LMFDB - Embedded newform 1575.4.a.bb.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,-2,0,14,0,0,-21,-66,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.22952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 14 $ x^3 - x^2 - 18*x + 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.770205$ 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)

$f(q)$	$=$	$q-0.229795 q^{2} -7.94719 q^{4} -7.00000 q^{7} +3.66459 q^{8} -51.0059 q^{11} -46.0867 q^{13} +1.60857 q^{14} +62.7354 q^{16} -72.7575 q^{17} -123.876 q^{19} +11.7209 q^{22} -156.124 q^{23} +10.5905 q^{26} +55.6304 q^{28} +191.155 q^{29} -116.298 q^{31} -43.7331 q^{32} +16.7194 q^{34} -83.1918 q^{37} +28.4660 q^{38} +466.186 q^{41} -422.695 q^{43} +405.354 q^{44} +35.8766 q^{46} -268.534 q^{47} +49.0000 q^{49} +366.260 q^{52} -310.545 q^{53} -25.6521 q^{56} -43.9266 q^{58} -709.612 q^{59} +402.425 q^{61} +26.7247 q^{62} -491.834 q^{64} +114.306 q^{67} +578.218 q^{68} +214.461 q^{71} -402.954 q^{73} +19.1171 q^{74} +984.463 q^{76} +357.041 q^{77} -1376.43 q^{79} -107.127 q^{82} -1152.33 q^{83} +97.1334 q^{86} -186.916 q^{88} +366.795 q^{89} +322.607 q^{91} +1240.75 q^{92} +61.7080 q^{94} +1068.22 q^{97} -11.2600 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 14 q^{4} - 21 q^{7} - 66 q^{8} - 20 q^{11} + 14 q^{14} + 114 q^{16} - 234 q^{17} - 82 q^{19} + 236 q^{22} - 30 q^{23} + 76 q^{26} - 98 q^{28} - 32 q^{29} - 362 q^{31} - 430 q^{32} + 596 q^{34}+ \cdots - 98 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 14 * q^4 - 21 * q^7 - 66 * q^8 - 20 * q^11 + 14 * q^14 + 114 * q^16 - 234 * q^17 - 82 * q^19 + 236 * q^22 - 30 * q^23 + 76 * q^26 - 98 * q^28 - 32 * q^29 - 362 * q^31 - 430 * q^32 + 596 * q^34 - 302 * q^37 - 160 * q^38 + 626 * q^41 + 68 * q^43 + 300 * q^44 - 412 * q^46 - 936 * q^47 + 147 * q^49 + 684 * q^52 - 298 * q^53 + 462 * q^56 + 1184 * q^58 + 428 * q^59 + 1050 * q^61 + 608 * q^62 - 154 * q^64 + 52 * q^67 - 1964 * q^68 + 416 * q^71 + 312 * q^73 + 1668 * q^74 + 1712 * q^76 + 140 * q^77 - 488 * q^79 + 1476 * q^82 - 948 * q^83 - 1928 * q^86 - 124 * q^88 - 142 * q^89 + 3220 * q^92 + 328 * q^94 + 3052 * q^97 - 98 * 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 $

$2$	−0.229795	−0.0812450	−0.0406225	−	0.999175i	$-0.512934\pi$
$2$	−0.229795	−0.0812450	−0.0406225	+	0.999175i	$0.512934\pi$
$3$	0	0
$3$	0	0
$4$	−7.94719	−0.993399
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	3.66459	0.161954
$9$	0	0
$10$	0	0
$11$	−51.0059	−1.39808	−0.699039	−	0.715083i	$-0.746389\pi$
$11$	−51.0059	−1.39808	−0.699039	+	0.715083i	$0.746389\pi$
$12$	0	0
$13$	−46.0867	−0.983243	−0.491622	−	0.870809i	$-0.663596\pi$
$13$	−46.0867	−0.983243	−0.491622	+	0.870809i	$0.663596\pi$
$14$	1.60857	0.0307077
$15$	0	0
$16$	62.7354	0.980241
$17$	−72.7575	−1.03802	−0.519009	−	0.854769i	$-0.673699\pi$
$17$	−72.7575	−1.03802	−0.519009	+	0.854769i	$0.673699\pi$
$18$	0	0
$19$	−123.876	−1.49574	−0.747868	−	0.663847i	$-0.768922\pi$
$19$	−123.876	−1.49574	−0.747868	+	0.663847i	$0.768922\pi$
$20$	0	0
$21$	0	0
$22$	11.7209	0.113587
$23$	−156.124	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$23$	−156.124	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$24$	0	0
$25$	0	0
$26$	10.5905	0.0798835
$27$	0	0
$28$	55.6304	0.375470
$29$	191.155	1.22402	0.612011	−	0.790850i	$-0.290361\pi$
$29$	191.155	1.22402	0.612011	+	0.790850i	$0.290361\pi$
$30$	0	0
$31$	−116.298	−0.673798	−0.336899	−	0.941541i	$-0.609378\pi$
$31$	−116.298	−0.673798	−0.336899	+	0.941541i	$0.609378\pi$
$32$	−43.7331	−0.241593
$33$	0	0
$34$	16.7194	0.0843337
$35$	0	0
$36$	0	0
$37$	−83.1918	−0.369639	−0.184820	−	0.982772i	$-0.559170\pi$
$37$	−83.1918	−0.369639	−0.184820	+	0.982772i	$0.559170\pi$
$38$	28.4660	0.121521
$39$	0	0
$40$	0	0
$41$	466.186	1.77576	0.887879	−	0.460077i	$-0.152178\pi$
$41$	466.186	1.77576	0.887879	+	0.460077i	$0.152178\pi$
$42$	0	0
$43$	−422.695	−1.49908	−0.749539	−	0.661960i	$-0.769725\pi$
$43$	−422.695	−1.49908	−0.749539	+	0.661960i	$0.769725\pi$
$44$	405.354	1.38885
$45$	0	0
$46$	35.8766	0.114994
$47$	−268.534	−0.833400	−0.416700	−	0.909044i	$-0.636813\pi$
$47$	−268.534	−0.833400	−0.416700	+	0.909044i	$0.636813\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	366.260	0.976753
$53$	−310.545	−0.804841	−0.402421	−	0.915455i	$-0.631831\pi$
$53$	−310.545	−0.804841	−0.402421	+	0.915455i	$0.631831\pi$
$54$	0	0
$55$	0	0
$56$	−25.6521	−0.0612127
$57$	0	0
$58$	−43.9266	−0.0994455
$59$	−709.612	−1.56582	−0.782912	−	0.622133i	$-0.786266\pi$
$59$	−709.612	−1.56582	−0.782912	+	0.622133i	$0.786266\pi$
$60$	0	0
$61$	402.425	0.844675	0.422338	−	0.906439i	$-0.361210\pi$
$61$	402.425	0.844675	0.422338	+	0.906439i	$0.361210\pi$
$62$	26.7247	0.0547427
$63$	0	0
$64$	−491.834	−0.960613
$65$	0	0
$66$	0	0
$67$	114.306	0.208428	0.104214	−	0.994555i	$-0.466767\pi$
$67$	114.306	0.208428	0.104214	+	0.994555i	$0.466767\pi$
$68$	578.218	1.03117
$69$	0	0
$70$	0	0
$71$	214.461	0.358476	0.179238	−	0.983806i	$-0.442637\pi$
$71$	214.461	0.358476	0.179238	+	0.983806i	$0.442637\pi$
$72$	0	0
$73$	−402.954	−0.646058	−0.323029	−	0.946389i	$-0.604701\pi$
$73$	−402.954	−0.646058	−0.323029	+	0.946389i	$0.604701\pi$
$74$	19.1171	0.0300313
$75$	0	0
$76$	984.463	1.48586
$77$	357.041	0.528424
$78$	0	0
$79$	−1376.43	−1.96026	−0.980128	−	0.198367i	$-0.936436\pi$
$79$	−1376.43	−1.96026	−0.980128	+	0.198367i	$0.936436\pi$
$80$	0	0
$81$	0	0
$82$	−107.127	−0.144271
$83$	−1152.33	−1.52391	−0.761955	−	0.647630i	$-0.775760\pi$
$83$	−1152.33	−1.52391	−0.761955	+	0.647630i	$0.775760\pi$
$84$	0	0
$85$	0	0
$86$	97.1334	0.121793
$87$	0	0
$88$	−186.916	−0.226424
$89$	366.795	0.436856	0.218428	−	0.975853i	$-0.429907\pi$
$89$	366.795	0.436856	0.218428	+	0.975853i	$0.429907\pi$
$90$	0	0
$91$	322.607	0.371631
$92$	1240.75	1.40605
$93$	0	0
$94$	61.7080	0.0677095
$95$	0	0
$96$	0	0
$97$	1068.22	1.11816	0.559079	−	0.829115i	$-0.311155\pi$
$97$	1068.22	1.11816	0.559079	+	0.829115i	$0.311155\pi$
$98$	−11.2600	−0.0116064
$99$	0	0
$100$	0	0
$101$	−393.710	−0.387877	−0.193939	−	0.981014i	$-0.562126\pi$
$101$	−393.710	−0.387877	−0.193939	+	0.981014i	$0.562126\pi$
$102$	0	0
$103$	1419.35	1.35779	0.678895	−	0.734235i	$-0.262459\pi$
$103$	1419.35	1.35779	0.678895	+	0.734235i	$0.262459\pi$
$104$	−168.889	−0.159240
$105$	0	0
$106$	71.3617	0.0653893
$107$	−2178.53	−1.96829	−0.984143	−	0.177377i	$-0.943239\pi$
$107$	−2178.53	−1.96829	−0.984143	+	0.177377i	$0.943239\pi$
$108$	0	0
$109$	249.368	0.219130	0.109565	−	0.993980i	$-0.465054\pi$
$109$	249.368	0.219130	0.109565	+	0.993980i	$0.465054\pi$
$110$	0	0
$111$	0	0
$112$	−439.148	−0.370496
$113$	805.709	0.670750	0.335375	−	0.942085i	$-0.391137\pi$
$113$	805.709	0.670750	0.335375	+	0.942085i	$0.391137\pi$
$114$	0	0
$115$	0	0
$116$	−1519.15	−1.21594
$117$	0	0
$118$	163.066	0.127215
$119$	509.303	0.392334
$120$	0	0
$121$	1270.60	0.954623
$122$	−92.4753	−0.0686256
$123$	0	0
$124$	924.242	0.669350
$125$	0	0
$126$	0	0
$127$	458.507	0.320362	0.160181	−	0.987088i	$-0.448792\pi$
$127$	458.507	0.320362	0.160181	+	0.987088i	$0.448792\pi$
$128$	462.886	0.319638
$129$	0	0
$130$	0	0
$131$	−1048.12	−0.699041	−0.349521	−	0.936929i	$-0.613656\pi$
$131$	−1048.12	−0.699041	−0.349521	+	0.936929i	$0.613656\pi$
$132$	0	0
$133$	867.129	0.565335
$134$	−26.2670	−0.0169338
$135$	0	0
$136$	−266.627	−0.168111
$137$	1559.36	0.972445	0.486223	−	0.873835i	$-0.338374\pi$
$137$	1559.36	0.972445	0.486223	+	0.873835i	$0.338374\pi$
$138$	0	0
$139$	230.969	0.140939	0.0704696	−	0.997514i	$-0.477550\pi$
$139$	230.969	0.140939	0.0704696	+	0.997514i	$0.477550\pi$
$140$	0	0
$141$	0	0
$142$	−49.2821	−0.0291244
$143$	2350.70	1.37465
$144$	0	0
$145$	0	0
$146$	92.5970	0.0524889
$147$	0	0
$148$	661.142	0.367199
$149$	−1754.58	−0.964701	−0.482351	−	0.875978i	$-0.660217\pi$
$149$	−1754.58	−0.964701	−0.482351	+	0.875978i	$0.660217\pi$
$150$	0	0
$151$	1707.28	0.920107	0.460054	−	0.887891i	$-0.347830\pi$
$151$	1707.28	0.920107	0.460054	+	0.887891i	$0.347830\pi$
$152$	−453.953	−0.242240
$153$	0	0
$154$	−82.0465	−0.0429318
$155$	0	0
$156$	0	0
$157$	1815.33	0.922798	0.461399	−	0.887193i	$-0.347348\pi$
$157$	1815.33	0.922798	0.461399	+	0.887193i	$0.347348\pi$
$158$	316.297	0.159261
$159$	0	0
$160$	0	0
$161$	1092.87	0.534969
$162$	0	0
$163$	−1763.58	−0.847449	−0.423724	−	0.905791i	$-0.639277\pi$
$163$	−1763.58	−0.847449	−0.423724	+	0.905791i	$0.639277\pi$
$164$	−3704.87	−1.76404
$165$	0	0
$166$	264.800	0.123810
$167$	1184.52	0.548866	0.274433	−	0.961606i	$-0.411510\pi$
$167$	1184.52	0.548866	0.274433	+	0.961606i	$0.411510\pi$
$168$	0	0
$169$	−73.0133	−0.0332332
$170$	0	0
$171$	0	0
$172$	3359.24	1.48918
$173$	−3232.75	−1.42070	−0.710351	−	0.703847i	$-0.751464\pi$
$173$	−3232.75	−1.42070	−0.710351	+	0.703847i	$0.751464\pi$
$174$	0	0
$175$	0	0
$176$	−3199.88	−1.37045
$177$	0	0
$178$	−84.2878	−0.0354924
$179$	1520.16	0.634760	0.317380	−	0.948298i	$-0.397197\pi$
$179$	1520.16	0.634760	0.317380	+	0.948298i	$0.397197\pi$
$180$	0	0
$181$	2173.37	0.892514	0.446257	−	0.894905i	$-0.352757\pi$
$181$	2173.37	0.892514	0.446257	+	0.894905i	$0.352757\pi$
$182$	−74.1336	−0.0301931
$183$	0	0
$184$	−572.130	−0.229228
$185$	0	0
$186$	0	0
$187$	3711.06	1.45123
$188$	2134.10	0.827899
$189$	0	0
$190$	0	0
$191$	1195.92	0.453057	0.226528	−	0.974005i	$-0.427262\pi$
$191$	1195.92	0.453057	0.226528	+	0.974005i	$0.427262\pi$
$192$	0	0
$193$	4678.84	1.74503	0.872514	−	0.488588i	$-0.162488\pi$
$193$	4678.84	1.74503	0.872514	+	0.488588i	$0.162488\pi$
$194$	−245.472	−0.0908446
$195$	0	0
$196$	−389.413	−0.141914
$197$	4802.92	1.73702	0.868512	−	0.495668i	$-0.165077\pi$
$197$	4802.92	1.73702	0.868512	+	0.495668i	$0.165077\pi$
$198$	0	0
$199$	1348.64	0.480414	0.240207	−	0.970722i	$-0.422785\pi$
$199$	1348.64	0.480414	0.240207	+	0.970722i	$0.422785\pi$
$200$	0	0
$201$	0	0
$202$	90.4727	0.0315131
$203$	−1338.09	−0.462636
$204$	0	0
$205$	0	0
$206$	−326.159	−0.110314
$207$	0	0
$208$	−2891.27	−0.963815
$209$	6318.38	2.09116
$210$	0	0
$211$	5550.35	1.81091	0.905455	−	0.424442i	$-0.139530\pi$
$211$	5550.35	1.81091	0.905455	+	0.424442i	$0.139530\pi$
$212$	2467.96	0.799529
$213$	0	0
$214$	500.617	0.159913
$215$	0	0
$216$	0	0
$217$	814.086	0.254672
$218$	−57.3037	−0.0178032
$219$	0	0
$220$	0	0
$221$	3353.16	1.02062
$222$	0	0
$223$	771.304	0.231616	0.115808	−	0.993272i	$-0.463054\pi$
$223$	771.304	0.231616	0.115808	+	0.993272i	$0.463054\pi$
$224$	306.131	0.0913137
$225$	0	0
$226$	−185.148	−0.0544951
$227$	−5509.84	−1.61102	−0.805508	−	0.592585i	$-0.798108\pi$
$227$	−5509.84	−1.61102	−0.805508	+	0.592585i	$0.798108\pi$
$228$	0	0
$229$	−1378.43	−0.397771	−0.198885	−	0.980023i	$-0.563732\pi$
$229$	−1378.43	−0.397771	−0.198885	+	0.980023i	$0.563732\pi$
$230$	0	0
$231$	0	0
$232$	700.506	0.198235
$233$	−2540.67	−0.714354	−0.357177	−	0.934037i	$-0.616261\pi$
$233$	−2540.67	−0.714354	−0.357177	+	0.934037i	$0.616261\pi$
$234$	0	0
$235$	0	0
$236$	5639.42	1.55549
$237$	0	0
$238$	−117.035	−0.0318751
$239$	−5082.14	−1.37547	−0.687733	−	0.725964i	$-0.741394\pi$
$239$	−5082.14	−1.37547	−0.687733	+	0.725964i	$0.741394\pi$
$240$	0	0
$241$	681.272	0.182094	0.0910468	−	0.995847i	$-0.470979\pi$
$241$	681.272	0.182094	0.0910468	+	0.995847i	$0.470979\pi$
$242$	−291.979	−0.0775583
$243$	0	0
$244$	−3198.15	−0.839100
$245$	0	0
$246$	0	0
$247$	5709.02	1.47067
$248$	−426.185	−0.109124
$249$	0	0
$250$	0	0
$251$	1044.10	0.262561	0.131280	−	0.991345i	$-0.458091\pi$
$251$	1044.10	0.262561	0.131280	+	0.991345i	$0.458091\pi$
$252$	0	0
$253$	7963.24	1.97883
$254$	−105.363	−0.0260278
$255$	0	0
$256$	3828.30	0.934644
$257$	−5205.91	−1.26356	−0.631781	−	0.775147i	$-0.717676\pi$
$257$	−5205.91	−1.26356	−0.631781	+	0.775147i	$0.717676\pi$
$258$	0	0
$259$	582.343	0.139710
$260$	0	0
$261$	0	0
$262$	240.853	0.0567936
$263$	2984.55	0.699754	0.349877	−	0.936796i	$-0.386223\pi$
$263$	2984.55	0.699754	0.349877	+	0.936796i	$0.386223\pi$
$264$	0	0
$265$	0	0
$266$	−199.262	−0.0459306
$267$	0	0
$268$	−908.412	−0.207053
$269$	3507.88	0.795089	0.397545	−	0.917583i	$-0.369862\pi$
$269$	3507.88	0.795089	0.397545	+	0.917583i	$0.369862\pi$
$270$	0	0
$271$	1612.27	0.361396	0.180698	−	0.983539i	$-0.442164\pi$
$271$	1612.27	0.361396	0.180698	+	0.983539i	$0.442164\pi$
$272$	−4564.48	−1.01751
$273$	0	0
$274$	−358.334	−0.0790063
$275$	0	0
$276$	0	0
$277$	−7326.89	−1.58928	−0.794639	−	0.607082i	$-0.792340\pi$
$277$	−7326.89	−1.58928	−0.794639	+	0.607082i	$0.792340\pi$
$278$	−53.0756	−0.0114506
$279$	0	0
$280$	0	0
$281$	−3499.91	−0.743016	−0.371508	−	0.928430i	$-0.621159\pi$
$281$	−3499.91	−0.743016	−0.371508	+	0.928430i	$0.621159\pi$
$282$	0	0
$283$	−6261.51	−1.31522	−0.657612	−	0.753357i	$-0.728433\pi$
$283$	−6261.51	−1.31522	−0.657612	+	0.753357i	$0.728433\pi$
$284$	−1704.36	−0.356110
$285$	0	0
$286$	−540.179	−0.111683
$287$	−3263.30	−0.671173
$288$	0	0
$289$	380.660	0.0774802
$290$	0	0
$291$	0	0
$292$	3202.35	0.641793
$293$	−5860.41	−1.16849	−0.584247	−	0.811576i	$-0.698610\pi$
$293$	−5860.41	−1.16849	−0.584247	+	0.811576i	$0.698610\pi$
$294$	0	0
$295$	0	0
$296$	−304.864	−0.0598644
$297$	0	0
$298$	403.194	0.0783771
$299$	7195.24	1.39168
$300$	0	0
$301$	2958.86	0.566598
$302$	−392.324	−0.0747541
$303$	0	0
$304$	−7771.39	−1.46618
$305$	0	0
$306$	0	0
$307$	3135.86	0.582974	0.291487	−	0.956575i	$-0.405850\pi$
$307$	3135.86	0.582974	0.291487	+	0.956575i	$0.405850\pi$
$308$	−2837.48	−0.524936
$309$	0	0
$310$	0	0
$311$	−5903.72	−1.07643	−0.538214	−	0.842808i	$-0.680901\pi$
$311$	−5903.72	−1.07643	−0.538214	+	0.842808i	$0.680901\pi$
$312$	0	0
$313$	−6127.89	−1.10661	−0.553304	−	0.832979i	$-0.686633\pi$
$313$	−6127.89	−1.10661	−0.553304	+	0.832979i	$0.686633\pi$
$314$	−417.155	−0.0749727
$315$	0	0
$316$	10938.7	1.94732
$317$	−2846.55	−0.504348	−0.252174	−	0.967682i	$-0.581146\pi$
$317$	−2846.55	−0.504348	−0.252174	+	0.967682i	$0.581146\pi$
$318$	0	0
$319$	−9750.04	−1.71128
$320$	0	0
$321$	0	0
$322$	−251.136	−0.0434635
$323$	9012.88	1.55260
$324$	0	0
$325$	0	0
$326$	405.262	0.0688509
$327$	0	0
$328$	1708.38	0.287590
$329$	1879.74	0.314995
$330$	0	0
$331$	−2166.44	−0.359753	−0.179877	−	0.983689i	$-0.557570\pi$
$331$	−2166.44	−0.359753	−0.179877	+	0.983689i	$0.557570\pi$
$332$	9157.79	1.51385
$333$	0	0
$334$	−272.197	−0.0445926
$335$	0	0
$336$	0	0
$337$	4382.45	0.708390	0.354195	−	0.935172i	$-0.384755\pi$
$337$	4382.45	0.708390	0.354195	+	0.935172i	$0.384755\pi$
$338$	16.7781	0.00270003
$339$	0	0
$340$	0	0
$341$	5931.88	0.942022
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−1549.00	−0.242781
$345$	0	0
$346$	742.871	0.115425
$347$	5931.63	0.917656	0.458828	−	0.888525i	$-0.348269\pi$
$347$	5931.63	0.917656	0.458828	+	0.888525i	$0.348269\pi$
$348$	0	0
$349$	1340.59	0.205616	0.102808	−	0.994701i	$-0.467217\pi$
$349$	1340.59	0.205616	0.102808	+	0.994701i	$0.467217\pi$
$350$	0	0
$351$	0	0
$352$	2230.64	0.337766
$353$	−716.976	−0.108104	−0.0540521	−	0.998538i	$-0.517214\pi$
$353$	−716.976	−0.108104	−0.0540521	+	0.998538i	$0.517214\pi$
$354$	0	0
$355$	0	0
$356$	−2914.99	−0.433973
$357$	0	0
$358$	−349.326	−0.0515711
$359$	3962.70	0.582572	0.291286	−	0.956636i	$-0.405917\pi$
$359$	3962.70	0.582572	0.291286	+	0.956636i	$0.405917\pi$
$360$	0	0
$361$	8486.14	1.23723
$362$	−499.430	−0.0725123
$363$	0	0
$364$	−2563.82	−0.369178
$365$	0	0
$366$	0	0
$367$	7257.92	1.03232	0.516158	−	0.856493i	$-0.327362\pi$
$367$	7257.92	1.03232	0.516158	+	0.856493i	$0.327362\pi$
$368$	−9794.50	−1.38743
$369$	0	0
$370$	0	0
$371$	2173.81	0.304201
$372$	0	0
$373$	−1949.21	−0.270580	−0.135290	−	0.990806i	$-0.543197\pi$
$373$	−1949.21	−0.270580	−0.135290	+	0.990806i	$0.543197\pi$
$374$	−852.786	−0.117905
$375$	0	0
$376$	−984.070	−0.134972
$377$	−8809.71	−1.20351
$378$	0	0
$379$	10147.5	1.37531	0.687655	−	0.726038i	$-0.258640\pi$
$379$	10147.5	1.37531	0.687655	+	0.726038i	$0.258640\pi$
$380$	0	0
$381$	0	0
$382$	−274.817	−0.0368086
$383$	−1593.28	−0.212566	−0.106283	−	0.994336i	$-0.533895\pi$
$383$	−1593.28	−0.212566	−0.106283	+	0.994336i	$0.533895\pi$
$384$	0	0
$385$	0	0
$386$	−1075.18	−0.141775
$387$	0	0
$388$	−8489.35	−1.11078
$389$	−10208.7	−1.33060	−0.665300	−	0.746576i	$-0.731696\pi$
$389$	−10208.7	−1.33060	−0.665300	+	0.746576i	$0.731696\pi$
$390$	0	0
$391$	11359.2	1.46920
$392$	179.565	0.0231362
$393$	0	0
$394$	−1103.69	−0.141124
$395$	0	0
$396$	0	0
$397$	8550.26	1.08092	0.540460	−	0.841370i	$-0.318250\pi$
$397$	8550.26	1.08092	0.540460	+	0.841370i	$0.318250\pi$
$398$	−309.911	−0.0390312
$399$	0	0
$400$	0	0
$401$	6422.21	0.799775	0.399888	−	0.916564i	$-0.369049\pi$
$401$	6422.21	0.799775	0.399888	+	0.916564i	$0.369049\pi$
$402$	0	0
$403$	5359.79	0.662507
$404$	3128.89	0.385317
$405$	0	0
$406$	307.486	0.0375869
$407$	4243.27	0.516785
$408$	0	0
$409$	−8536.12	−1.03199	−0.515995	−	0.856591i	$-0.672578\pi$
$409$	−8536.12	−1.03199	−0.515995	+	0.856591i	$0.672578\pi$
$410$	0	0
$411$	0	0
$412$	−11279.8	−1.34883
$413$	4967.28	0.591826
$414$	0	0
$415$	0	0
$416$	2015.51	0.237545
$417$	0	0
$418$	−1451.94	−0.169896
$419$	−9948.69	−1.15997	−0.579983	−	0.814629i	$-0.696941\pi$
$419$	−9948.69	−1.15997	−0.579983	+	0.814629i	$0.696941\pi$
$420$	0	0
$421$	−7989.63	−0.924918	−0.462459	−	0.886641i	$-0.653033\pi$
$421$	−7989.63	−0.924918	−0.462459	+	0.886641i	$0.653033\pi$
$422$	−1275.45	−0.147127
$423$	0	0
$424$	−1138.02	−0.130347
$425$	0	0
$426$	0	0
$427$	−2816.97	−0.319257
$428$	17313.2	1.95529
$429$	0	0
$430$	0	0
$431$	−9893.26	−1.10566	−0.552832	−	0.833292i	$-0.686453\pi$
$431$	−9893.26	−1.10566	−0.552832	+	0.833292i	$0.686453\pi$
$432$	0	0
$433$	10225.7	1.13491	0.567456	−	0.823404i	$-0.307928\pi$
$433$	10225.7	1.13491	0.567456	+	0.823404i	$0.307928\pi$
$434$	−187.073	−0.0206908
$435$	0	0
$436$	−1981.78	−0.217684
$437$	19339.9	2.11706
$438$	0	0
$439$	3871.07	0.420857	0.210428	−	0.977609i	$-0.432514\pi$
$439$	3871.07	0.420857	0.210428	+	0.977609i	$0.432514\pi$
$440$	0	0
$441$	0	0
$442$	−770.540	−0.0829205
$443$	−394.626	−0.0423233	−0.0211617	−	0.999776i	$-0.506736\pi$
$443$	−394.626	−0.0423233	−0.0211617	+	0.999776i	$0.506736\pi$
$444$	0	0
$445$	0	0
$446$	−177.242	−0.0188176
$447$	0	0
$448$	3442.84	0.363078
$449$	−14437.4	−1.51747	−0.758735	−	0.651400i	$-0.774182\pi$
$449$	−14437.4	−1.51747	−0.758735	+	0.651400i	$0.774182\pi$
$450$	0	0
$451$	−23778.3	−2.48265
$452$	−6403.13	−0.666323
$453$	0	0
$454$	1266.14	0.130887
$455$	0	0
$456$	0	0
$457$	−10334.8	−1.05786	−0.528928	−	0.848667i	$-0.677406\pi$
$457$	−10334.8	−1.05786	−0.528928	+	0.848667i	$0.677406\pi$
$458$	316.758	0.0323169
$459$	0	0
$460$	0	0
$461$	14401.9	1.45502	0.727511	−	0.686096i	$-0.240677\pi$
$461$	14401.9	1.45502	0.727511	+	0.686096i	$0.240677\pi$
$462$	0	0
$463$	−1265.08	−0.126984	−0.0634919	−	0.997982i	$-0.520224\pi$
$463$	−1265.08	−0.126984	−0.0634919	+	0.997982i	$0.520224\pi$
$464$	11992.2	1.19984
$465$	0	0
$466$	583.833	0.0580377
$467$	13178.9	1.30588	0.652941	−	0.757409i	$-0.273535\pi$
$467$	13178.9	1.30588	0.652941	+	0.757409i	$0.273535\pi$
$468$	0	0
$469$	−800.142	−0.0787785
$470$	0	0
$471$	0	0
$472$	−2600.44	−0.253591
$473$	21559.9	2.09583
$474$	0	0
$475$	0	0
$476$	−4047.53	−0.389744
$477$	0	0
$478$	1167.85	0.111750
$479$	883.505	0.0842763	0.0421382	−	0.999112i	$-0.486583\pi$
$479$	883.505	0.0842763	0.0421382	+	0.999112i	$0.486583\pi$
$480$	0	0
$481$	3834.04	0.363445
$482$	−156.553	−0.0147942
$483$	0	0
$484$	−10097.7	−0.948322
$485$	0	0
$486$	0	0
$487$	−4959.35	−0.461458	−0.230729	−	0.973018i	$-0.574111\pi$
$487$	−4959.35	−0.461458	−0.230729	+	0.973018i	$0.574111\pi$
$488$	1474.72	0.136798
$489$	0	0
$490$	0	0
$491$	−18088.8	−1.66260	−0.831301	−	0.555822i	$-0.812404\pi$
$491$	−18088.8	−1.66260	−0.831301	+	0.555822i	$0.812404\pi$
$492$	0	0
$493$	−13908.0	−1.27056
$494$	−1311.91	−0.119485
$495$	0	0
$496$	−7296.00	−0.660484
$497$	−1501.22	−0.135491
$498$	0	0
$499$	9428.10	0.845811	0.422906	−	0.906174i	$-0.361010\pi$
$499$	9428.10	0.845811	0.422906	+	0.906174i	$0.361010\pi$
$500$	0	0
$501$	0	0
$502$	−239.928	−0.0213317
$503$	−8543.95	−0.757367	−0.378684	−	0.925526i	$-0.623623\pi$
$503$	−8543.95	−0.757367	−0.378684	+	0.925526i	$0.623623\pi$
$504$	0	0
$505$	0	0
$506$	−1829.92	−0.160770
$507$	0	0
$508$	−3643.85	−0.318247
$509$	−4081.74	−0.355442	−0.177721	−	0.984081i	$-0.556872\pi$
$509$	−4081.74	−0.355442	−0.177721	+	0.984081i	$0.556872\pi$
$510$	0	0
$511$	2820.68	0.244187
$512$	−4582.81	−0.395573
$513$	0	0
$514$	1196.29	0.102658
$515$	0	0
$516$	0	0
$517$	13696.8	1.16516
$518$	−133.820	−0.0113508
$519$	0	0
$520$	0	0
$521$	−2593.89	−0.218120	−0.109060	−	0.994035i	$-0.534784\pi$
$521$	−2593.89	−0.218120	−0.109060	+	0.994035i	$0.534784\pi$
$522$	0	0
$523$	3216.08	0.268890	0.134445	−	0.990921i	$-0.457075\pi$
$523$	3216.08	0.268890	0.134445	+	0.990921i	$0.457075\pi$
$524$	8329.59	0.694427
$525$	0	0
$526$	−685.836	−0.0568515
$527$	8461.55	0.699414
$528$	0	0
$529$	12207.7	1.00334
$530$	0	0
$531$	0	0
$532$	−6891.24	−0.561604
$533$	−21485.0	−1.74600
$534$	0	0
$535$	0	0
$536$	418.885	0.0337557
$537$	0	0
$538$	−806.094	−0.0645970
$539$	−2499.29	−0.199725
$540$	0	0
$541$	−14686.4	−1.16713	−0.583567	−	0.812065i	$-0.698343\pi$
$541$	−14686.4	−1.16713	−0.583567	+	0.812065i	$0.698343\pi$
$542$	−370.492	−0.0293616
$543$	0	0
$544$	3181.91	0.250778
$545$	0	0
$546$	0	0
$547$	−8296.56	−0.648510	−0.324255	−	0.945970i	$-0.605114\pi$
$547$	−8296.56	−0.648510	−0.324255	+	0.945970i	$0.605114\pi$
$548$	−12392.5	−0.966027
$549$	0	0
$550$	0	0
$551$	−23679.4	−1.83081
$552$	0	0
$553$	9634.99	0.740907
$554$	1683.69	0.129121
$555$	0	0
$556$	−1835.56	−0.140009
$557$	−19711.8	−1.49949	−0.749746	−	0.661726i	$-0.769824\pi$
$557$	−19711.8	−1.49949	−0.749746	+	0.661726i	$0.769824\pi$
$558$	0	0
$559$	19480.6	1.47396
$560$	0	0
$561$	0	0
$562$	804.264	0.0603663
$563$	11500.3	0.860891	0.430446	−	0.902617i	$-0.358356\pi$
$563$	11500.3	0.860891	0.430446	+	0.902617i	$0.358356\pi$
$564$	0	0
$565$	0	0
$566$	1438.87	0.106855
$567$	0	0
$568$	785.911	0.0580565
$569$	6191.04	0.456137	0.228068	−	0.973645i	$-0.426759\pi$
$569$	6191.04	0.456137	0.228068	+	0.973645i	$0.426759\pi$
$570$	0	0
$571$	−15992.6	−1.17210	−0.586050	−	0.810275i	$-0.699318\pi$
$571$	−15992.6	−1.17210	−0.586050	+	0.810275i	$0.699318\pi$
$572$	−18681.4	−1.36558
$573$	0	0
$574$	749.892	0.0545295
$575$	0	0
$576$	0	0
$577$	−8285.26	−0.597781	−0.298891	−	0.954287i	$-0.596617\pi$
$577$	−8285.26	−0.597781	−0.298891	+	0.954287i	$0.596617\pi$
$578$	−87.4740	−0.00629488
$579$	0	0
$580$	0	0
$581$	8066.31	0.575984
$582$	0	0
$583$	15839.6	1.12523
$584$	−1476.66	−0.104631
$585$	0	0
$586$	1346.70	0.0949343
$587$	−5419.34	−0.381056	−0.190528	−	0.981682i	$-0.561020\pi$
$587$	−5419.34	−0.381056	−0.190528	+	0.981682i	$0.561020\pi$
$588$	0	0
$589$	14406.5	1.00782
$590$	0	0
$591$	0	0
$592$	−5219.08	−0.362336
$593$	−16791.7	−1.16282	−0.581412	−	0.813609i	$-0.697500\pi$
$593$	−16791.7	−1.16282	−0.581412	+	0.813609i	$0.697500\pi$
$594$	0	0
$595$	0	0
$596$	13944.0	0.958333
$597$	0	0
$598$	−1653.43	−0.113067
$599$	3005.34	0.205000	0.102500	−	0.994733i	$-0.467316\pi$
$599$	3005.34	0.205000	0.102500	+	0.994733i	$0.467316\pi$
$600$	0	0
$601$	−1088.83	−0.0739007	−0.0369503	−	0.999317i	$-0.511764\pi$
$601$	−1088.83	−0.0739007	−0.0369503	+	0.999317i	$0.511764\pi$
$602$	−679.934	−0.0460333
$603$	0	0
$604$	−13568.1	−0.914034
$605$	0	0
$606$	0	0
$607$	2227.51	0.148949	0.0744743	−	0.997223i	$-0.476272\pi$
$607$	2227.51	0.148949	0.0744743	+	0.997223i	$0.476272\pi$
$608$	5417.46	0.361360
$609$	0	0
$610$	0	0
$611$	12375.9	0.819434
$612$	0	0
$613$	20003.8	1.31802	0.659010	−	0.752134i	$-0.270976\pi$
$613$	20003.8	1.31802	0.659010	+	0.752134i	$0.270976\pi$
$614$	−720.606	−0.0473637
$615$	0	0
$616$	1308.41	0.0855802
$617$	5297.13	0.345631	0.172816	−	0.984954i	$-0.444714\pi$
$617$	5297.13	0.345631	0.172816	+	0.984954i	$0.444714\pi$
$618$	0	0
$619$	24122.7	1.56636	0.783178	−	0.621797i	$-0.213597\pi$
$619$	24122.7	1.56636	0.783178	+	0.621797i	$0.213597\pi$
$620$	0	0
$621$	0	0
$622$	1356.65	0.0874544
$623$	−2567.57	−0.165116
$624$	0	0
$625$	0	0
$626$	1408.16	0.0899064
$627$	0	0
$628$	−14426.8	−0.916707
$629$	6052.83	0.383692
$630$	0	0
$631$	−13099.2	−0.826423	−0.413211	−	0.910635i	$-0.635593\pi$
$631$	−13099.2	−0.826423	−0.413211	+	0.910635i	$0.635593\pi$
$632$	−5044.05	−0.317471
$633$	0	0
$634$	654.125	0.0409757
$635$	0	0
$636$	0	0
$637$	−2258.25	−0.140463
$638$	2240.51	0.139033
$639$	0	0
$640$	0	0
$641$	−7992.28	−0.492475	−0.246237	−	0.969210i	$-0.579194\pi$
$641$	−7992.28	−0.492475	−0.246237	+	0.969210i	$0.579194\pi$
$642$	0	0
$643$	7618.97	0.467283	0.233642	−	0.972323i	$-0.424936\pi$
$643$	7618.97	0.467283	0.233642	+	0.972323i	$0.424936\pi$
$644$	−8685.23	−0.531438
$645$	0	0
$646$	−2071.12	−0.126141
$647$	−5037.17	−0.306076	−0.153038	−	0.988220i	$-0.548906\pi$
$647$	−5037.17	−0.306076	−0.153038	+	0.988220i	$0.548906\pi$
$648$	0	0
$649$	36194.4	2.18914
$650$	0	0
$651$	0	0
$652$	14015.5	0.841855
$653$	8818.74	0.528490	0.264245	−	0.964456i	$-0.414877\pi$
$653$	8818.74	0.528490	0.264245	+	0.964456i	$0.414877\pi$
$654$	0	0
$655$	0	0
$656$	29246.4	1.74067
$657$	0	0
$658$	−431.956	−0.0255918
$659$	2787.13	0.164752	0.0823758	−	0.996601i	$-0.473749\pi$
$659$	2787.13	0.164752	0.0823758	+	0.996601i	$0.473749\pi$
$660$	0	0
$661$	−14848.3	−0.873726	−0.436863	−	0.899528i	$-0.643910\pi$
$661$	−14848.3	−0.873726	−0.436863	+	0.899528i	$0.643910\pi$
$662$	497.838	0.0292281
$663$	0	0
$664$	−4222.82	−0.246803
$665$	0	0
$666$	0	0
$667$	−29843.9	−1.73247
$668$	−9413.58	−0.545243
$669$	0	0
$670$	0	0
$671$	−20526.0	−1.18092
$672$	0	0
$673$	2238.59	0.128219	0.0641093	−	0.997943i	$-0.479579\pi$
$673$	2238.59	0.128219	0.0641093	+	0.997943i	$0.479579\pi$
$674$	−1007.07	−0.0575531
$675$	0	0
$676$	580.250	0.0330138
$677$	5505.03	0.312519	0.156259	−	0.987716i	$-0.450056\pi$
$677$	5505.03	0.312519	0.156259	+	0.987716i	$0.450056\pi$
$678$	0	0
$679$	−7477.53	−0.422624
$680$	0	0
$681$	0	0
$682$	−1363.12	−0.0765345
$683$	−8638.56	−0.483961	−0.241980	−	0.970281i	$-0.577797\pi$
$683$	−8638.56	−0.483961	−0.241980	+	0.970281i	$0.577797\pi$
$684$	0	0
$685$	0	0
$686$	78.8198	0.00438682
$687$	0	0
$688$	−26518.0	−1.46946
$689$	14312.0	0.791355
$690$	0	0
$691$	−32838.2	−1.80785	−0.903925	−	0.427691i	$-0.859327\pi$
$691$	−32838.2	−1.80785	−0.903925	+	0.427691i	$0.859327\pi$
$692$	25691.3	1.41132
$693$	0	0
$694$	−1363.06	−0.0745549
$695$	0	0
$696$	0	0
$697$	−33918.6	−1.84327
$698$	−308.061	−0.0167052
$699$	0	0
$700$	0	0
$701$	32942.3	1.77491	0.887455	−	0.460894i	$-0.152471\pi$
$701$	32942.3	1.77491	0.887455	+	0.460894i	$0.152471\pi$
$702$	0	0
$703$	10305.4	0.552883
$704$	25086.4	1.34301
$705$	0	0
$706$	164.758	0.00878292
$707$	2755.97	0.146604
$708$	0	0
$709$	−34495.5	−1.82723	−0.913614	−	0.406583i	$-0.866720\pi$
$709$	−34495.5	−1.82723	−0.913614	+	0.406583i	$0.866720\pi$
$710$	0	0
$711$	0	0
$712$	1344.15	0.0707505
$713$	18156.9	0.953690
$714$	0	0
$715$	0	0
$716$	−12081.0	−0.630570
$717$	0	0
$718$	−910.611	−0.0473310
$719$	−6884.11	−0.357071	−0.178535	−	0.983933i	$-0.557136\pi$
$719$	−6884.11	−0.357071	−0.178535	+	0.983933i	$0.557136\pi$
$720$	0	0
$721$	−9935.43	−0.513196
$722$	−1950.08	−0.100518
$723$	0	0
$724$	−17272.2	−0.886623
$725$	0	0
$726$	0	0
$727$	−13950.4	−0.711682	−0.355841	−	0.934546i	$-0.615806\pi$
$727$	−13950.4	−0.711682	−0.355841	+	0.934546i	$0.615806\pi$
$728$	1182.22	0.0601870
$729$	0	0
$730$	0	0
$731$	30754.2	1.55607
$732$	0	0
$733$	4055.50	0.204357	0.102178	−	0.994766i	$-0.467419\pi$
$733$	4055.50	0.204357	0.102178	+	0.994766i	$0.467419\pi$
$734$	−1667.84	−0.0838705
$735$	0	0
$736$	6827.77	0.341950
$737$	−5830.28	−0.291399
$738$	0	0
$739$	−11122.2	−0.553638	−0.276819	−	0.960922i	$-0.589280\pi$
$739$	−11122.2	−0.553638	−0.276819	+	0.960922i	$0.589280\pi$
$740$	0	0
$741$	0	0
$742$	−499.532	−0.0247148
$743$	−4496.34	−0.222012	−0.111006	−	0.993820i	$-0.535407\pi$
$743$	−4496.34	−0.222012	−0.111006	+	0.993820i	$0.535407\pi$
$744$	0	0
$745$	0	0
$746$	447.920	0.0219833
$747$	0	0
$748$	−29492.6	−1.44165
$749$	15249.7	0.743942
$750$	0	0
$751$	1524.74	0.0740862	0.0370431	−	0.999314i	$-0.488206\pi$
$751$	1524.74	0.0740862	0.0370431	+	0.999314i	$0.488206\pi$
$752$	−16846.6	−0.816933
$753$	0	0
$754$	2024.43	0.0977791
$755$	0	0
$756$	0	0
$757$	−19727.4	−0.947166	−0.473583	−	0.880749i	$-0.657040\pi$
$757$	−19727.4	−0.947166	−0.473583	+	0.880749i	$0.657040\pi$
$758$	−2331.85	−0.111737
$759$	0	0
$760$	0	0
$761$	−6327.41	−0.301404	−0.150702	−	0.988579i	$-0.548153\pi$
$761$	−6327.41	−0.301404	−0.150702	+	0.988579i	$0.548153\pi$
$762$	0	0
$763$	−1745.58	−0.0828233
$764$	−9504.22	−0.450066
$765$	0	0
$766$	366.128	0.0172699
$767$	32703.7	1.53959
$768$	0	0
$769$	6909.63	0.324015	0.162008	−	0.986790i	$-0.448203\pi$
$769$	6909.63	0.324015	0.162008	+	0.986790i	$0.448203\pi$
$770$	0	0
$771$	0	0
$772$	−37183.7	−1.73351
$773$	−15447.7	−0.718777	−0.359388	−	0.933188i	$-0.617015\pi$
$773$	−15447.7	−0.718777	−0.359388	+	0.933188i	$0.617015\pi$
$774$	0	0
$775$	0	0
$776$	3914.59	0.181090
$777$	0	0
$778$	2345.92	0.108105
$779$	−57749.1	−2.65607
$780$	0	0
$781$	−10938.8	−0.501178
$782$	−2610.29	−0.119365
$783$	0	0
$784$	3074.04	0.140034
$785$	0	0
$786$	0	0
$787$	−28152.9	−1.27515	−0.637575	−	0.770388i	$-0.720062\pi$
$787$	−28152.9	−1.27515	−0.637575	+	0.770388i	$0.720062\pi$
$788$	−38169.7	−1.72556
$789$	0	0
$790$	0	0
$791$	−5639.97	−0.253520
$792$	0	0
$793$	−18546.4	−0.830521
$794$	−1964.81	−0.0878193
$795$	0	0
$796$	−10717.9	−0.477243
$797$	−10603.4	−0.471259	−0.235629	−	0.971843i	$-0.575715\pi$
$797$	−10603.4	−0.471259	−0.235629	+	0.971843i	$0.575715\pi$
$798$	0	0
$799$	19537.9	0.865083
$800$	0	0
$801$	0	0
$802$	−1475.79	−0.0649777
$803$	20553.0	0.903239
$804$	0	0
$805$	0	0
$806$	−1231.66	−0.0538253
$807$	0	0
$808$	−1442.79	−0.0628181
$809$	8022.99	0.348669	0.174335	−	0.984686i	$-0.444223\pi$
$809$	8022.99	0.348669	0.174335	+	0.984686i	$0.444223\pi$
$810$	0	0
$811$	−33370.2	−1.44487	−0.722433	−	0.691441i	$-0.756976\pi$
$811$	−33370.2	−1.44487	−0.722433	+	0.691441i	$0.756976\pi$
$812$	10634.0	0.459583
$813$	0	0
$814$	−975.085	−0.0419861
$815$	0	0
$816$	0	0
$817$	52361.5	2.24223
$818$	1961.56	0.0838440
$819$	0	0
$820$	0	0
$821$	−29452.2	−1.25200	−0.625999	−	0.779824i	$-0.715308\pi$
$821$	−29452.2	−1.25200	−0.625999	+	0.779824i	$0.715308\pi$
$822$	0	0
$823$	3399.31	0.143976	0.0719882	−	0.997405i	$-0.477066\pi$
$823$	3399.31	0.143976	0.0719882	+	0.997405i	$0.477066\pi$
$824$	5201.33	0.219899
$825$	0	0
$826$	−1141.46	−0.0480829
$827$	−18959.8	−0.797217	−0.398608	−	0.917121i	$-0.630507\pi$
$827$	−18959.8	−0.797217	−0.398608	+	0.917121i	$0.630507\pi$
$828$	0	0
$829$	−3162.26	−0.132485	−0.0662424	−	0.997804i	$-0.521101\pi$
$829$	−3162.26	−0.132485	−0.0662424	+	0.997804i	$0.521101\pi$
$830$	0	0
$831$	0	0
$832$	22667.0	0.944516
$833$	−3565.12	−0.148288
$834$	0	0
$835$	0	0
$836$	−50213.4	−2.07735
$837$	0	0
$838$	2286.16	0.0942413
$839$	10141.2	0.417296	0.208648	−	0.977991i	$-0.433094\pi$
$839$	10141.2	0.417296	0.208648	+	0.977991i	$0.433094\pi$
$840$	0	0
$841$	12151.3	0.498228
$842$	1835.98	0.0751450
$843$	0	0
$844$	−44109.7	−1.79896
$845$	0	0
$846$	0	0
$847$	−8894.22	−0.360814
$848$	−19482.2	−0.788939
$849$	0	0
$850$	0	0
$851$	12988.2	0.523185
$852$	0	0
$853$	−32027.8	−1.28559	−0.642796	−	0.766038i	$-0.722226\pi$
$853$	−32027.8	−1.28559	−0.642796	+	0.766038i	$0.722226\pi$
$854$	647.327	0.0259380
$855$	0	0
$856$	−7983.43	−0.318771
$857$	−49716.8	−1.98167	−0.990837	−	0.135067i	$-0.956875\pi$
$857$	−49716.8	−1.98167	−0.990837	+	0.135067i	$0.956875\pi$
$858$	0	0
$859$	40559.0	1.61101	0.805504	−	0.592590i	$-0.201895\pi$
$859$	40559.0	1.61101	0.805504	+	0.592590i	$0.201895\pi$
$860$	0	0
$861$	0	0
$862$	2273.43	0.0898297
$863$	41262.0	1.62755	0.813774	−	0.581182i	$-0.197410\pi$
$863$	41262.0	1.62755	0.813774	+	0.581182i	$0.197410\pi$
$864$	0	0
$865$	0	0
$866$	−2349.83	−0.0922059
$867$	0	0
$868$	−6469.70	−0.252991
$869$	70205.9	2.74059
$870$	0	0
$871$	−5267.99	−0.204936
$872$	913.834	0.0354889
$873$	0	0
$874$	−4444.23	−0.172000
$875$	0	0
$876$	0	0
$877$	25132.2	0.967677	0.483838	−	0.875157i	$-0.339242\pi$
$877$	25132.2	0.967677	0.483838	+	0.875157i	$0.339242\pi$
$878$	−889.554	−0.0341925
$879$	0	0
$880$	0	0
$881$	−42008.7	−1.60648	−0.803240	−	0.595655i	$-0.796893\pi$
$881$	−42008.7	−1.60648	−0.803240	+	0.595655i	$0.796893\pi$
$882$	0	0
$883$	2280.54	0.0869153	0.0434576	−	0.999055i	$-0.486163\pi$
$883$	2280.54	0.0869153	0.0434576	+	0.999055i	$0.486163\pi$
$884$	−26648.2	−1.01389
$885$	0	0
$886$	90.6832	0.00343856
$887$	22386.4	0.847419	0.423709	−	0.905798i	$-0.360728\pi$
$887$	22386.4	0.847419	0.423709	+	0.905798i	$0.360728\pi$
$888$	0	0
$889$	−3209.55	−0.121085
$890$	0	0
$891$	0	0
$892$	−6129.70	−0.230087
$893$	33264.8	1.24655
$894$	0	0
$895$	0	0
$896$	−3240.20	−0.120812
$897$	0	0
$898$	3317.65	0.123287
$899$	−22230.9	−0.824743
$900$	0	0
$901$	22594.5	0.835439
$902$	5464.14	0.201703
$903$	0	0
$904$	2952.60	0.108630
$905$	0	0
$906$	0	0
$907$	−19570.3	−0.716451	−0.358226	−	0.933635i	$-0.616618\pi$
$907$	−19570.3	−0.716451	−0.358226	+	0.933635i	$0.616618\pi$
$908$	43787.7	1.60038
$909$	0	0
$910$	0	0
$911$	−15605.6	−0.567550	−0.283775	−	0.958891i	$-0.591587\pi$
$911$	−15605.6	−0.567550	−0.283775	+	0.958891i	$0.591587\pi$
$912$	0	0
$913$	58775.6	2.13055
$914$	2374.88	0.0859454
$915$	0	0
$916$	10954.7	0.395145
$917$	7336.82	0.264213
$918$	0	0
$919$	−905.249	−0.0324934	−0.0162467	−	0.999868i	$-0.505172\pi$
$919$	−905.249	−0.0324934	−0.0162467	+	0.999868i	$0.505172\pi$
$920$	0	0
$921$	0	0
$922$	−3309.50	−0.118213
$923$	−9883.79	−0.352469
$924$	0	0
$925$	0	0
$926$	290.711	0.0103168
$927$	0	0
$928$	−8359.80	−0.295715
$929$	−44804.2	−1.58232	−0.791161	−	0.611608i	$-0.790523\pi$
$929$	−44804.2	−1.58232	−0.791161	+	0.611608i	$0.790523\pi$
$930$	0	0
$931$	−6069.90	−0.213677
$932$	20191.2	0.709639
$933$	0	0
$934$	−3028.45	−0.106096
$935$	0	0
$936$	0	0
$937$	−22958.8	−0.800461	−0.400230	−	0.916415i	$-0.631070\pi$
$937$	−22958.8	−0.800461	−0.400230	+	0.916415i	$0.631070\pi$
$938$	183.869	0.00640036
$939$	0	0
$940$	0	0
$941$	27191.5	0.941994	0.470997	−	0.882135i	$-0.343894\pi$
$941$	27191.5	0.941994	0.470997	+	0.882135i	$0.343894\pi$
$942$	0	0
$943$	−72782.8	−2.51340
$944$	−44517.8	−1.53488
$945$	0	0
$946$	−4954.38	−0.170276
$947$	45681.4	1.56752	0.783762	−	0.621061i	$-0.213298\pi$
$947$	45681.4	1.56752	0.783762	+	0.621061i	$0.213298\pi$
$948$	0	0
$949$	18570.8	0.635232
$950$	0	0
$951$	0	0
$952$	1866.39	0.0635399
$953$	−21670.3	−0.736591	−0.368295	−	0.929709i	$-0.620058\pi$
$953$	−21670.3	−0.736591	−0.368295	+	0.929709i	$0.620058\pi$
$954$	0	0
$955$	0	0
$956$	40388.7	1.36639
$957$	0	0
$958$	−203.025	−0.00684703
$959$	−10915.5	−0.367550
$960$	0	0
$961$	−16265.8	−0.545997
$962$	−881.045	−0.0295281
$963$	0	0
$964$	−5414.20	−0.180892
$965$	0	0
$966$	0	0
$967$	46935.4	1.56085	0.780424	−	0.625250i	$-0.215003\pi$
$967$	46935.4	1.56085	0.780424	+	0.625250i	$0.215003\pi$
$968$	4656.24	0.154605
$969$	0	0
$970$	0	0
$971$	−6475.64	−0.214020	−0.107010	−	0.994258i	$-0.534128\pi$
$971$	−6475.64	−0.214020	−0.107010	+	0.994258i	$0.534128\pi$
$972$	0	0
$973$	−1616.78	−0.0532700
$974$	1139.64	0.0374911
$975$	0	0
$976$	25246.3	0.827985
$977$	−1025.22	−0.0335717	−0.0167859	−	0.999859i	$-0.505343\pi$
$977$	−1025.22	−0.0335717	−0.0167859	+	0.999859i	$0.505343\pi$
$978$	0	0
$979$	−18708.7	−0.610759
$980$	0	0
$981$	0	0
$982$	4156.73	0.135078
$983$	7379.49	0.239440	0.119720	−	0.992808i	$-0.461800\pi$
$983$	7379.49	0.239440	0.119720	+	0.992808i	$0.461800\pi$
$984$	0	0
$985$	0	0
$986$	3195.99	0.103226
$987$	0	0
$988$	−45370.7	−1.46096
$989$	65992.8	2.12179
$990$	0	0
$991$	−27747.3	−0.889428	−0.444714	−	0.895673i	$-0.646695\pi$
$991$	−27747.3	−0.889428	−0.444714	+	0.895673i	$0.646695\pi$
$992$	5086.07	0.162785
$993$	0	0
$994$	344.975	0.0110080
$995$	0	0
$996$	0	0
$997$	−48509.6	−1.54094	−0.770469	−	0.637478i	$-0.779978\pi$
$997$	−48509.6	−1.54094	−0.770469	+	0.637478i	$0.779978\pi$
$998$	−2166.54	−0.0687179
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bb.1.2		3
3.2	odd	2		1575.4.a.be.1.2		3
5.4	even	2		315.4.a.o.1.2	yes	3
15.14	odd	2		315.4.a.n.1.2	✓	3
35.34	odd	2		2205.4.a.bl.1.2		3
105.104	even	2		2205.4.a.bk.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.n.1.2	✓	3	15.14	odd	2
315.4.a.o.1.2	yes	3	5.4	even	2
1575.4.a.bb.1.2		3	1.1	even	1	trivial
1575.4.a.be.1.2		3	3.2	odd	2
2205.4.a.bk.1.2		3	105.104	even	2
2205.4.a.bl.1.2		3	35.34	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-0.229795 q^{2} -7.94719 q^{4} -7.00000 q^{7} +3.66459 q^{8} -51.0059 q^{11} -46.0867 q^{13} +1.60857 q^{14} +62.7354 q^{16} -72.7575 q^{17} -123.876 q^{19} +11.7209 q^{22} -156.124 q^{23} +10.5905 q^{26} +55.6304 q^{28} +191.155 q^{29} -116.298 q^{31} -43.7331 q^{32} +16.7194 q^{34} -83.1918 q^{37} +28.4660 q^{38} +466.186 q^{41} -422.695 q^{43} +405.354 q^{44} +35.8766 q^{46} -268.534 q^{47} +49.0000 q^{49} +366.260 q^{52} -310.545 q^{53} -25.6521 q^{56} -43.9266 q^{58} -709.612 q^{59} +402.425 q^{61} +26.7247 q^{62} -491.834 q^{64} +114.306 q^{67} +578.218 q^{68} +214.461 q^{71} -402.954 q^{73} +19.1171 q^{74} +984.463 q^{76} +357.041 q^{77} -1376.43 q^{79} -107.127 q^{82} -1152.33 q^{83} +97.1334 q^{86} -186.916 q^{88} +366.795 q^{89} +322.607 q^{91} +1240.75 q^{92} +61.7080 q^{94} +1068.22 q^{97} -11.2600 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 14 q^{4} - 21 q^{7} - 66 q^{8} - 20 q^{11} + 14 q^{14} + 114 q^{16} - 234 q^{17} - 82 q^{19} + 236 q^{22} - 30 q^{23} + 76 q^{26} - 98 q^{28} - 32 q^{29} - 362 q^{31} - 430 q^{32} + 596 q^{34}+ \cdots - 98 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 14 * q^4 - 21 * q^7 - 66 * q^8 - 20 * q^11 + 14 * q^14 + 114 * q^16 - 234 * q^17 - 82 * q^19 + 236 * q^22 - 30 * q^23 + 76 * q^26 - 98 * q^28 - 32 * q^29 - 362 * q^31 - 430 * q^32 + 596 * q^34 - 302 * q^37 - 160 * q^38 + 626 * q^41 + 68 * q^43 + 300 * q^44 - 412 * q^46 - 936 * q^47 + 147 * q^49 + 684 * q^52 - 298 * q^53 + 462 * q^56 + 1184 * q^58 + 428 * q^59 + 1050 * q^61 + 608 * q^62 - 154 * q^64 + 52 * q^67 - 1964 * q^68 + 416 * q^71 + 312 * q^73 + 1668 * q^74 + 1712 * q^76 + 140 * q^77 - 488 * q^79 + 1476 * q^82 - 948 * q^83 - 1928 * q^86 - 124 * q^88 - 142 * q^89 + 3220 * q^92 + 328 * q^94 + 3052 * q^97 - 98 * q^98

Introduction

L-functions

Modular forms

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Database

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