LMFDB - Embedded newform 1575.4.a.s.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ 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)

$f(q)$	$=$	$q-3.70156 q^{2} +5.70156 q^{4} +7.00000 q^{7} +8.50781 q^{8} +60.0156 q^{11} +0.387503 q^{13} -25.9109 q^{14} -77.1047 q^{16} +35.4922 q^{17} -6.08907 q^{19} -222.152 q^{22} +31.5078 q^{23} -1.43437 q^{26} +39.9109 q^{28} +292.942 q^{29} +130.303 q^{31} +217.345 q^{32} -131.377 q^{34} +219.989 q^{37} +22.5391 q^{38} +447.795 q^{41} +210.020 q^{43} +342.183 q^{44} -116.628 q^{46} -457.769 q^{47} +49.0000 q^{49} +2.20937 q^{52} -144.334 q^{53} +59.5547 q^{56} -1084.34 q^{58} -767.328 q^{59} +667.884 q^{61} -482.325 q^{62} -187.680 q^{64} -77.4593 q^{67} +202.361 q^{68} +906.573 q^{71} -1029.72 q^{73} -814.303 q^{74} -34.7172 q^{76} +420.109 q^{77} -690.764 q^{79} -1657.54 q^{82} +979.408 q^{83} -777.403 q^{86} +510.602 q^{88} +910.927 q^{89} +2.71252 q^{91} +179.644 q^{92} +1694.46 q^{94} +11.1751 q^{97} -181.377 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8} + 56 q^{11} + 52 q^{13} - 7 q^{14} - 135 q^{16} + 103 q^{17} - 57 q^{19} - 233 q^{22} + 31 q^{23} + 138 q^{26} + 35 q^{28} + 413 q^{29} - 162 q^{31} + 249 q^{32}+ \cdots - 49 q^{98}+O(q^{100}) $ 2 * q - q^2 + 5 * q^4 + 14 * q^7 - 15 * q^8 + 56 * q^11 + 52 * q^13 - 7 * q^14 - 135 * q^16 + 103 * q^17 - 57 * q^19 - 233 * q^22 + 31 * q^23 + 138 * q^26 + 35 * q^28 + 413 * q^29 - 162 * q^31 + 249 * q^32 + 51 * q^34 + 75 * q^37 - 115 * q^38 + 505 * q^41 - 73 * q^43 + 345 * q^44 - 118 * q^46 - 224 * q^47 + 98 * q^49 - 34 * q^52 + 262 * q^53 - 105 * q^56 - 760 * q^58 - 190 * q^59 + 990 * q^61 - 1272 * q^62 + 361 * q^64 + 908 * q^67 + 155 * q^68 - 127 * q^71 - 337 * q^73 - 1206 * q^74 + q^76 + 392 * q^77 - 1119 * q^79 - 1503 * q^82 + 1517 * q^83 - 1542 * q^86 + 605 * q^88 + 1713 * q^89 + 364 * q^91 + 180 * q^92 + 2326 * q^94 + 1764 * q^97 - 49 * q^98

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 $

$2$	−3.70156	−1.30870	−0.654350	−	0.756192i	$-0.727058\pi$
$2$	−3.70156	−1.30870	−0.654350	+	0.756192i	$0.727058\pi$
$3$	0	0
$3$	0	0
$4$	5.70156	0.712695
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.50781	0.375996
$9$	0	0
$10$	0	0
$11$	60.0156	1.64504	0.822518	−	0.568739i	$-0.192569\pi$
$11$	60.0156	1.64504	0.822518	+	0.568739i	$0.192569\pi$
$12$	0	0
$13$	0.387503	0.00826723	0.00413362	−	0.999991i	$-0.498684\pi$
$13$	0.387503	0.00826723	0.00413362	+	0.999991i	$0.498684\pi$
$14$	−25.9109	−0.494642
$15$	0	0
$16$	−77.1047	−1.20476
$17$	35.4922	0.506360	0.253180	−	0.967419i	$-0.418524\pi$
$17$	35.4922	0.506360	0.253180	+	0.967419i	$0.418524\pi$
$18$	0	0
$19$	−6.08907	−0.0735225	−0.0367612	−	0.999324i	$-0.511704\pi$
$19$	−6.08907	−0.0735225	−0.0367612	+	0.999324i	$0.511704\pi$
$20$	0	0
$21$	0	0
$22$	−222.152	−2.15286
$23$	31.5078	0.285645	0.142822	−	0.989748i	$-0.454382\pi$
$23$	31.5078	0.285645	0.142822	+	0.989748i	$0.454382\pi$
$24$	0	0
$25$	0	0
$26$	−1.43437	−0.0108193
$27$	0	0
$28$	39.9109	0.269373
$29$	292.942	1.87579	0.937896	−	0.346915i	$-0.112771\pi$
$29$	292.942	1.87579	0.937896	+	0.346915i	$0.112771\pi$
$30$	0	0
$31$	130.303	0.754940	0.377470	−	0.926022i	$-0.376794\pi$
$31$	130.303	0.754940	0.377470	+	0.926022i	$0.376794\pi$
$32$	217.345	1.20067
$33$	0	0
$34$	−131.377	−0.662673
$35$	0	0
$36$	0	0
$37$	219.989	0.977459	0.488729	−	0.872435i	$-0.337461\pi$
$37$	219.989	0.977459	0.488729	+	0.872435i	$0.337461\pi$
$38$	22.5391	0.0962189
$39$	0	0
$40$	0	0
$41$	447.795	1.70570	0.852852	−	0.522153i	$-0.174871\pi$
$41$	447.795	1.70570	0.852852	+	0.522153i	$0.174871\pi$
$42$	0	0
$43$	210.020	0.744832	0.372416	−	0.928066i	$-0.378529\pi$
$43$	210.020	0.744832	0.372416	+	0.928066i	$0.378529\pi$
$44$	342.183	1.17241
$45$	0	0
$46$	−116.628	−0.373823
$47$	−457.769	−1.42069	−0.710345	−	0.703854i	$-0.751461\pi$
$47$	−457.769	−1.42069	−0.710345	+	0.703854i	$0.751461\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	2.20937	0.00589202
$53$	−144.334	−0.374073	−0.187036	−	0.982353i	$-0.559888\pi$
$53$	−144.334	−0.374073	−0.187036	+	0.982353i	$0.559888\pi$
$54$	0	0
$55$	0	0
$56$	59.5547	0.142113
$57$	0	0
$58$	−1084.34	−2.45485
$59$	−767.328	−1.69318	−0.846590	−	0.532246i	$-0.821348\pi$
$59$	−767.328	−1.69318	−0.846590	+	0.532246i	$0.821348\pi$
$60$	0	0
$61$	667.884	1.40187	0.700933	−	0.713227i	$-0.252767\pi$
$61$	667.884	1.40187	0.700933	+	0.713227i	$0.252767\pi$
$62$	−482.325	−0.987989
$63$	0	0
$64$	−187.680	−0.366562
$65$	0	0
$66$	0	0
$67$	−77.4593	−0.141241	−0.0706206	−	0.997503i	$-0.522498\pi$
$67$	−77.4593	−0.141241	−0.0706206	+	0.997503i	$0.522498\pi$
$68$	202.361	0.360880
$69$	0	0
$70$	0	0
$71$	906.573	1.51536	0.757679	−	0.652627i	$-0.226333\pi$
$71$	906.573	1.51536	0.757679	+	0.652627i	$0.226333\pi$
$72$	0	0
$73$	−1029.72	−1.65095	−0.825477	−	0.564436i	$-0.809094\pi$
$73$	−1029.72	−1.65095	−0.825477	+	0.564436i	$0.809094\pi$
$74$	−814.303	−1.27920
$75$	0	0
$76$	−34.7172	−0.0523991
$77$	420.109	0.621765
$78$	0	0
$79$	−690.764	−0.983760	−0.491880	−	0.870663i	$-0.663690\pi$
$79$	−690.764	−0.983760	−0.491880	+	0.870663i	$0.663690\pi$
$80$	0	0
$81$	0	0
$82$	−1657.54	−2.23225
$83$	979.408	1.29523	0.647614	−	0.761968i	$-0.275767\pi$
$83$	979.408	1.29523	0.647614	+	0.761968i	$0.275767\pi$
$84$	0	0
$85$	0	0
$86$	−777.403	−0.974762
$87$	0	0
$88$	510.602	0.618526
$89$	910.927	1.08492	0.542461	−	0.840081i	$-0.317493\pi$
$89$	910.927	1.08492	0.542461	+	0.840081i	$0.317493\pi$
$90$	0	0
$91$	2.71252	0.00312472
$92$	179.644	0.203578
$93$	0	0
$94$	1694.46	1.85926
$95$	0	0
$96$	0	0
$97$	11.1751	0.0116975	0.00584876	−	0.999983i	$-0.498138\pi$
$97$	11.1751	0.0116975	0.00584876	+	0.999983i	$0.498138\pi$
$98$	−181.377	−0.186957
$99$	0	0
$100$	0	0
$101$	−675.850	−0.665837	−0.332919	−	0.942956i	$-0.608033\pi$
$101$	−675.850	−0.665837	−0.332919	+	0.942956i	$0.608033\pi$
$102$	0	0
$103$	−1528.88	−1.46257	−0.731286	−	0.682071i	$-0.761079\pi$
$103$	−1528.88	−1.46257	−0.731286	+	0.682071i	$0.761079\pi$
$104$	3.29680	0.00310844
$105$	0	0
$106$	534.263	0.489549
$107$	−701.595	−0.633886	−0.316943	−	0.948445i	$-0.602656\pi$
$107$	−701.595	−0.633886	−0.316943	+	0.948445i	$0.602656\pi$
$108$	0	0
$109$	−597.130	−0.524722	−0.262361	−	0.964970i	$-0.584501\pi$
$109$	−597.130	−0.524722	−0.262361	+	0.964970i	$0.584501\pi$
$110$	0	0
$111$	0	0
$112$	−539.733	−0.455357
$113$	−88.1187	−0.0733585	−0.0366792	−	0.999327i	$-0.511678\pi$
$113$	−88.1187	−0.0733585	−0.0366792	+	0.999327i	$0.511678\pi$
$114$	0	0
$115$	0	0
$116$	1670.23	1.33687
$117$	0	0
$118$	2840.31	2.21586
$119$	248.445	0.191386
$120$	0	0
$121$	2270.87	1.70614
$122$	−2472.22	−1.83462
$123$	0	0
$124$	742.931	0.538042
$125$	0	0
$126$	0	0
$127$	−2119.35	−1.48080	−0.740401	−	0.672166i	$-0.765364\pi$
$127$	−2119.35	−1.48080	−0.740401	+	0.672166i	$0.765364\pi$
$128$	−1044.05	−0.720955
$129$	0	0
$130$	0	0
$131$	264.294	0.176271	0.0881353	−	0.996109i	$-0.471909\pi$
$131$	264.294	0.176271	0.0881353	+	0.996109i	$0.471909\pi$
$132$	0	0
$133$	−42.6235	−0.0277889
$134$	286.720	0.184842
$135$	0	0
$136$	301.961	0.190389
$137$	−468.227	−0.291995	−0.145997	−	0.989285i	$-0.546639\pi$
$137$	−468.227	−0.291995	−0.145997	+	0.989285i	$0.546639\pi$
$138$	0	0
$139$	956.908	0.583913	0.291956	−	0.956432i	$-0.405694\pi$
$139$	956.908	0.583913	0.291956	+	0.956432i	$0.405694\pi$
$140$	0	0
$141$	0	0
$142$	−3355.74	−1.98315
$143$	23.2562	0.0135999
$144$	0	0
$145$	0	0
$146$	3811.57	2.16060
$147$	0	0
$148$	1254.28	0.696630
$149$	382.967	0.210563	0.105282	−	0.994442i	$-0.466426\pi$
$149$	382.967	0.210563	0.105282	+	0.994442i	$0.466426\pi$
$150$	0	0
$151$	1302.98	0.702218	0.351109	−	0.936335i	$-0.385805\pi$
$151$	1302.98	0.702218	0.351109	+	0.936335i	$0.385805\pi$
$152$	−51.8046	−0.0276441
$153$	0	0
$154$	−1555.06	−0.813704
$155$	0	0
$156$	0	0
$157$	−3203.32	−1.62836	−0.814182	−	0.580610i	$-0.802814\pi$
$157$	−3203.32	−1.62836	−0.814182	+	0.580610i	$0.802814\pi$
$158$	2556.91	1.28745
$159$	0	0
$160$	0	0
$161$	220.555	0.107964
$162$	0	0
$163$	1368.81	0.657752	0.328876	−	0.944373i	$-0.393330\pi$
$163$	1368.81	0.657752	0.328876	+	0.944373i	$0.393330\pi$
$164$	2553.13	1.21565
$165$	0	0
$166$	−3625.34	−1.69507
$167$	−3073.34	−1.42409	−0.712043	−	0.702136i	$-0.752230\pi$
$167$	−3073.34	−1.42409	−0.712043	+	0.702136i	$0.752230\pi$
$168$	0	0
$169$	−2196.85	−0.999932
$170$	0	0
$171$	0	0
$172$	1197.44	0.530839
$173$	2402.16	1.05568	0.527841	−	0.849343i	$-0.323002\pi$
$173$	2402.16	1.05568	0.527841	+	0.849343i	$0.323002\pi$
$174$	0	0
$175$	0	0
$176$	−4627.49	−1.98187
$177$	0	0
$178$	−3371.85	−1.41984
$179$	1146.25	0.478628	0.239314	−	0.970942i	$-0.423077\pi$
$179$	1146.25	0.478628	0.239314	+	0.970942i	$0.423077\pi$
$180$	0	0
$181$	−475.847	−0.195411	−0.0977056	−	0.995215i	$-0.531150\pi$
$181$	−475.847	−0.195411	−0.0977056	+	0.995215i	$0.531150\pi$
$182$	−10.0406	−0.00408932
$183$	0	0
$184$	268.062	0.107401
$185$	0	0
$186$	0	0
$187$	2130.09	0.832980
$188$	−2610.00	−1.01252
$189$	0	0
$190$	0	0
$191$	−990.003	−0.375048	−0.187524	−	0.982260i	$-0.560046\pi$
$191$	−990.003	−0.375048	−0.187524	+	0.982260i	$0.560046\pi$
$192$	0	0
$193$	1392.88	0.519492	0.259746	−	0.965677i	$-0.416361\pi$
$193$	1392.88	0.519492	0.259746	+	0.965677i	$0.416361\pi$
$194$	−41.3653	−0.0153085
$195$	0	0
$196$	279.377	0.101814
$197$	3583.39	1.29597	0.647985	−	0.761653i	$-0.275612\pi$
$197$	3583.39	1.29597	0.647985	+	0.761653i	$0.275612\pi$
$198$	0	0
$199$	623.947	0.222263	0.111132	−	0.993806i	$-0.464552\pi$
$199$	623.947	0.222263	0.111132	+	0.993806i	$0.464552\pi$
$200$	0	0
$201$	0	0
$202$	2501.70	0.871381
$203$	2050.60	0.708983
$204$	0	0
$205$	0	0
$206$	5659.24	1.91407
$207$	0	0
$208$	−29.8783	−0.00996004
$209$	−365.439	−0.120947
$210$	0	0
$211$	2266.48	0.739482	0.369741	−	0.929135i	$-0.379446\pi$
$211$	2266.48	0.739482	0.369741	+	0.929135i	$0.379446\pi$
$212$	−822.931	−0.266600
$213$	0	0
$214$	2597.00	0.829566
$215$	0	0
$216$	0	0
$217$	912.122	0.285340
$218$	2210.31	0.686703
$219$	0	0
$220$	0	0
$221$	13.7533	0.00418620
$222$	0	0
$223$	2118.67	0.636217	0.318108	−	0.948054i	$-0.396952\pi$
$223$	2118.67	0.636217	0.318108	+	0.948054i	$0.396952\pi$
$224$	1521.42	0.453812
$225$	0	0
$226$	326.177	0.0960042
$227$	−2790.62	−0.815948	−0.407974	−	0.912994i	$-0.633765\pi$
$227$	−2790.62	−0.815948	−0.407974	+	0.912994i	$0.633765\pi$
$228$	0	0
$229$	−5813.77	−1.67766	−0.838832	−	0.544390i	$-0.816761\pi$
$229$	−5813.77	−1.67766	−0.838832	+	0.544390i	$0.816761\pi$
$230$	0	0
$231$	0	0
$232$	2492.30	0.705290
$233$	−1936.18	−0.544392	−0.272196	−	0.962242i	$-0.587750\pi$
$233$	−1936.18	−0.544392	−0.272196	+	0.962242i	$0.587750\pi$
$234$	0	0
$235$	0	0
$236$	−4374.97	−1.20672
$237$	0	0
$238$	−919.636	−0.250467
$239$	2755.79	0.745845	0.372923	−	0.927862i	$-0.378356\pi$
$239$	2755.79	0.745845	0.372923	+	0.927862i	$0.378356\pi$
$240$	0	0
$241$	−4025.23	−1.07588	−0.537942	−	0.842982i	$-0.680798\pi$
$241$	−4025.23	−1.07588	−0.537942	+	0.842982i	$0.680798\pi$
$242$	−8405.78	−2.23283
$243$	0	0
$244$	3807.98	0.999103
$245$	0	0
$246$	0	0
$247$	−2.35953	−0.000607827 0
$248$	1108.59	0.283854
$249$	0	0
$250$	0	0
$251$	−7166.64	−1.80221	−0.901104	−	0.433602i	$-0.857242\pi$
$251$	−7166.64	−1.80221	−0.901104	+	0.433602i	$0.857242\pi$
$252$	0	0
$253$	1890.96	0.469896
$254$	7844.90	1.93792
$255$	0	0
$256$	5366.07	1.31008
$257$	−647.756	−0.157221	−0.0786107	−	0.996905i	$-0.525048\pi$
$257$	−647.756	−0.157221	−0.0786107	+	0.996905i	$0.525048\pi$
$258$	0	0
$259$	1539.92	0.369445
$260$	0	0
$261$	0	0
$262$	−978.300	−0.230685
$263$	6131.33	1.43754	0.718771	−	0.695246i	$-0.244705\pi$
$263$	6131.33	1.43754	0.718771	+	0.695246i	$0.244705\pi$
$264$	0	0
$265$	0	0
$266$	157.773	0.0363673
$267$	0	0
$268$	−441.639	−0.100662
$269$	975.631	0.221135	0.110567	−	0.993869i	$-0.464733\pi$
$269$	975.631	0.221135	0.110567	+	0.993869i	$0.464733\pi$
$270$	0	0
$271$	4672.83	1.04743	0.523716	−	0.851893i	$-0.324545\pi$
$271$	4672.83	1.04743	0.523716	+	0.851893i	$0.324545\pi$
$272$	−2736.61	−0.610043
$273$	0	0
$274$	1733.17	0.382134
$275$	0	0
$276$	0	0
$277$	2390.16	0.518451	0.259226	−	0.965817i	$-0.416533\pi$
$277$	2390.16	0.518451	0.259226	+	0.965817i	$0.416533\pi$
$278$	−3542.05	−0.764166
$279$	0	0
$280$	0	0
$281$	3321.28	0.705092	0.352546	−	0.935794i	$-0.385316\pi$
$281$	3321.28	0.705092	0.352546	+	0.935794i	$0.385316\pi$
$282$	0	0
$283$	8260.15	1.73504	0.867518	−	0.497406i	$-0.165714\pi$
$283$	8260.15	1.73504	0.867518	+	0.497406i	$0.165714\pi$
$284$	5168.88	1.07999
$285$	0	0
$286$	−86.0844	−0.0177982
$287$	3134.57	0.644696
$288$	0	0
$289$	−3653.30	−0.743600
$290$	0	0
$291$	0	0
$292$	−5871.01	−1.17663
$293$	628.249	0.125265	0.0626326	−	0.998037i	$-0.480050\pi$
$293$	628.249	0.125265	0.0626326	+	0.998037i	$0.480050\pi$
$294$	0	0
$295$	0	0
$296$	1871.63	0.367520
$297$	0	0
$298$	−1417.58	−0.275564
$299$	12.2094	0.00236149
$300$	0	0
$301$	1470.14	0.281520
$302$	−4823.06	−0.918993
$303$	0	0
$304$	469.495	0.0885770
$305$	0	0
$306$	0	0
$307$	25.2045	0.00468565	0.00234283	−	0.999997i	$-0.499254\pi$
$307$	25.2045	0.00468565	0.00234283	+	0.999997i	$0.499254\pi$
$308$	2395.28	0.443129
$309$	0	0
$310$	0	0
$311$	9436.18	1.72050	0.860252	−	0.509870i	$-0.170306\pi$
$311$	9436.18	1.72050	0.860252	+	0.509870i	$0.170306\pi$
$312$	0	0
$313$	−2778.60	−0.501776	−0.250888	−	0.968016i	$-0.580723\pi$
$313$	−2778.60	−0.501776	−0.250888	+	0.968016i	$0.580723\pi$
$314$	11857.3	2.13104
$315$	0	0
$316$	−3938.43	−0.701121
$317$	2980.21	0.528030	0.264015	−	0.964519i	$-0.414953\pi$
$317$	2980.21	0.528030	0.264015	+	0.964519i	$0.414953\pi$
$318$	0	0
$319$	17581.1	3.08575
$320$	0	0
$321$	0	0
$322$	−816.397	−0.141292
$323$	−216.114	−0.0372289
$324$	0	0
$325$	0	0
$326$	−5066.74	−0.860800
$327$	0	0
$328$	3809.76	0.641337
$329$	−3204.38	−0.536970
$330$	0	0
$331$	2588.16	0.429782	0.214891	−	0.976638i	$-0.431060\pi$
$331$	2588.16	0.429782	0.214891	+	0.976638i	$0.431060\pi$
$332$	5584.15	0.923103
$333$	0	0
$334$	11376.2	1.86370
$335$	0	0
$336$	0	0
$337$	5284.64	0.854221	0.427110	−	0.904199i	$-0.359532\pi$
$337$	5284.64	0.854221	0.427110	+	0.904199i	$0.359532\pi$
$338$	8131.78	1.30861
$339$	0	0
$340$	0	0
$341$	7820.22	1.24190
$342$	0	0
$343$	343.000	0.0539949
$344$	1786.81	0.280054
$345$	0	0
$346$	−8891.75	−1.38157
$347$	11787.7	1.82363	0.911813	−	0.410606i	$-0.134683\pi$
$347$	11787.7	1.82363	0.911813	+	0.410606i	$0.134683\pi$
$348$	0	0
$349$	8765.61	1.34445	0.672224	−	0.740347i	$-0.265339\pi$
$349$	8765.61	1.34445	0.672224	+	0.740347i	$0.265339\pi$
$350$	0	0
$351$	0	0
$352$	13044.1	1.97515
$353$	−8578.29	−1.29342	−0.646709	−	0.762737i	$-0.723855\pi$
$353$	−8578.29	−1.29342	−0.646709	+	0.762737i	$0.723855\pi$
$354$	0	0
$355$	0	0
$356$	5193.70	0.773218
$357$	0	0
$358$	−4242.90	−0.626380
$359$	−3730.38	−0.548417	−0.274209	−	0.961670i	$-0.588416\pi$
$359$	−3730.38	−0.548417	−0.274209	+	0.961670i	$0.588416\pi$
$360$	0	0
$361$	−6821.92	−0.994594
$362$	1761.38	0.255735
$363$	0	0
$364$	15.4656	0.00222697
$365$	0	0
$366$	0	0
$367$	−515.769	−0.0733594	−0.0366797	−	0.999327i	$-0.511678\pi$
$367$	−515.769	−0.0733594	−0.0366797	+	0.999327i	$0.511678\pi$
$368$	−2429.40	−0.344134
$369$	0	0
$370$	0	0
$371$	−1010.34	−0.141386
$372$	0	0
$373$	10922.0	1.51614	0.758071	−	0.652173i	$-0.226142\pi$
$373$	10922.0	1.51614	0.758071	+	0.652173i	$0.226142\pi$
$374$	−7884.64	−1.09012
$375$	0	0
$376$	−3894.61	−0.534173
$377$	113.516	0.0155076
$378$	0	0
$379$	−8403.14	−1.13889	−0.569446	−	0.822029i	$-0.692842\pi$
$379$	−8403.14	−1.13889	−0.569446	+	0.822029i	$0.692842\pi$
$380$	0	0
$381$	0	0
$382$	3664.56	0.490825
$383$	3030.68	0.404336	0.202168	−	0.979351i	$-0.435201\pi$
$383$	3030.68	0.404336	0.202168	+	0.979351i	$0.435201\pi$
$384$	0	0
$385$	0	0
$386$	−5155.85	−0.679859
$387$	0	0
$388$	63.7155	0.00833677
$389$	−1403.25	−0.182899	−0.0914497	−	0.995810i	$-0.529150\pi$
$389$	−1403.25	−0.182899	−0.0914497	+	0.995810i	$0.529150\pi$
$390$	0	0
$391$	1118.28	0.144639
$392$	416.883	0.0537137
$393$	0	0
$394$	−13264.1	−1.69604
$395$	0	0
$396$	0	0
$397$	55.9118	0.00706835	0.00353417	−	0.999994i	$-0.498875\pi$
$397$	55.9118	0.00706835	0.00353417	+	0.999994i	$0.498875\pi$
$398$	−2309.58	−0.290876
$399$	0	0
$400$	0	0
$401$	3730.18	0.464529	0.232264	−	0.972653i	$-0.425387\pi$
$401$	3730.18	0.464529	0.232264	+	0.972653i	$0.425387\pi$
$402$	0	0
$403$	50.4928	0.00624126
$404$	−3853.40	−0.474539
$405$	0	0
$406$	−7590.41	−0.927846
$407$	13202.8	1.60795
$408$	0	0
$409$	−1968.50	−0.237985	−0.118993	−	0.992895i	$-0.537967\pi$
$409$	−1968.50	−0.237985	−0.118993	+	0.992895i	$0.537967\pi$
$410$	0	0
$411$	0	0
$412$	−8716.99	−1.04237
$413$	−5371.30	−0.639962
$414$	0	0
$415$	0	0
$416$	84.2220	0.00992625
$417$	0	0
$418$	1352.70	0.158283
$419$	13208.4	1.54003	0.770015	−	0.638026i	$-0.220249\pi$
$419$	13208.4	1.54003	0.770015	+	0.638026i	$0.220249\pi$
$420$	0	0
$421$	−7485.74	−0.866586	−0.433293	−	0.901253i	$-0.642648\pi$
$421$	−7485.74	−0.866586	−0.433293	+	0.901253i	$0.642648\pi$
$422$	−8389.50	−0.967760
$423$	0	0
$424$	−1227.97	−0.140650
$425$	0	0
$426$	0	0
$427$	4675.19	0.529856
$428$	−4000.19	−0.451767
$429$	0	0
$430$	0	0
$431$	−10623.4	−1.18726	−0.593631	−	0.804737i	$-0.702306\pi$
$431$	−10623.4	−1.18726	−0.593631	+	0.804737i	$0.702306\pi$
$432$	0	0
$433$	7268.73	0.806727	0.403364	−	0.915040i	$-0.367841\pi$
$433$	7268.73	0.806727	0.403364	+	0.915040i	$0.367841\pi$
$434$	−3376.28	−0.373425
$435$	0	0
$436$	−3404.57	−0.373967
$437$	−191.853	−0.0210013
$438$	0	0
$439$	−743.352	−0.0808161	−0.0404080	−	0.999183i	$-0.512866\pi$
$439$	−743.352	−0.0808161	−0.0404080	+	0.999183i	$0.512866\pi$
$440$	0	0
$441$	0	0
$442$	−50.9088	−0.00547847
$443$	−10857.8	−1.16449	−0.582246	−	0.813013i	$-0.697826\pi$
$443$	−10857.8	−1.16449	−0.582246	+	0.813013i	$0.697826\pi$
$444$	0	0
$445$	0	0
$446$	−7842.37	−0.832617
$447$	0	0
$448$	−1313.76	−0.138547
$449$	11162.9	1.17329	0.586645	−	0.809844i	$-0.300448\pi$
$449$	11162.9	1.17329	0.586645	+	0.809844i	$0.300448\pi$
$450$	0	0
$451$	26874.7	2.80594
$452$	−502.414	−0.0522822
$453$	0	0
$454$	10329.7	1.06783
$455$	0	0
$456$	0	0
$457$	13451.0	1.37683	0.688414	−	0.725318i	$-0.258307\pi$
$457$	13451.0	1.37683	0.688414	+	0.725318i	$0.258307\pi$
$458$	21520.0	2.19556
$459$	0	0
$460$	0	0
$461$	9139.46	0.923356	0.461678	−	0.887048i	$-0.347248\pi$
$461$	9139.46	0.923356	0.461678	+	0.887048i	$0.347248\pi$
$462$	0	0
$463$	−13122.5	−1.31718	−0.658588	−	0.752504i	$-0.728846\pi$
$463$	−13122.5	−1.31718	−0.658588	+	0.752504i	$0.728846\pi$
$464$	−22587.2	−2.25988
$465$	0	0
$466$	7166.89	0.712446
$467$	−11921.2	−1.18126	−0.590629	−	0.806943i	$-0.701120\pi$
$467$	−11921.2	−1.18126	−0.590629	+	0.806943i	$0.701120\pi$
$468$	0	0
$469$	−542.215	−0.0533842
$470$	0	0
$471$	0	0
$472$	−6528.28	−0.636628
$473$	12604.5	1.22528
$474$	0	0
$475$	0	0
$476$	1416.53	0.136400
$477$	0	0
$478$	−10200.7	−0.976088
$479$	−825.281	−0.0787224	−0.0393612	−	0.999225i	$-0.512532\pi$
$479$	−825.281	−0.0787224	−0.0393612	+	0.999225i	$0.512532\pi$
$480$	0	0
$481$	85.2464	0.00808088
$482$	14899.6	1.40801
$483$	0	0
$484$	12947.5	1.21596
$485$	0	0
$486$	0	0
$487$	−6678.32	−0.621404	−0.310702	−	0.950507i	$-0.600564\pi$
$487$	−6678.32	−0.621404	−0.310702	+	0.950507i	$0.600564\pi$
$488$	5682.23	0.527096
$489$	0	0
$490$	0	0
$491$	3098.65	0.284807	0.142403	−	0.989809i	$-0.454517\pi$
$491$	3098.65	0.284807	0.142403	+	0.989809i	$0.454517\pi$
$492$	0	0
$493$	10397.2	0.949827
$494$	8.73395	0.000795464 0
$495$	0	0
$496$	−10047.0	−0.909522
$497$	6346.01	0.572752
$498$	0	0
$499$	−2265.10	−0.203206	−0.101603	−	0.994825i	$-0.532397\pi$
$499$	−2265.10	−0.203206	−0.101603	+	0.994825i	$0.532397\pi$
$500$	0	0
$501$	0	0
$502$	26527.8	2.35855
$503$	9980.43	0.884703	0.442351	−	0.896842i	$-0.354144\pi$
$503$	9980.43	0.884703	0.442351	+	0.896842i	$0.354144\pi$
$504$	0	0
$505$	0	0
$506$	−6999.51	−0.614953
$507$	0	0
$508$	−12083.6	−1.05536
$509$	4956.16	0.431588	0.215794	−	0.976439i	$-0.430766\pi$
$509$	4956.16	0.431588	0.215794	+	0.976439i	$0.430766\pi$
$510$	0	0
$511$	−7208.04	−0.624002
$512$	−11510.4	−0.993541
$513$	0	0
$514$	2397.71	0.205756
$515$	0	0
$516$	0	0
$517$	−27473.3	−2.33709
$518$	−5700.12	−0.483492
$519$	0	0
$520$	0	0
$521$	−3442.97	−0.289519	−0.144759	−	0.989467i	$-0.546241\pi$
$521$	−3442.97	−0.289519	−0.144759	+	0.989467i	$0.546241\pi$
$522$	0	0
$523$	−11109.9	−0.928878	−0.464439	−	0.885605i	$-0.653744\pi$
$523$	−11109.9	−0.928878	−0.464439	+	0.885605i	$0.653744\pi$
$524$	1506.89	0.125627
$525$	0	0
$526$	−22695.5	−1.88131
$527$	4624.74	0.382271
$528$	0	0
$529$	−11174.3	−0.918407
$530$	0	0
$531$	0	0
$532$	−243.020	−0.0198050
$533$	173.522	0.0141015
$534$	0	0
$535$	0	0
$536$	−659.009	−0.0531061
$537$	0	0
$538$	−3611.36	−0.289399
$539$	2940.77	0.235005
$540$	0	0
$541$	3680.65	0.292502	0.146251	−	0.989248i	$-0.453279\pi$
$541$	3680.65	0.292502	0.146251	+	0.989248i	$0.453279\pi$
$542$	−17296.8	−1.37078
$543$	0	0
$544$	7714.06	0.607974
$545$	0	0
$546$	0	0
$547$	14657.4	1.14572	0.572858	−	0.819655i	$-0.305835\pi$
$547$	14657.4	1.14572	0.572858	+	0.819655i	$0.305835\pi$
$548$	−2669.62	−0.208103
$549$	0	0
$550$	0	0
$551$	−1783.74	−0.137913
$552$	0	0
$553$	−4835.35	−0.371826
$554$	−8847.33	−0.678497
$555$	0	0
$556$	5455.87	0.416152
$557$	6547.61	0.498081	0.249040	−	0.968493i	$-0.419885\pi$
$557$	6547.61	0.498081	0.249040	+	0.968493i	$0.419885\pi$
$558$	0	0
$559$	81.3835	0.00615770
$560$	0	0
$561$	0	0
$562$	−12293.9	−0.922754
$563$	11983.3	0.897047	0.448523	−	0.893771i	$-0.351950\pi$
$563$	11983.3	0.897047	0.448523	+	0.893771i	$0.351950\pi$
$564$	0	0
$565$	0	0
$566$	−30575.5	−2.27064
$567$	0	0
$568$	7712.95	0.569768
$569$	−19164.8	−1.41200	−0.706002	−	0.708210i	$-0.749503\pi$
$569$	−19164.8	−1.41200	−0.706002	+	0.708210i	$0.749503\pi$
$570$	0	0
$571$	14618.8	1.07142	0.535708	−	0.844403i	$-0.320045\pi$
$571$	14618.8	1.07142	0.535708	+	0.844403i	$0.320045\pi$
$572$	132.597	0.00969258
$573$	0	0
$574$	−11602.8	−0.843713
$575$	0	0
$576$	0	0
$577$	12446.8	0.898033	0.449017	−	0.893523i	$-0.351774\pi$
$577$	12446.8	0.898033	0.449017	+	0.893523i	$0.351774\pi$
$578$	13522.9	0.973149
$579$	0	0
$580$	0	0
$581$	6855.85	0.489550
$582$	0	0
$583$	−8662.32	−0.615363
$584$	−8760.66	−0.620752
$585$	0	0
$586$	−2325.50	−0.163935
$587$	7467.53	0.525073	0.262537	−	0.964922i	$-0.415441\pi$
$587$	7467.53	0.525073	0.262537	+	0.964922i	$0.415441\pi$
$588$	0	0
$589$	−793.424	−0.0555050
$590$	0	0
$591$	0	0
$592$	−16962.2	−1.17760
$593$	−26703.9	−1.84924	−0.924619	−	0.380892i	$-0.875617\pi$
$593$	−26703.9	−1.84924	−0.924619	+	0.380892i	$0.875617\pi$
$594$	0	0
$595$	0	0
$596$	2183.51	0.150067
$597$	0	0
$598$	−45.1938	−0.00309048
$599$	−6896.60	−0.470430	−0.235215	−	0.971943i	$-0.575579\pi$
$599$	−6896.60	−0.470430	−0.235215	+	0.971943i	$0.575579\pi$
$600$	0	0
$601$	18156.5	1.23231	0.616154	−	0.787626i	$-0.288690\pi$
$601$	18156.5	1.23231	0.616154	+	0.787626i	$0.288690\pi$
$602$	−5441.82	−0.368425
$603$	0	0
$604$	7429.02	0.500468
$605$	0	0
$606$	0	0
$607$	−2074.50	−0.138717	−0.0693587	−	0.997592i	$-0.522095\pi$
$607$	−2074.50	−0.138717	−0.0693587	+	0.997592i	$0.522095\pi$
$608$	−1323.43	−0.0882766
$609$	0	0
$610$	0	0
$611$	−177.387	−0.0117452
$612$	0	0
$613$	12159.4	0.801162	0.400581	−	0.916261i	$-0.368808\pi$
$613$	12159.4	0.801162	0.400581	+	0.916261i	$0.368808\pi$
$614$	−93.2959	−0.00613211
$615$	0	0
$616$	3574.21	0.233781
$617$	4359.84	0.284474	0.142237	−	0.989833i	$-0.454570\pi$
$617$	4359.84	0.284474	0.142237	+	0.989833i	$0.454570\pi$
$618$	0	0
$619$	−29046.6	−1.88608	−0.943038	−	0.332684i	$-0.892046\pi$
$619$	−29046.6	−1.88608	−0.943038	+	0.332684i	$0.892046\pi$
$620$	0	0
$621$	0	0
$622$	−34928.6	−2.25162
$623$	6376.49	0.410062
$624$	0	0
$625$	0	0
$626$	10285.2	0.656674
$627$	0	0
$628$	−18264.0	−1.16053
$629$	7807.89	0.494946
$630$	0	0
$631$	−14708.4	−0.927946	−0.463973	−	0.885849i	$-0.653577\pi$
$631$	−14708.4	−0.927946	−0.463973	+	0.885849i	$0.653577\pi$
$632$	−5876.89	−0.369889
$633$	0	0
$634$	−11031.4	−0.691033
$635$	0	0
$636$	0	0
$637$	18.9876	0.00118103
$638$	−65077.6	−4.03832
$639$	0	0
$640$	0	0
$641$	24048.7	1.48185	0.740925	−	0.671588i	$-0.234388\pi$
$641$	24048.7	1.48185	0.740925	+	0.671588i	$0.234388\pi$
$642$	0	0
$643$	19196.1	1.17732	0.588661	−	0.808380i	$-0.299655\pi$
$643$	19196.1	1.17732	0.588661	+	0.808380i	$0.299655\pi$
$644$	1257.51	0.0769452
$645$	0	0
$646$	799.960	0.0487214
$647$	−7185.82	−0.436636	−0.218318	−	0.975878i	$-0.570057\pi$
$647$	−7185.82	−0.436636	−0.218318	+	0.975878i	$0.570057\pi$
$648$	0	0
$649$	−46051.7	−2.78534
$650$	0	0
$651$	0	0
$652$	7804.36	0.468777
$653$	−403.329	−0.0241707	−0.0120854	−	0.999927i	$-0.503847\pi$
$653$	−403.329	−0.0241707	−0.0120854	+	0.999927i	$0.503847\pi$
$654$	0	0
$655$	0	0
$656$	−34527.1	−2.05497
$657$	0	0
$658$	11861.2	0.702733
$659$	−20527.9	−1.21343	−0.606716	−	0.794919i	$-0.707513\pi$
$659$	−20527.9	−1.21343	−0.606716	+	0.794919i	$0.707513\pi$
$660$	0	0
$661$	−4372.75	−0.257307	−0.128654	−	0.991690i	$-0.541066\pi$
$661$	−4372.75	−0.257307	−0.128654	+	0.991690i	$0.541066\pi$
$662$	−9580.22	−0.562456
$663$	0	0
$664$	8332.62	0.487000
$665$	0	0
$666$	0	0
$667$	9229.97	0.535811
$668$	−17522.8	−1.01494
$669$	0	0
$670$	0	0
$671$	40083.5	2.30612
$672$	0	0
$673$	−19902.7	−1.13996	−0.569980	−	0.821659i	$-0.693049\pi$
$673$	−19902.7	−1.13996	−0.569980	+	0.821659i	$0.693049\pi$
$674$	−19561.4	−1.11792
$675$	0	0
$676$	−12525.5	−0.712647
$677$	−9714.40	−0.551484	−0.275742	−	0.961232i	$-0.588924\pi$
$677$	−9714.40	−0.551484	−0.275742	+	0.961232i	$0.588924\pi$
$678$	0	0
$679$	78.2257	0.00442125
$680$	0	0
$681$	0	0
$682$	−28947.0	−1.62528
$683$	88.0227	0.00493132	0.00246566	−	0.999997i	$-0.499215\pi$
$683$	88.0227	0.00493132	0.00246566	+	0.999997i	$0.499215\pi$
$684$	0	0
$685$	0	0
$686$	−1269.64	−0.0706631
$687$	0	0
$688$	−16193.5	−0.897345
$689$	−55.9300	−0.00309254
$690$	0	0
$691$	3722.48	0.204934	0.102467	−	0.994736i	$-0.467326\pi$
$691$	3722.48	0.204934	0.102467	+	0.994736i	$0.467326\pi$
$692$	13696.1	0.752380
$693$	0	0
$694$	−43633.0	−2.38658
$695$	0	0
$696$	0	0
$697$	15893.2	0.863700
$698$	−32446.5	−1.75948
$699$	0	0
$700$	0	0
$701$	−13167.2	−0.709443	−0.354722	−	0.934972i	$-0.615424\pi$
$701$	−13167.2	−0.709443	−0.354722	+	0.934972i	$0.615424\pi$
$702$	0	0
$703$	−1339.53	−0.0718652
$704$	−11263.7	−0.603007
$705$	0	0
$706$	31753.1	1.69270
$707$	−4730.95	−0.251663
$708$	0	0
$709$	−14539.6	−0.770166	−0.385083	−	0.922882i	$-0.625827\pi$
$709$	−14539.6	−0.770166	−0.385083	+	0.922882i	$0.625827\pi$
$710$	0	0
$711$	0	0
$712$	7749.99	0.407926
$713$	4105.57	0.215645
$714$	0	0
$715$	0	0
$716$	6535.39	0.341116
$717$	0	0
$718$	13808.2	0.717713
$719$	−26509.9	−1.37504	−0.687519	−	0.726166i	$-0.741300\pi$
$719$	−26509.9	−1.37504	−0.687519	+	0.726166i	$0.741300\pi$
$720$	0	0
$721$	−10702.1	−0.552800
$722$	25251.8	1.30163
$723$	0	0
$724$	−2713.07	−0.139269
$725$	0	0
$726$	0	0
$727$	11761.8	0.600030	0.300015	−	0.953934i	$-0.403008\pi$
$727$	11761.8	0.600030	0.300015	+	0.953934i	$0.403008\pi$
$728$	23.0776	0.00117488
$729$	0	0
$730$	0	0
$731$	7454.08	0.377153
$732$	0	0
$733$	−6458.47	−0.325442	−0.162721	−	0.986672i	$-0.552027\pi$
$733$	−6458.47	−0.325442	−0.162721	+	0.986672i	$0.552027\pi$
$734$	1909.15	0.0960055
$735$	0	0
$736$	6848.07	0.342967
$737$	−4648.77	−0.232347
$738$	0	0
$739$	33663.0	1.67566	0.837830	−	0.545932i	$-0.183824\pi$
$739$	33663.0	1.67566	0.837830	+	0.545932i	$0.183824\pi$
$740$	0	0
$741$	0	0
$742$	3739.84	0.185032
$743$	19979.4	0.986506	0.493253	−	0.869886i	$-0.335808\pi$
$743$	19979.4	0.986506	0.493253	+	0.869886i	$0.335808\pi$
$744$	0	0
$745$	0	0
$746$	−40428.5	−1.98417
$747$	0	0
$748$	12144.8	0.593661
$749$	−4911.17	−0.239586
$750$	0	0
$751$	−31850.0	−1.54757	−0.773785	−	0.633449i	$-0.781639\pi$
$751$	−31850.0	−1.54757	−0.773785	+	0.633449i	$0.781639\pi$
$752$	35296.1	1.71159
$753$	0	0
$754$	−420.186	−0.0202948
$755$	0	0
$756$	0	0
$757$	−39396.6	−1.89154	−0.945769	−	0.324840i	$-0.894690\pi$
$757$	−39396.6	−1.89154	−0.945769	+	0.324840i	$0.894690\pi$
$758$	31104.7	1.49047
$759$	0	0
$760$	0	0
$761$	7170.87	0.341582	0.170791	−	0.985307i	$-0.445368\pi$
$761$	7170.87	0.341582	0.170791	+	0.985307i	$0.445368\pi$
$762$	0	0
$763$	−4179.91	−0.198326
$764$	−5644.57	−0.267295
$765$	0	0
$766$	−11218.3	−0.529154
$767$	−297.342	−0.0139979
$768$	0	0
$769$	27056.7	1.26878	0.634388	−	0.773015i	$-0.281252\pi$
$769$	27056.7	1.26878	0.634388	+	0.773015i	$0.281252\pi$
$770$	0	0
$771$	0	0
$772$	7941.62	0.370240
$773$	−40087.8	−1.86527	−0.932637	−	0.360815i	$-0.882499\pi$
$773$	−40087.8	−1.86527	−0.932637	+	0.360815i	$0.882499\pi$
$774$	0	0
$775$	0	0
$776$	95.0757	0.00439822
$777$	0	0
$778$	5194.23	0.239360
$779$	−2726.65	−0.125408
$780$	0	0
$781$	54408.6	2.49282
$782$	−4139.39	−0.189289
$783$	0	0
$784$	−3778.13	−0.172109
$785$	0	0
$786$	0	0
$787$	−23501.4	−1.06447	−0.532233	−	0.846598i	$-0.678647\pi$
$787$	−23501.4	−1.06447	−0.532233	+	0.846598i	$0.678647\pi$
$788$	20430.9	0.923632
$789$	0	0
$790$	0	0
$791$	−616.831	−0.0277269
$792$	0	0
$793$	258.807	0.0115896
$794$	−206.961	−0.00925035
$795$	0	0
$796$	3557.47	0.158406
$797$	1635.92	0.0727066	0.0363533	−	0.999339i	$-0.488426\pi$
$797$	1635.92	0.0727066	0.0363533	+	0.999339i	$0.488426\pi$
$798$	0	0
$799$	−16247.2	−0.719381
$800$	0	0
$801$	0	0
$802$	−13807.5	−0.607929
$803$	−61799.3	−2.71588
$804$	0	0
$805$	0	0
$806$	−186.902	−0.00816794
$807$	0	0
$808$	−5750.00	−0.250352
$809$	33824.6	1.46998	0.734988	−	0.678080i	$-0.237188\pi$
$809$	33824.6	1.46998	0.734988	+	0.678080i	$0.237188\pi$
$810$	0	0
$811$	−3636.30	−0.157445	−0.0787224	−	0.996897i	$-0.525084\pi$
$811$	−3636.30	−0.157445	−0.0787224	+	0.996897i	$0.525084\pi$
$812$	11691.6	0.505289
$813$	0	0
$814$	−48870.9	−2.10433
$815$	0	0
$816$	0	0
$817$	−1278.83	−0.0547619
$818$	7286.52	0.311451
$819$	0	0
$820$	0	0
$821$	−2417.39	−0.102762	−0.0513808	−	0.998679i	$-0.516362\pi$
$821$	−2417.39	−0.102762	−0.0513808	+	0.998679i	$0.516362\pi$
$822$	0	0
$823$	15752.2	0.667178	0.333589	−	0.942719i	$-0.391740\pi$
$823$	15752.2	0.667178	0.333589	+	0.942719i	$0.391740\pi$
$824$	−13007.4	−0.549920
$825$	0	0
$826$	19882.2	0.837518
$827$	−4850.49	−0.203952	−0.101976	−	0.994787i	$-0.532516\pi$
$827$	−4850.49	−0.203952	−0.101976	+	0.994787i	$0.532516\pi$
$828$	0	0
$829$	33160.3	1.38927	0.694636	−	0.719362i	$-0.255566\pi$
$829$	33160.3	1.38927	0.694636	+	0.719362i	$0.255566\pi$
$830$	0	0
$831$	0	0
$832$	−72.7264	−0.00303045
$833$	1739.12	0.0723371
$834$	0	0
$835$	0	0
$836$	−2083.57	−0.0861984
$837$	0	0
$838$	−48891.7	−2.01544
$839$	−14966.6	−0.615857	−0.307929	−	0.951409i	$-0.599636\pi$
$839$	−14966.6	−0.615857	−0.307929	+	0.951409i	$0.599636\pi$
$840$	0	0
$841$	61426.1	2.51860
$842$	27708.9	1.13410
$843$	0	0
$844$	12922.5	0.527025
$845$	0	0
$846$	0	0
$847$	15896.1	0.644861
$848$	11128.9	0.450668
$849$	0	0
$850$	0	0
$851$	6931.37	0.279206
$852$	0	0
$853$	−5806.35	−0.233066	−0.116533	−	0.993187i	$-0.537178\pi$
$853$	−5806.35	−0.233066	−0.116533	+	0.993187i	$0.537178\pi$
$854$	−17305.5	−0.693422
$855$	0	0
$856$	−5969.04	−0.238338
$857$	9191.52	0.366367	0.183183	−	0.983079i	$-0.441360\pi$
$857$	9191.52	0.366367	0.183183	+	0.983079i	$0.441360\pi$
$858$	0	0
$859$	15029.2	0.596962	0.298481	−	0.954416i	$-0.403520\pi$
$859$	15029.2	0.596962	0.298481	+	0.954416i	$0.403520\pi$
$860$	0	0
$861$	0	0
$862$	39323.1	1.55377
$863$	−17147.5	−0.676369	−0.338185	−	0.941080i	$-0.609813\pi$
$863$	−17147.5	−0.676369	−0.338185	+	0.941080i	$0.609813\pi$
$864$	0	0
$865$	0	0
$866$	−26905.7	−1.05576
$867$	0	0
$868$	5200.52	0.203361
$869$	−41456.6	−1.61832
$870$	0	0
$871$	−30.0157	−0.00116767
$872$	−5080.27	−0.197293
$873$	0	0
$874$	710.156	0.0274844
$875$	0	0
$876$	0	0
$877$	33483.6	1.28924	0.644618	−	0.764505i	$-0.277016\pi$
$877$	33483.6	1.28924	0.644618	+	0.764505i	$0.277016\pi$
$878$	2751.56	0.105764
$879$	0	0
$880$	0	0
$881$	10695.6	0.409019	0.204509	−	0.978865i	$-0.434440\pi$
$881$	10695.6	0.409019	0.204509	+	0.978865i	$0.434440\pi$
$882$	0	0
$883$	−31934.0	−1.21706	−0.608531	−	0.793530i	$-0.708241\pi$
$883$	−31934.0	−1.21706	−0.608531	+	0.793530i	$0.708241\pi$
$884$	78.4155	0.00298348
$885$	0	0
$886$	40190.9	1.52397
$887$	10655.2	0.403345	0.201672	−	0.979453i	$-0.435362\pi$
$887$	10655.2	0.403345	0.201672	+	0.979453i	$0.435362\pi$
$888$	0	0
$889$	−14835.4	−0.559690
$890$	0	0
$891$	0	0
$892$	12079.7	0.453429
$893$	2787.38	0.104453
$894$	0	0
$895$	0	0
$896$	−7308.38	−0.272495
$897$	0	0
$898$	−41320.0	−1.53549
$899$	38171.3	1.41611
$900$	0	0
$901$	−5122.74	−0.189415
$902$	−99478.4	−3.67214
$903$	0	0
$904$	−749.697	−0.0275825
$905$	0	0
$906$	0	0
$907$	20737.2	0.759171	0.379585	−	0.925157i	$-0.376067\pi$
$907$	20737.2	0.759171	0.379585	+	0.925157i	$0.376067\pi$
$908$	−15910.9	−0.581522
$909$	0	0
$910$	0	0
$911$	−11734.8	−0.426774	−0.213387	−	0.976968i	$-0.568450\pi$
$911$	−11734.8	−0.426774	−0.213387	+	0.976968i	$0.568450\pi$
$912$	0	0
$913$	58779.8	2.13070
$914$	−49789.7	−1.80185
$915$	0	0
$916$	−33147.6	−1.19566
$917$	1850.06	0.0666240
$918$	0	0
$919$	−21922.8	−0.786906	−0.393453	−	0.919345i	$-0.628720\pi$
$919$	−21922.8	−0.786906	−0.393453	+	0.919345i	$0.628720\pi$
$920$	0	0
$921$	0	0
$922$	−33830.3	−1.20840
$923$	351.300	0.0125278
$924$	0	0
$925$	0	0
$926$	48573.6	1.72379
$927$	0	0
$928$	63669.6	2.25222
$929$	36647.7	1.29427	0.647133	−	0.762377i	$-0.275968\pi$
$929$	36647.7	1.29427	0.647133	+	0.762377i	$0.275968\pi$
$930$	0	0
$931$	−298.364	−0.0105032
$932$	−11039.2	−0.387986
$933$	0	0
$934$	44127.1	1.54591
$935$	0	0
$936$	0	0
$937$	−24959.7	−0.870222	−0.435111	−	0.900377i	$-0.643291\pi$
$937$	−24959.7	−0.870222	−0.435111	+	0.900377i	$0.643291\pi$
$938$	2007.04	0.0698638
$939$	0	0
$940$	0	0
$941$	−17086.7	−0.591935	−0.295968	−	0.955198i	$-0.595642\pi$
$941$	−17086.7	−0.591935	−0.295968	+	0.955198i	$0.595642\pi$
$942$	0	0
$943$	14109.0	0.487226
$944$	59164.6	2.03988
$945$	0	0
$946$	−46656.3	−1.60352
$947$	54034.9	1.85417	0.927085	−	0.374852i	$-0.122307\pi$
$947$	54034.9	1.85417	0.927085	+	0.374852i	$0.122307\pi$
$948$	0	0
$949$	−399.020	−0.0136488
$950$	0	0
$951$	0	0
$952$	2113.73	0.0719603
$953$	−24377.9	−0.828622	−0.414311	−	0.910135i	$-0.635977\pi$
$953$	−24377.9	−0.828622	−0.414311	+	0.910135i	$0.635977\pi$
$954$	0	0
$955$	0	0
$956$	15712.3	0.531561
$957$	0	0
$958$	3054.83	0.103024
$959$	−3277.59	−0.110364
$960$	0	0
$961$	−12812.1	−0.430066
$962$	−315.545	−0.0105754
$963$	0	0
$964$	−22950.1	−0.766777
$965$	0	0
$966$	0	0
$967$	32668.1	1.08639	0.543194	−	0.839607i	$-0.317215\pi$
$967$	32668.1	1.08639	0.543194	+	0.839607i	$0.317215\pi$
$968$	19320.2	0.641502
$969$	0	0
$970$	0	0
$971$	−24623.2	−0.813795	−0.406898	−	0.913474i	$-0.633389\pi$
$971$	−24623.2	−0.813795	−0.406898	+	0.913474i	$0.633389\pi$
$972$	0	0
$973$	6698.36	0.220698
$974$	24720.2	0.813231
$975$	0	0
$976$	−51497.0	−1.68891
$977$	−18320.6	−0.599926	−0.299963	−	0.953951i	$-0.596974\pi$
$977$	−18320.6	−0.599926	−0.299963	+	0.953951i	$0.596974\pi$
$978$	0	0
$979$	54669.8	1.78473
$980$	0	0
$981$	0	0
$982$	−11469.8	−0.372727
$983$	2176.33	0.0706145	0.0353072	−	0.999377i	$-0.488759\pi$
$983$	2176.33	0.0706145	0.0353072	+	0.999377i	$0.488759\pi$
$984$	0	0
$985$	0	0
$986$	−38485.7	−1.24304
$987$	0	0
$988$	−13.4530	−0.000433196 0
$989$	6617.28	0.212758
$990$	0	0
$991$	−21487.2	−0.688761	−0.344381	−	0.938830i	$-0.611911\pi$
$991$	−21487.2	−0.688761	−0.344381	+	0.938830i	$0.611911\pi$
$992$	28320.8	0.906437
$993$	0	0
$994$	−23490.2	−0.749560
$995$	0	0
$996$	0	0
$997$	19083.0	0.606182	0.303091	−	0.952962i	$-0.401981\pi$
$997$	19083.0	0.606182	0.303091	+	0.952962i	$0.401981\pi$
$998$	8384.41	0.265936
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.s.1.1		2
3.2	odd	2		175.4.a.e.1.2	yes	2
5.4	even	2		1575.4.a.v.1.2		2
15.2	even	4		175.4.b.d.99.4		4
15.8	even	4		175.4.b.d.99.1		4
15.14	odd	2		175.4.a.d.1.1	✓	2
21.20	even	2		1225.4.a.t.1.2		2
105.104	even	2		1225.4.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.d.1.1	✓	2	15.14	odd	2
175.4.a.e.1.2	yes	2	3.2	odd	2
175.4.b.d.99.1		4	15.8	even	4
175.4.b.d.99.4		4	15.2	even	4
1225.4.a.r.1.1		2	105.104	even	2
1225.4.a.t.1.2		2	21.20	even	2
1575.4.a.s.1.1		2	1.1	even	1	trivial
1575.4.a.v.1.2		2	5.4	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-3.70156 q^{2} +5.70156 q^{4} +7.00000 q^{7} +8.50781 q^{8} +60.0156 q^{11} +0.387503 q^{13} -25.9109 q^{14} -77.1047 q^{16} +35.4922 q^{17} -6.08907 q^{19} -222.152 q^{22} +31.5078 q^{23} -1.43437 q^{26} +39.9109 q^{28} +292.942 q^{29} +130.303 q^{31} +217.345 q^{32} -131.377 q^{34} +219.989 q^{37} +22.5391 q^{38} +447.795 q^{41} +210.020 q^{43} +342.183 q^{44} -116.628 q^{46} -457.769 q^{47} +49.0000 q^{49} +2.20937 q^{52} -144.334 q^{53} +59.5547 q^{56} -1084.34 q^{58} -767.328 q^{59} +667.884 q^{61} -482.325 q^{62} -187.680 q^{64} -77.4593 q^{67} +202.361 q^{68} +906.573 q^{71} -1029.72 q^{73} -814.303 q^{74} -34.7172 q^{76} +420.109 q^{77} -690.764 q^{79} -1657.54 q^{82} +979.408 q^{83} -777.403 q^{86} +510.602 q^{88} +910.927 q^{89} +2.71252 q^{91} +179.644 q^{92} +1694.46 q^{94} +11.1751 q^{97} -181.377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8} + 56 q^{11} + 52 q^{13} - 7 q^{14} - 135 q^{16} + 103 q^{17} - 57 q^{19} - 233 q^{22} + 31 q^{23} + 138 q^{26} + 35 q^{28} + 413 q^{29} - 162 q^{31} + 249 q^{32}+ \cdots - 49 q^{98}+O(q^{100}) \) 2 * q - q^2 + 5 * q^4 + 14 * q^7 - 15 * q^8 + 56 * q^11 + 52 * q^13 - 7 * q^14 - 135 * q^16 + 103 * q^17 - 57 * q^19 - 233 * q^22 + 31 * q^23 + 138 * q^26 + 35 * q^28 + 413 * q^29 - 162 * q^31 + 249 * q^32 + 51 * q^34 + 75 * q^37 - 115 * q^38 + 505 * q^41 - 73 * q^43 + 345 * q^44 - 118 * q^46 - 224 * q^47 + 98 * q^49 - 34 * q^52 + 262 * q^53 - 105 * q^56 - 760 * q^58 - 190 * q^59 + 990 * q^61 - 1272 * q^62 + 361 * q^64 + 908 * q^67 + 155 * q^68 - 127 * q^71 - 337 * q^73 - 1206 * q^74 + q^76 + 392 * q^77 - 1119 * q^79 - 1503 * q^82 + 1517 * q^83 - 1542 * q^86 + 605 * q^88 + 1713 * q^89 + 364 * q^91 + 180 * q^92 + 2326 * q^94 + 1764 * q^97 - 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more