.0	−1.64143	−0.820717	−	0.571334i	$-0.806426\pi$
$863$	−41614.0	−1.64143	−0.820717	+	0.571334i	$0.806426\pi$
$864$	0	0
$865$	0	0
$866$	−38759.2	−1.52089
$867$	0	0
$868$	−3092.35	−0.120923
$869$	−2211.06	−0.0863120
$870$	0	0
$871$	17723.7	0.689489
$872$	−3934.90	−0.152813
$873$	0	0
$874$	54963.3	2.12719
$875$	0	0
$876$	0	0
$877$	37061.0	1.42698	0.713490	−	0.700665i	$-0.247113\pi$
$877$	37061.0	1.42698	0.713490	+	0.700665i	$0.247113\pi$
$878$	27560.7	1.05937
$879$	0	0
$880$	0	0
$881$	25468.7	0.973962	0.486981	−	0.873412i	$-0.338098\pi$
$881$	25468.7	0.973962	0.486981	+	0.873412i	$0.338098\pi$
$882$	0	0
$883$	−34428.3	−1.31212	−0.656062	−	0.754707i	$-0.727779\pi$
$883$	−34428.3	−1.31212	−0.656062	+	0.754707i	$0.727779\pi$
$884$	−4129.88	−0.157130
$885$	0	0
$886$	31222.6	1.18391
$887$	41295.4	1.56321	0.781603	−	0.623777i	$-0.214403\pi$
$887$	41295.4	1.56321	0.781603	+	0.623777i	$0.214403\pi$
$888$	0	0
$889$	−11471.5	−0.432782
$890$	0	0
$891$	0	0
$892$	−25567.2	−0.959702
$893$	−9218.37	−0.345443
$894$	0	0
$895$	0	0
$896$	9671.26	0.360596
$897$	0	0
$898$	−11039.3	−0.410230
$899$	13406.7	0.497373
$900$	0	0
$901$	−1761.30	−0.0651247
$902$	−3888.40	−0.143536
$903$	0	0
$904$	−22216.2	−0.817368
$905$	0	0
$906$	0	0
$907$	53733.8	1.96715	0.983573	−	0.180509i	$-0.0577746\pi$
$907$	53733.8	1.96715	0.983573	+	0.180509i	$0.0577746\pi$
$908$	1316.50	0.0481162
$909$	0	0
$910$	0	0
$911$	−24296.8	−0.883634	−0.441817	−	0.897105i	$-0.645666\pi$
$911$	−24296.8	−0.883634	−0.441817	+	0.897105i	$0.645666\pi$
$912$	0	0
$913$	−1469.55	−0.0532696
$914$	21807.2	0.789187
$915$	0	0
$916$	13007.6	0.469197
$917$	4186.99	0.150782
$918$	0	0
$919$	4280.46	0.153645	0.0768223	−	0.997045i	$-0.475523\pi$
$919$	4280.46	0.153645	0.0768223	+	0.997045i	$0.475523\pi$
$920$	0	0
$921$	0	0
$922$	37083.6	1.32460
$923$	43307.9	1.54442
$924$	0	0
$925$	0	0
$926$	40168.9	1.42552
$927$	0	0
$928$	26840.7	0.949450
$929$	31884.5	1.12604	0.563022	−	0.826442i	$-0.309639\pi$
$929$	31884.5	1.12604	0.563022	+	0.826442i	$0.309639\pi$
$930$	0	0
$931$	4284.90	0.150840
$932$	27082.4	0.951839
$933$	0	0
$934$	53149.6	1.86200
$935$	0	0
$936$	0	0
$937$	44523.1	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$937$	44523.1	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$938$	−8100.76	−0.281982
$939$	0	0
$940$	0	0
$941$	−46374.9	−1.60657	−0.803283	−	0.595598i	$-0.796915\pi$
$941$	−46374.9	−1.60657	−0.803283	+	0.595598i	$0.796915\pi$
$942$	0	0
$943$	−37107.9	−1.28144
$944$	16730.6	0.576836
$945$	0	0
$946$	−9645.09	−0.331489
$947$	−20348.2	−0.698234	−0.349117	−	0.937079i	$-0.613518\pi$
$947$	−20348.2	−0.698234	−0.349117	+	0.937079i	$0.613518\pi$
$948$	0	0
$949$	17204.8	0.588507
$950$	0	0
$951$	0	0
$952$	−1335.85	−0.0454782
$953$	−45012.9	−1.53002	−0.765010	−	0.644018i	$-0.777266\pi$
$953$	−45012.9	−1.53002	−0.765010	+	0.644018i	$0.777266\pi$
$954$	0	0
$955$	0	0
$956$	−7440.02	−0.251702
$957$	0	0
$958$	16647.4	0.561435
$959$	−7036.76	−0.236944
$960$	0	0
$961$	−20898.5	−0.701503
$962$	3366.79	0.112838
$963$	0	0
$964$	−20284.6	−0.677721
$965$	0	0
$966$	0	0
$967$	40305.8	1.34038	0.670190	−	0.742190i	$-0.266213\pi$
$967$	40305.8	1.34038	0.670190	+	0.742190i	$0.266213\pi$
$968$	15397.8	0.511265
$969$	0	0
$970$	0	0
$971$	−33991.8	−1.12343	−0.561713	−	0.827332i	$-0.689858\pi$
$971$	−33991.8	−1.12343	−0.561713	+	0.827332i	$0.689858\pi$
$972$	0	0
$973$	−13125.0	−0.432445
$974$	−60837.2	−2.00139
$975$	0	0
$976$	53628.2	1.75881
$977$	−18219.0	−0.596600	−0.298300	−	0.954472i	$-0.596420\pi$
$977$	−18219.0	−0.596600	−0.298300	+	0.954472i	$0.596420\pi$
$978$	0	0
$979$	4377.73	0.142914
$980$	0	0
$981$	0	0
$982$	−64834.1	−2.10686
$983$	7676.89	0.249089	0.124545	−	0.992214i	$-0.460253\pi$
$983$	7676.89	0.249089	0.124545	+	0.992214i	$0.460253\pi$
$984$	0	0
$985$	0	0
$986$	−8183.55	−0.264318
$987$	0	0
$988$	−22345.4	−0.719537
$989$	−92045.3	−2.95942
$990$	0	0
$991$	50585.6	1.62150	0.810748	−	0.585395i	$-0.199061\pi$
$991$	50585.6	1.62150	0.810748	+	0.585395i	$0.199061\pi$
$992$	17803.2	0.569810
$993$	0	0
$994$	−19794.2	−0.631625
$995$	0	0
$996$	0	0
$997$	53060.5	1.68550	0.842750	−	0.538305i	$-0.180935\pi$
$997$	53060.5	1.68550	0.842750	+	0.538305i	$0.180935\pi$
$998$	25320.8	0.803121
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.x.1.2		2
3.2	odd	2		525.4.a.j.1.1	✓	2
5.4	even	2		1575.4.a.o.1.1		2
15.2	even	4		525.4.d.m.274.1		4
15.8	even	4		525.4.d.m.274.4		4
15.14	odd	2		525.4.a.m.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.j.1.1	✓	2	3.2	odd	2
525.4.a.m.1.2	yes	2	15.14	odd	2
525.4.d.m.274.1		4	15.2	even	4
525.4.d.m.274.4		4	15.8	even	4
1575.4.a.o.1.1		2	5.4	even	2
1575.4.a.x.1.2		2	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +O(q^{10})\)	\(q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +5.19224 q^{11} -54.5464 q^{13} +24.9309 q^{14} -79.5312 q^{16} +16.1619 q^{17} +87.4470 q^{19} +18.4924 q^{22} +176.477 q^{23} -194.270 q^{26} +32.7926 q^{28} -142.170 q^{29} -94.3002 q^{31} -188.793 q^{32} +57.5616 q^{34} -17.3305 q^{37} +311.447 q^{38} -210.270 q^{41} -521.570 q^{43} +24.3239 q^{44} +628.533 q^{46} -105.417 q^{47} +49.0000 q^{49} -255.531 q^{52} -108.978 q^{53} -82.6543 q^{56} -506.348 q^{58} -210.365 q^{59} -674.304 q^{61} -335.855 q^{62} -36.1449 q^{64} -324.929 q^{67} +75.7131 q^{68} -793.965 q^{71} -315.417 q^{73} -61.7235 q^{74} +409.659 q^{76} +36.3457 q^{77} -425.840 q^{79} -748.887 q^{82} -283.029 q^{83} -1857.60 q^{86} -61.3087 q^{88} +843.131 q^{89} -381.825 q^{91} +826.736 q^{92} -375.447 q^{94} -1537.33 q^{97} +174.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 - 3 * q^4 + 14 * q^7 - 3 * q^8	\( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + 31 q^{11} - 39 q^{13} + 21 q^{14} - 23 q^{16} - 79 q^{17} - 56 q^{19} + 4 q^{22} + 254 q^{23} - 203 q^{26} - 21 q^{28} + 62 q^{29} - 135 q^{31} - 291 q^{32} + 111 q^{34} - 113 q^{37} + 392 q^{38} - 235 q^{41} - 804 q^{43} - 174 q^{44} + 585 q^{46} + 152 q^{47} + 98 q^{49} - 375 q^{52} + 149 q^{53} - 21 q^{56} - 621 q^{58} + 441 q^{59} - 223 q^{61} - 313 q^{62} - 431 q^{64} - 1157 q^{67} + 807 q^{68} - 619 q^{71} - 268 q^{73} - 8 q^{74} + 1512 q^{76} + 217 q^{77} - 427 q^{79} - 735 q^{82} + 1211 q^{83} - 1699 q^{86} + 166 q^{88} - 466 q^{89} - 273 q^{91} + 231 q^{92} - 520 q^{94} - 172 q^{97} + 147 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 - 3 * q^4 + 14 * q^7 - 3 * q^8 + 31 * q^11 - 39 * q^13 + 21 * q^14 - 23 * q^16 - 79 * q^17 - 56 * q^19 + 4 * q^22 + 254 * q^23 - 203 * q^26 - 21 * q^28 + 62 * q^29 - 135 * q^31 - 291 * q^32 + 111 * q^34 - 113 * q^37 + 392 * q^38 - 235 * q^41 - 804 * q^43 - 174 * q^44 + 585 * q^46 + 152 * q^47 + 98 * q^49 - 375 * q^52 + 149 * q^53 - 21 * q^56 - 621 * q^58 + 441 * q^59 - 223 * q^61 - 313 * q^62 - 431 * q^64 - 1157 * q^67 + 807 * q^68 - 619 * q^71 - 268 * q^73 - 8 * q^74 + 1512 * q^76 + 217 * q^77 - 427 * q^79 - 735 * q^82 + 1211 * q^83 - 1699 * q^86 + 166 * q^88 - 466 * q^89 - 273 * q^91 + 231 * q^92 - 520 * q^94 - 172 * q^97 + 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more