LMFDB - Embedded newform 1575.4.a.p.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ 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)

$f(q)$	$=$	$q-5.27492 q^{2} +19.8248 q^{4} -7.00000 q^{7} -62.3746 q^{8} +O(q^{10})$	$q-5.27492 q^{2} +19.8248 q^{4} -7.00000 q^{7} -62.3746 q^{8} -34.7492 q^{11} +37.2990 q^{13} +36.9244 q^{14} +170.423 q^{16} -10.5498 q^{17} -58.5980 q^{19} +183.299 q^{22} -125.347 q^{23} -196.749 q^{26} -138.773 q^{28} +35.4020 q^{29} +291.794 q^{31} -399.969 q^{32} +55.6495 q^{34} +259.897 q^{37} +309.100 q^{38} +338.248 q^{41} -6.80397 q^{43} -688.894 q^{44} +661.196 q^{46} +250.694 q^{47} +49.0000 q^{49} +739.444 q^{52} -536.900 q^{53} +436.622 q^{56} -186.743 q^{58} +35.8904 q^{59} +57.7940 q^{61} -1539.19 q^{62} +746.423 q^{64} -481.691 q^{67} -209.148 q^{68} -363.752 q^{71} -581.299 q^{73} -1370.94 q^{74} -1161.69 q^{76} +243.244 q^{77} -693.691 q^{79} -1784.23 q^{82} +1334.39 q^{83} +35.8904 q^{86} +2167.47 q^{88} +353.038 q^{89} -261.093 q^{91} -2484.98 q^{92} -1322.39 q^{94} -1445.88 q^{97} -258.471 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8	$ 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8 + 6 * q^11 - 16 * q^13 + 21 * q^14 + 137 * q^16 - 6 * q^17 + 64 * q^19 + 276 * q^22 + 6 * q^23 - 318 * q^26 - 119 * q^28 + 252 * q^29 + 40 * q^31 - 279 * q^32 + 66 * q^34 + 248 * q^37 + 588 * q^38 + 450 * q^41 - 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 98 * q^49 + 890 * q^52 - 1104 * q^53 + 609 * q^56 + 306 * q^58 - 804 * q^59 - 428 * q^61 - 2112 * q^62 + 1289 * q^64 - 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 - 1508 * q^76 - 42 * q^77 - 572 * q^79 - 1530 * q^82 + 1944 * q^83 - 804 * q^86 + 1164 * q^88 - 366 * q^89 + 112 * q^91 - 2856 * q^92 - 1920 * q^94 - 808 * 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 $

$2$	−5.27492	−1.86496	−0.932482	−	0.361215i	$-0.882362\pi$
$2$	−5.27492	−1.86496	−0.932482	+	0.361215i	$0.882362\pi$
$3$	0	0
$3$	0	0
$4$	19.8248	2.47809
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−62.3746	−2.75659
$9$	0	0
$10$	0	0
$11$	−34.7492	−0.952479	−0.476240	−	0.879316i	$-0.658000\pi$
$11$	−34.7492	−0.952479	−0.476240	+	0.879316i	$0.658000\pi$
$12$	0	0
$13$	37.2990	0.795760	0.397880	−	0.917437i	$-0.369746\pi$
$13$	37.2990	0.795760	0.397880	+	0.917437i	$0.369746\pi$
$14$	36.9244	0.704890
$15$	0	0
$16$	170.423	2.66286
$17$	−10.5498	−0.150512	−0.0752562	−	0.997164i	$-0.523977\pi$
$17$	−10.5498	−0.150512	−0.0752562	+	0.997164i	$0.523977\pi$
$18$	0	0
$19$	−58.5980	−0.707542	−0.353771	−	0.935332i	$-0.615101\pi$
$19$	−58.5980	−0.707542	−0.353771	+	0.935332i	$0.615101\pi$
$20$	0	0
$21$	0	0
$22$	183.299	1.77634
$23$	−125.347	−1.13638	−0.568189	−	0.822898i	$-0.692356\pi$
$23$	−125.347	−1.13638	−0.568189	+	0.822898i	$0.692356\pi$
$24$	0	0
$25$	0	0
$26$	−196.749	−1.48406
$27$	0	0
$28$	−138.773	−0.936631
$29$	35.4020	0.226689	0.113345	−	0.993556i	$-0.463844\pi$
$29$	35.4020	0.226689	0.113345	+	0.993556i	$0.463844\pi$
$30$	0	0
$31$	291.794	1.69057	0.845286	−	0.534313i	$-0.179430\pi$
$31$	291.794	1.69057	0.845286	+	0.534313i	$0.179430\pi$
$32$	−399.969	−2.20954
$33$	0	0
$34$	55.6495	0.280700
$35$	0	0
$36$	0	0
$37$	259.897	1.15478	0.577389	−	0.816469i	$-0.304072\pi$
$37$	259.897	1.15478	0.577389	+	0.816469i	$0.304072\pi$
$38$	309.100	1.31954
$39$	0	0
$40$	0	0
$41$	338.248	1.28842	0.644212	−	0.764847i	$-0.277185\pi$
$41$	338.248	1.28842	0.644212	+	0.764847i	$0.277185\pi$
$42$	0	0
$43$	−6.80397	−0.0241301	−0.0120651	−	0.999927i	$-0.503841\pi$
$43$	−6.80397	−0.0241301	−0.0120651	+	0.999927i	$0.503841\pi$
$44$	−688.894	−2.36033
$45$	0	0
$46$	661.196	2.11931
$47$	250.694	0.778033	0.389016	−	0.921231i	$-0.372815\pi$
$47$	250.694	0.778033	0.389016	+	0.921231i	$0.372815\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	739.444	1.97197
$53$	−536.900	−1.39149	−0.695745	−	0.718289i	$-0.744925\pi$
$53$	−536.900	−1.39149	−0.695745	+	0.718289i	$0.744925\pi$
$54$	0	0
$55$	0	0
$56$	436.622	1.04189
$57$	0	0
$58$	−186.743	−0.422767
$59$	35.8904	0.0791955	0.0395977	−	0.999216i	$-0.487392\pi$
$59$	35.8904	0.0791955	0.0395977	+	0.999216i	$0.487392\pi$
$60$	0	0
$61$	57.7940	0.121308	0.0606538	−	0.998159i	$-0.480681\pi$
$61$	57.7940	0.121308	0.0606538	+	0.998159i	$0.480681\pi$
$62$	−1539.19	−3.15286
$63$	0	0
$64$	746.423	1.45786
$65$	0	0
$66$	0	0
$67$	−481.691	−0.878327	−0.439164	−	0.898407i	$-0.644725\pi$
$67$	−481.691	−0.878327	−0.439164	+	0.898407i	$0.644725\pi$
$68$	−209.148	−0.372984
$69$	0	0
$70$	0	0
$71$	−363.752	−0.608021	−0.304010	−	0.952669i	$-0.598326\pi$
$71$	−363.752	−0.608021	−0.304010	+	0.952669i	$0.598326\pi$
$72$	0	0
$73$	−581.299	−0.931999	−0.465999	−	0.884785i	$-0.654305\pi$
$73$	−581.299	−0.931999	−0.465999	+	0.884785i	$0.654305\pi$
$74$	−1370.94	−2.15362
$75$	0	0
$76$	−1161.69	−1.75336
$77$	243.244	0.360003
$78$	0	0
$79$	−693.691	−0.987928	−0.493964	−	0.869482i	$-0.664453\pi$
$79$	−693.691	−0.987928	−0.493964	+	0.869482i	$0.664453\pi$
$80$	0	0
$81$	0	0
$82$	−1784.23	−2.40287
$83$	1334.39	1.76468	0.882341	−	0.470611i	$-0.155967\pi$
$83$	1334.39	1.76468	0.882341	+	0.470611i	$0.155967\pi$
$84$	0	0
$85$	0	0
$86$	35.8904	0.0450019
$87$	0	0
$88$	2167.47	2.62560
$89$	353.038	0.420472	0.210236	−	0.977651i	$-0.432577\pi$
$89$	353.038	0.420472	0.210236	+	0.977651i	$0.432577\pi$
$90$	0	0
$91$	−261.093	−0.300769
$92$	−2484.98	−2.81605
$93$	0	0
$94$	−1322.39	−1.45100
$95$	0	0
$96$	0	0
$97$	−1445.88	−1.51347	−0.756735	−	0.653722i	$-0.773207\pi$
$97$	−1445.88	−1.51347	−0.756735	+	0.653722i	$0.773207\pi$
$98$	−258.471	−0.266424
$99$	0	0
$100$	0	0
$101$	−474.852	−0.467817	−0.233909	−	0.972259i	$-0.575152\pi$
$101$	−474.852	−0.467817	−0.233909	+	0.972259i	$0.575152\pi$
$102$	0	0
$103$	1999.59	1.91287	0.956433	−	0.291951i	$-0.0943044\pi$
$103$	1999.59	1.91287	0.956433	+	0.291951i	$0.0943044\pi$
$104$	−2326.51	−2.19359
$105$	0	0
$106$	2832.10	2.59508
$107$	1166.74	1.05414	0.527068	−	0.849823i	$-0.323291\pi$
$107$	1166.74	1.05414	0.527068	+	0.849823i	$0.323291\pi$
$108$	0	0
$109$	−1337.18	−1.17503	−0.587515	−	0.809213i	$-0.699894\pi$
$109$	−1337.18	−1.17503	−0.587515	+	0.809213i	$0.699894\pi$
$110$	0	0
$111$	0	0
$112$	−1192.96	−1.00646
$113$	906.578	0.754723	0.377361	−	0.926066i	$-0.376831\pi$
$113$	906.578	0.754723	0.377361	+	0.926066i	$0.376831\pi$
$114$	0	0
$115$	0	0
$116$	701.836	0.561757
$117$	0	0
$118$	−189.319	−0.147697
$119$	73.8488	0.0568883
$120$	0	0
$121$	−123.495	−0.0927836
$122$	−304.859	−0.226235
$123$	0	0
$124$	5784.74	4.18940
$125$	0	0
$126$	0	0
$127$	1714.89	1.19820	0.599101	−	0.800674i	$-0.295525\pi$
$127$	1714.89	1.19820	0.599101	+	0.800674i	$0.295525\pi$
$128$	−737.564	−0.509313
$129$	0	0
$130$	0	0
$131$	−470.611	−0.313874	−0.156937	−	0.987609i	$-0.550162\pi$
$131$	−470.611	−0.313874	−0.156937	+	0.987609i	$0.550162\pi$
$132$	0	0
$133$	410.186	0.267426
$134$	2540.88	1.63805
$135$	0	0
$136$	658.042	0.414901
$137$	−443.910	−0.276831	−0.138415	−	0.990374i	$-0.544201\pi$
$137$	−443.910	−0.276831	−0.138415	+	0.990374i	$0.544201\pi$
$138$	0	0
$139$	1669.98	1.01904	0.509518	−	0.860460i	$-0.329824\pi$
$139$	1669.98	1.01904	0.509518	+	0.860460i	$0.329824\pi$
$140$	0	0
$141$	0	0
$142$	1918.76	1.13394
$143$	−1296.11	−0.757945
$144$	0	0
$145$	0	0
$146$	3066.30	1.73814
$147$	0	0
$148$	5152.39	2.86165
$149$	−743.871	−0.408995	−0.204497	−	0.978867i	$-0.565556\pi$
$149$	−743.871	−0.408995	−0.204497	+	0.978867i	$0.565556\pi$
$150$	0	0
$151$	606.764	0.327005	0.163503	−	0.986543i	$-0.447721\pi$
$151$	606.764	0.327005	0.163503	+	0.986543i	$0.447721\pi$
$152$	3655.03	1.95041
$153$	0	0
$154$	−1283.09	−0.671393
$155$	0	0
$156$	0	0
$157$	−3114.78	−1.58336	−0.791678	−	0.610939i	$-0.790792\pi$
$157$	−3114.78	−1.58336	−0.791678	+	0.610939i	$0.790792\pi$
$158$	3659.16	1.84245
$159$	0	0
$160$	0	0
$161$	877.430	0.429511
$162$	0	0
$163$	−2413.07	−1.15955	−0.579774	−	0.814777i	$-0.696859\pi$
$163$	−2413.07	−1.15955	−0.579774	+	0.814777i	$0.696859\pi$
$164$	6705.67	3.19284
$165$	0	0
$166$	−7038.81	−3.29107
$167$	−610.475	−0.282874	−0.141437	−	0.989947i	$-0.545172\pi$
$167$	−610.475	−0.282874	−0.141437	+	0.989947i	$0.545172\pi$
$168$	0	0
$169$	−805.784	−0.366766
$170$	0	0
$171$	0	0
$172$	−134.887	−0.0597968
$173$	3793.81	1.66727	0.833636	−	0.552315i	$-0.186255\pi$
$173$	3793.81	1.66727	0.833636	+	0.552315i	$0.186255\pi$
$174$	0	0
$175$	0	0
$176$	−5922.05	−2.53631
$177$	0	0
$178$	−1862.25	−0.784165
$179$	2804.68	1.17112	0.585562	−	0.810627i	$-0.300874\pi$
$179$	2804.68	1.17112	0.585562	+	0.810627i	$0.300874\pi$
$180$	0	0
$181$	3106.04	1.27553	0.637763	−	0.770232i	$-0.279860\pi$
$181$	3106.04	1.27553	0.637763	+	0.770232i	$0.279860\pi$
$182$	1377.24	0.560924
$183$	0	0
$184$	7818.48	3.13253
$185$	0	0
$186$	0	0
$187$	366.598	0.143360
$188$	4969.95	1.92804
$189$	0	0
$190$	0	0
$191$	−261.952	−0.0992365	−0.0496182	−	0.998768i	$-0.515800\pi$
$191$	−261.952	−0.0992365	−0.0496182	+	0.998768i	$0.515800\pi$
$192$	0	0
$193$	−4051.07	−1.51089	−0.755447	−	0.655210i	$-0.772580\pi$
$193$	−4051.07	−1.51089	−0.755447	+	0.655210i	$0.772580\pi$
$194$	7626.88	2.82257
$195$	0	0
$196$	971.413	0.354013
$197$	−2874.83	−1.03971	−0.519855	−	0.854254i	$-0.674014\pi$
$197$	−2874.83	−1.03971	−0.519855	+	0.854254i	$0.674014\pi$
$198$	0	0
$199$	−3066.97	−1.09252	−0.546261	−	0.837615i	$-0.683949\pi$
$199$	−3066.97	−1.09252	−0.546261	+	0.837615i	$0.683949\pi$
$200$	0	0
$201$	0	0
$202$	2504.81	0.872463
$203$	−247.814	−0.0856804
$204$	0	0
$205$	0	0
$206$	−10547.7	−3.56743
$207$	0	0
$208$	6356.60	2.11899
$209$	2036.23	0.673919
$210$	0	0
$211$	595.422	0.194268	0.0971340	−	0.995271i	$-0.469032\pi$
$211$	595.422	0.194268	0.0971340	+	0.995271i	$0.469032\pi$
$212$	−10643.9	−3.44824
$213$	0	0
$214$	−6154.44	−1.96593
$215$	0	0
$216$	0	0
$217$	−2042.56	−0.638976
$218$	7053.49	2.19139
$219$	0	0
$220$	0	0
$221$	−393.498	−0.119772
$222$	0	0
$223$	3779.79	1.13504	0.567520	−	0.823360i	$-0.307903\pi$
$223$	3779.79	1.13504	0.567520	+	0.823360i	$0.307903\pi$
$224$	2799.79	0.835127
$225$	0	0
$226$	−4782.12	−1.40753
$227$	1827.62	0.534376	0.267188	−	0.963644i	$-0.413906\pi$
$227$	1827.62	0.534376	0.267188	+	0.963644i	$0.413906\pi$
$228$	0	0
$229$	−850.249	−0.245354	−0.122677	−	0.992447i	$-0.539148\pi$
$229$	−850.249	−0.245354	−0.122677	+	0.992447i	$0.539148\pi$
$230$	0	0
$231$	0	0
$232$	−2208.18	−0.624890
$233$	−6591.10	−1.85321	−0.926604	−	0.376039i	$-0.877286\pi$
$233$	−6591.10	−1.85321	−0.926604	+	0.376039i	$0.877286\pi$
$234$	0	0
$235$	0	0
$236$	711.518	0.196254
$237$	0	0
$238$	−389.547	−0.106095
$239$	182.556	0.0494083	0.0247042	−	0.999695i	$-0.492136\pi$
$239$	182.556	0.0494083	0.0247042	+	0.999695i	$0.492136\pi$
$240$	0	0
$241$	1523.90	0.407315	0.203657	−	0.979042i	$-0.434717\pi$
$241$	1523.90	0.407315	0.203657	+	0.979042i	$0.434717\pi$
$242$	651.426	0.173038
$243$	0	0
$244$	1145.75	0.300612
$245$	0	0
$246$	0	0
$247$	−2185.65	−0.563034
$248$	−18200.5	−4.66022
$249$	0	0
$250$	0	0
$251$	−2357.73	−0.592903	−0.296451	−	0.955048i	$-0.595803\pi$
$251$	−2357.73	−0.592903	−0.296451	+	0.955048i	$0.595803\pi$
$252$	0	0
$253$	4355.71	1.08238
$254$	−9045.89	−2.23460
$255$	0	0
$256$	−2080.79	−0.508006
$257$	2782.55	0.675372	0.337686	−	0.941259i	$-0.390356\pi$
$257$	2782.55	0.675372	0.337686	+	0.941259i	$0.390356\pi$
$258$	0	0
$259$	−1819.28	−0.436465
$260$	0	0
$261$	0	0
$262$	2482.44	0.585364
$263$	2043.78	0.479183	0.239591	−	0.970874i	$-0.422987\pi$
$263$	2043.78	0.479183	0.239591	+	0.970874i	$0.422987\pi$
$264$	0	0
$265$	0	0
$266$	−2163.70	−0.498740
$267$	0	0
$268$	−9549.41	−2.17658
$269$	−3452.84	−0.782614	−0.391307	−	0.920260i	$-0.627977\pi$
$269$	−3452.84	−0.782614	−0.391307	+	0.920260i	$0.627977\pi$
$270$	0	0
$271$	2644.29	0.592728	0.296364	−	0.955075i	$-0.404226\pi$
$271$	2644.29	0.592728	0.296364	+	0.955075i	$0.404226\pi$
$272$	−1797.93	−0.400793
$273$	0	0
$274$	2341.59	0.516280
$275$	0	0
$276$	0	0
$277$	−2679.49	−0.581208	−0.290604	−	0.956843i	$-0.593856\pi$
$277$	−2679.49	−0.581208	−0.290604	+	0.956843i	$0.593856\pi$
$278$	−8809.01	−1.90046
$279$	0	0
$280$	0	0
$281$	1019.69	0.216476	0.108238	−	0.994125i	$-0.465479\pi$
$281$	1019.69	0.216476	0.108238	+	0.994125i	$0.465479\pi$
$282$	0	0
$283$	−432.206	−0.0907844	−0.0453922	−	0.998969i	$-0.514454\pi$
$283$	−432.206	−0.0907844	−0.0453922	+	0.998969i	$0.514454\pi$
$284$	−7211.30	−1.50673
$285$	0	0
$286$	6836.87	1.41354
$287$	−2367.73	−0.486979
$288$	0	0
$289$	−4801.70	−0.977346
$290$	0	0
$291$	0	0
$292$	−11524.1	−2.30958
$293$	−2245.92	−0.447809	−0.223904	−	0.974611i	$-0.571880\pi$
$293$	−2245.92	−0.447809	−0.223904	+	0.974611i	$0.571880\pi$
$294$	0	0
$295$	0	0
$296$	−16211.0	−3.18325
$297$	0	0
$298$	3923.86	0.762761
$299$	−4675.33	−0.904284
$300$	0	0
$301$	47.6278	0.00912034
$302$	−3200.63	−0.609853
$303$	0	0
$304$	−9986.44	−1.88408
$305$	0	0
$306$	0	0
$307$	3197.08	0.594354	0.297177	−	0.954822i	$-0.403955\pi$
$307$	3197.08	0.594354	0.297177	+	0.954822i	$0.403955\pi$
$308$	4822.26	0.892122
$309$	0	0
$310$	0	0
$311$	3355.60	0.611829	0.305915	−	0.952059i	$-0.401038\pi$
$311$	3355.60	0.611829	0.305915	+	0.952059i	$0.401038\pi$
$312$	0	0
$313$	2256.39	0.407472	0.203736	−	0.979026i	$-0.434692\pi$
$313$	2256.39	0.407472	0.203736	+	0.979026i	$0.434692\pi$
$314$	16430.2	2.95290
$315$	0	0
$316$	−13752.3	−2.44818
$317$	−6139.19	−1.08773	−0.543866	−	0.839172i	$-0.683040\pi$
$317$	−6139.19	−1.08773	−0.543866	+	0.839172i	$0.683040\pi$
$318$	0	0
$319$	−1230.19	−0.215917
$320$	0	0
$321$	0	0
$322$	−4628.37	−0.801022
$323$	618.199	0.106494
$324$	0	0
$325$	0	0
$326$	12728.8	2.16252
$327$	0	0
$328$	−21098.0	−3.55166
$329$	−1754.86	−0.294069
$330$	0	0
$331$	7029.81	1.16735	0.583676	−	0.811987i	$-0.301614\pi$
$331$	7029.81	1.16735	0.583676	+	0.811987i	$0.301614\pi$
$332$	26454.0	4.37305
$333$	0	0
$334$	3220.21	0.527550
$335$	0	0
$336$	0	0
$337$	−10328.4	−1.66951	−0.834757	−	0.550619i	$-0.814392\pi$
$337$	−10328.4	−1.66951	−0.834757	+	0.550619i	$0.814392\pi$
$338$	4250.44	0.684005
$339$	0	0
$340$	0	0
$341$	−10139.6	−1.61024
$342$	0	0
$343$	−343.000	−0.0539949
$344$	424.395	0.0665170
$345$	0	0
$346$	−20012.0	−3.10940
$347$	1967.54	0.304389	0.152194	−	0.988351i	$-0.451366\pi$
$347$	1967.54	0.304389	0.152194	+	0.988351i	$0.451366\pi$
$348$	0	0
$349$	−4365.46	−0.669564	−0.334782	−	0.942296i	$-0.608663\pi$
$349$	−4365.46	−0.669564	−0.334782	+	0.942296i	$0.608663\pi$
$350$	0	0
$351$	0	0
$352$	13898.6	2.10454
$353$	−6071.59	−0.915462	−0.457731	−	0.889091i	$-0.651338\pi$
$353$	−6071.59	−0.915462	−0.457731	+	0.889091i	$0.651338\pi$
$354$	0	0
$355$	0	0
$356$	6998.90	1.04197
$357$	0	0
$358$	−14794.4	−2.18411
$359$	−9638.04	−1.41693	−0.708463	−	0.705748i	$-0.750611\pi$
$359$	−9638.04	−1.41693	−0.708463	+	0.705748i	$0.750611\pi$
$360$	0	0
$361$	−3425.27	−0.499384
$362$	−16384.1	−2.37881
$363$	0	0
$364$	−5176.10	−0.745334
$365$	0	0
$366$	0	0
$367$	−522.725	−0.0743488	−0.0371744	−	0.999309i	$-0.511836\pi$
$367$	−522.725	−0.0743488	−0.0371744	+	0.999309i	$0.511836\pi$
$368$	−21362.0	−3.02601
$369$	0	0
$370$	0	0
$371$	3758.30	0.525934
$372$	0	0
$373$	−3229.84	−0.448351	−0.224175	−	0.974549i	$-0.571969\pi$
$373$	−3229.84	−0.448351	−0.224175	+	0.974549i	$0.571969\pi$
$374$	−1933.77	−0.267361
$375$	0	0
$376$	−15637.0	−2.14472
$377$	1320.46	0.180390
$378$	0	0
$379$	6639.71	0.899892	0.449946	−	0.893056i	$-0.351443\pi$
$379$	6639.71	0.899892	0.449946	+	0.893056i	$0.351443\pi$
$380$	0	0
$381$	0	0
$382$	1381.77	0.185073
$383$	−14224.4	−1.89774	−0.948871	−	0.315664i	$-0.897773\pi$
$383$	−14224.4	−1.89774	−0.948871	+	0.315664i	$0.897773\pi$
$384$	0	0
$385$	0	0
$386$	21369.1	2.81777
$387$	0	0
$388$	−28664.2	−3.75052
$389$	−2921.82	−0.380828	−0.190414	−	0.981704i	$-0.560983\pi$
$389$	−2921.82	−0.380828	−0.190414	+	0.981704i	$0.560983\pi$
$390$	0	0
$391$	1322.39	0.171039
$392$	−3056.35	−0.393799
$393$	0	0
$394$	15164.5	1.93902
$395$	0	0
$396$	0	0
$397$	−811.940	−0.102645	−0.0513226	−	0.998682i	$-0.516344\pi$
$397$	−811.940	−0.102645	−0.0513226	+	0.998682i	$0.516344\pi$
$398$	16178.0	2.03751
$399$	0	0
$400$	0	0
$401$	−2338.63	−0.291237	−0.145618	−	0.989341i	$-0.546517\pi$
$401$	−2338.63	−0.291237	−0.145618	+	0.989341i	$0.546517\pi$
$402$	0	0
$403$	10883.6	1.34529
$404$	−9413.83	−1.15930
$405$	0	0
$406$	1307.20	0.159791
$407$	−9031.21	−1.09990
$408$	0	0
$409$	−2727.57	−0.329755	−0.164877	−	0.986314i	$-0.552723\pi$
$409$	−2727.57	−0.329755	−0.164877	+	0.986314i	$0.552723\pi$
$410$	0	0
$411$	0	0
$412$	39641.3	4.74026
$413$	−251.233	−0.0299331
$414$	0	0
$415$	0	0
$416$	−14918.5	−1.75826
$417$	0	0
$418$	−10741.0	−1.25684
$419$	−13306.3	−1.55144	−0.775721	−	0.631076i	$-0.782614\pi$
$419$	−13306.3	−1.55144	−0.775721	+	0.631076i	$0.782614\pi$
$420$	0	0
$421$	−11007.5	−1.27428	−0.637138	−	0.770750i	$-0.719882\pi$
$421$	−11007.5	−1.27428	−0.637138	+	0.770750i	$0.719882\pi$
$422$	−3140.80	−0.362303
$423$	0	0
$424$	33488.9	3.83577
$425$	0	0
$426$	0	0
$427$	−404.558	−0.0458500
$428$	23130.2	2.61225
$429$	0	0
$430$	0	0
$431$	6525.62	0.729300	0.364650	−	0.931145i	$-0.381189\pi$
$431$	6525.62	0.729300	0.364650	+	0.931145i	$0.381189\pi$
$432$	0	0
$433$	11716.3	1.30034	0.650171	−	0.759788i	$-0.274697\pi$
$433$	11716.3	1.30034	0.650171	+	0.759788i	$0.274697\pi$
$434$	10774.3	1.19167
$435$	0	0
$436$	−26509.2	−2.91183
$437$	7345.10	0.804036
$438$	0	0
$439$	−14611.4	−1.58853	−0.794264	−	0.607573i	$-0.792143\pi$
$439$	−14611.4	−1.58853	−0.794264	+	0.607573i	$0.792143\pi$
$440$	0	0
$441$	0	0
$442$	2075.67	0.223370
$443$	−15239.8	−1.63446	−0.817228	−	0.576314i	$-0.804490\pi$
$443$	−15239.8	−1.63446	−0.817228	+	0.576314i	$0.804490\pi$
$444$	0	0
$445$	0	0
$446$	−19938.1	−2.11681
$447$	0	0
$448$	−5224.96	−0.551018
$449$	−10678.8	−1.12241	−0.561206	−	0.827676i	$-0.689662\pi$
$449$	−10678.8	−1.12241	−0.561206	+	0.827676i	$0.689662\pi$
$450$	0	0
$451$	−11753.8	−1.22720
$452$	17972.7	1.87027
$453$	0	0
$454$	−9640.53	−0.996592
$455$	0	0
$456$	0	0
$457$	−4228.23	−0.432797	−0.216399	−	0.976305i	$-0.569431\pi$
$457$	−4228.23	−0.432797	−0.216399	+	0.976305i	$0.569431\pi$
$458$	4484.99	0.457577
$459$	0	0
$460$	0	0
$461$	−910.121	−0.0919492	−0.0459746	−	0.998943i	$-0.514639\pi$
$461$	−910.121	−0.0919492	−0.0459746	+	0.998943i	$0.514639\pi$
$462$	0	0
$463$	−4456.16	−0.447290	−0.223645	−	0.974671i	$-0.571796\pi$
$463$	−4456.16	−0.447290	−0.223645	+	0.974671i	$0.571796\pi$
$464$	6033.30	0.603640
$465$	0	0
$466$	34767.5	3.45617
$467$	−4429.42	−0.438907	−0.219453	−	0.975623i	$-0.570427\pi$
$467$	−4429.42	−0.438907	−0.219453	+	0.975623i	$0.570427\pi$
$468$	0	0
$469$	3371.84	0.331977
$470$	0	0
$471$	0	0
$472$	−2238.65	−0.218310
$473$	236.432	0.0229835
$474$	0	0
$475$	0	0
$476$	1464.03	0.140975
$477$	0	0
$478$	−962.970	−0.0921448
$479$	−2752.85	−0.262591	−0.131296	−	0.991343i	$-0.541914\pi$
$479$	−2752.85	−0.262591	−0.131296	+	0.991343i	$0.541914\pi$
$480$	0	0
$481$	9693.90	0.918927
$482$	−8038.43	−0.759628
$483$	0	0
$484$	−2448.26	−0.229927
$485$	0	0
$486$	0	0
$487$	670.598	0.0623977	0.0311989	−	0.999513i	$-0.490067\pi$
$487$	670.598	0.0623977	0.0311989	+	0.999513i	$0.490067\pi$
$488$	−3604.88	−0.334396
$489$	0	0
$490$	0	0
$491$	8244.70	0.757797	0.378898	−	0.925438i	$-0.376303\pi$
$491$	8244.70	0.757797	0.378898	+	0.925438i	$0.376303\pi$
$492$	0	0
$493$	−373.485	−0.0341195
$494$	11529.1	1.05004
$495$	0	0
$496$	49728.3	4.50175
$497$	2546.27	0.229810
$498$	0	0
$499$	8164.91	0.732488	0.366244	−	0.930519i	$-0.380644\pi$
$499$	8164.91	0.732488	0.366244	+	0.930519i	$0.380644\pi$
$500$	0	0
$501$	0	0
$502$	12436.8	1.10574
$503$	8175.59	0.724715	0.362357	−	0.932039i	$-0.381972\pi$
$503$	8175.59	0.724715	0.362357	+	0.932039i	$0.381972\pi$
$504$	0	0
$505$	0	0
$506$	−22976.0	−2.01859
$507$	0	0
$508$	33997.2	2.96926
$509$	878.448	0.0764961	0.0382480	−	0.999268i	$-0.487822\pi$
$509$	878.448	0.0764961	0.0382480	+	0.999268i	$0.487822\pi$
$510$	0	0
$511$	4069.09	0.352262
$512$	16876.5	1.45673
$513$	0	0
$514$	−14677.7	−1.25955
$515$	0	0
$516$	0	0
$517$	−8711.42	−0.741060
$518$	9596.55	0.813992
$519$	0	0
$520$	0	0
$521$	−11712.6	−0.984910	−0.492455	−	0.870338i	$-0.663900\pi$
$521$	−11712.6	−0.984910	−0.492455	+	0.870338i	$0.663900\pi$
$522$	0	0
$523$	7341.82	0.613834	0.306917	−	0.951736i	$-0.400703\pi$
$523$	7341.82	0.613834	0.306917	+	0.951736i	$0.400703\pi$
$524$	−9329.75	−0.777809
$525$	0	0
$526$	−10780.8	−0.893659
$527$	−3078.38	−0.254452
$528$	0	0
$529$	3544.92	0.291355
$530$	0	0
$531$	0	0
$532$	8131.84	0.662707
$533$	12616.3	1.02528
$534$	0	0
$535$	0	0
$536$	30045.3	2.42119
$537$	0	0
$538$	18213.4	1.45955
$539$	−1702.71	−0.136068
$540$	0	0
$541$	−15868.7	−1.26109	−0.630545	−	0.776153i	$-0.717169\pi$
$541$	−15868.7	−1.26109	−0.630545	+	0.776153i	$0.717169\pi$
$542$	−13948.4	−1.10542
$543$	0	0
$544$	4219.61	0.332563
$545$	0	0
$546$	0	0
$547$	−2315.26	−0.180975	−0.0904875	−	0.995898i	$-0.528843\pi$
$547$	−2315.26	−0.180975	−0.0904875	+	0.995898i	$0.528843\pi$
$548$	−8800.41	−0.686013
$549$	0	0
$550$	0	0
$551$	−2074.49	−0.160392
$552$	0	0
$553$	4855.84	0.373402
$554$	14134.1	1.08393
$555$	0	0
$556$	33106.9	2.52526
$557$	−4819.05	−0.366588	−0.183294	−	0.983058i	$-0.558676\pi$
$557$	−4819.05	−0.366588	−0.183294	+	0.983058i	$0.558676\pi$
$558$	0	0
$559$	−253.781	−0.0192018
$560$	0	0
$561$	0	0
$562$	−5378.79	−0.403720
$563$	2540.86	0.190203	0.0951017	−	0.995468i	$-0.469682\pi$
$563$	2540.86	0.190203	0.0951017	+	0.995468i	$0.469682\pi$
$564$	0	0
$565$	0	0
$566$	2279.85	0.169310
$567$	0	0
$568$	22688.9	1.67607
$569$	24220.0	1.78445	0.892227	−	0.451587i	$-0.149142\pi$
$569$	24220.0	1.78445	0.892227	+	0.451587i	$0.149142\pi$
$570$	0	0
$571$	−11772.1	−0.862778	−0.431389	−	0.902166i	$-0.641976\pi$
$571$	−11772.1	−0.862778	−0.431389	+	0.902166i	$0.641976\pi$
$572$	−25695.1	−1.87826
$573$	0	0
$574$	12489.6	0.908198
$575$	0	0
$576$	0	0
$577$	−10584.3	−0.763655	−0.381827	−	0.924234i	$-0.624705\pi$
$577$	−10584.3	−0.763655	−0.381827	+	0.924234i	$0.624705\pi$
$578$	25328.6	1.82272
$579$	0	0
$580$	0	0
$581$	−9340.74	−0.666987
$582$	0	0
$583$	18656.8	1.32536
$584$	36258.3	2.56914
$585$	0	0
$586$	11847.0	0.835148
$587$	−8712.63	−0.612621	−0.306311	−	0.951932i	$-0.599095\pi$
$587$	−8712.63	−0.612621	−0.306311	+	0.951932i	$0.599095\pi$
$588$	0	0
$589$	−17098.6	−1.19615
$590$	0	0
$591$	0	0
$592$	44292.4	3.07501
$593$	−15362.9	−1.06387	−0.531937	−	0.846784i	$-0.678536\pi$
$593$	−15362.9	−1.06387	−0.531937	+	0.846784i	$0.678536\pi$
$594$	0	0
$595$	0	0
$596$	−14747.0	−1.01353
$597$	0	0
$598$	24662.0	1.68646
$599$	−26003.8	−1.77377	−0.886883	−	0.461994i	$-0.847134\pi$
$599$	−26003.8	−1.77377	−0.886883	+	0.461994i	$0.847134\pi$
$600$	0	0
$601$	20567.7	1.39596	0.697982	−	0.716115i	$-0.254082\pi$
$601$	20567.7	1.39596	0.697982	+	0.716115i	$0.254082\pi$
$602$	−251.233	−0.0170091
$603$	0	0
$604$	12029.0	0.810349
$605$	0	0
$606$	0	0
$607$	−19642.1	−1.31342	−0.656711	−	0.754142i	$-0.728053\pi$
$607$	−19642.1	−1.31342	−0.656711	+	0.754142i	$0.728053\pi$
$608$	23437.4	1.56334
$609$	0	0
$610$	0	0
$611$	9350.65	0.619127
$612$	0	0
$613$	−8454.59	−0.557060	−0.278530	−	0.960428i	$-0.589847\pi$
$613$	−8454.59	−0.557060	−0.278530	+	0.960428i	$0.589847\pi$
$614$	−16864.3	−1.10845
$615$	0	0
$616$	−15172.3	−0.992383
$617$	−24168.4	−1.57696	−0.788479	−	0.615061i	$-0.789131\pi$
$617$	−24168.4	−1.57696	−0.788479	+	0.615061i	$0.789131\pi$
$618$	0	0
$619$	−2037.56	−0.132305	−0.0661523	−	0.997810i	$-0.521072\pi$
$619$	−2037.56	−0.132305	−0.0661523	+	0.997810i	$0.521072\pi$
$620$	0	0
$621$	0	0
$622$	−17700.5	−1.14104
$623$	−2471.27	−0.158923
$624$	0	0
$625$	0	0
$626$	−11902.3	−0.759921
$627$	0	0
$628$	−61749.8	−3.92370
$629$	−2741.87	−0.173808
$630$	0	0
$631$	12339.5	0.778489	0.389244	−	0.921135i	$-0.372736\pi$
$631$	12339.5	0.778489	0.389244	+	0.921135i	$0.372736\pi$
$632$	43268.7	2.72332
$633$	0	0
$634$	32383.7	2.02858
$635$	0	0
$636$	0	0
$637$	1827.65	0.113680
$638$	6489.15	0.402677
$639$	0	0
$640$	0	0
$641$	10222.6	0.629906	0.314953	−	0.949107i	$-0.398011\pi$
$641$	10222.6	0.629906	0.314953	+	0.949107i	$0.398011\pi$
$642$	0	0
$643$	1211.75	0.0743187	0.0371594	−	0.999309i	$-0.488169\pi$
$643$	1211.75	0.0743187	0.0371594	+	0.999309i	$0.488169\pi$
$644$	17394.8	1.06437
$645$	0	0
$646$	−3260.95	−0.198607
$647$	−2817.22	−0.171184	−0.0855922	−	0.996330i	$-0.527278\pi$
$647$	−2817.22	−0.171184	−0.0855922	+	0.996330i	$0.527278\pi$
$648$	0	0
$649$	−1247.16	−0.0754320
$650$	0	0
$651$	0	0
$652$	−47838.6	−2.87347
$653$	20986.2	1.25766	0.628831	−	0.777542i	$-0.283534\pi$
$653$	20986.2	1.25766	0.628831	+	0.777542i	$0.283534\pi$
$654$	0	0
$655$	0	0
$656$	57645.1	3.43089
$657$	0	0
$658$	9256.74	0.548428
$659$	2384.09	0.140927	0.0704635	−	0.997514i	$-0.477552\pi$
$659$	2384.09	0.140927	0.0704635	+	0.997514i	$0.477552\pi$
$660$	0	0
$661$	−7577.10	−0.445862	−0.222931	−	0.974834i	$-0.571562\pi$
$661$	−7577.10	−0.445862	−0.222931	+	0.974834i	$0.571562\pi$
$662$	−37081.7	−2.17707
$663$	0	0
$664$	−83232.2	−4.86451
$665$	0	0
$666$	0	0
$667$	−4437.54	−0.257605
$668$	−12102.5	−0.700989
$669$	0	0
$670$	0	0
$671$	−2008.30	−0.115543
$672$	0	0
$673$	−11724.6	−0.671547	−0.335774	−	0.941943i	$-0.608998\pi$
$673$	−11724.6	−0.671547	−0.335774	+	0.941943i	$0.608998\pi$
$674$	54481.7	3.11358
$675$	0	0
$676$	−15974.5	−0.908880
$677$	32304.3	1.83390	0.916952	−	0.398997i	$-0.130642\pi$
$677$	32304.3	1.83390	0.916952	+	0.398997i	$0.130642\pi$
$678$	0	0
$679$	10121.1	0.572038
$680$	0	0
$681$	0	0
$682$	53485.6	3.00303
$683$	33367.1	1.86934	0.934669	−	0.355519i	$-0.115696\pi$
$683$	33367.1	1.86934	0.934669	+	0.355519i	$0.115696\pi$
$684$	0	0
$685$	0	0
$686$	1809.30	0.100699
$687$	0	0
$688$	−1159.55	−0.0642551
$689$	−20025.8	−1.10729
$690$	0	0
$691$	−1043.67	−0.0574577	−0.0287288	−	0.999587i	$-0.509146\pi$
$691$	−1043.67	−0.0574577	−0.0287288	+	0.999587i	$0.509146\pi$
$692$	75211.3	4.13166
$693$	0	0
$694$	−10378.6	−0.567674
$695$	0	0
$696$	0	0
$697$	−3568.46	−0.193924
$698$	23027.4	1.24871
$699$	0	0
$700$	0	0
$701$	11305.7	0.609143	0.304572	−	0.952489i	$-0.401487\pi$
$701$	11305.7	0.609143	0.304572	+	0.952489i	$0.401487\pi$
$702$	0	0
$703$	−15229.4	−0.817055
$704$	−25937.6	−1.38858
$705$	0	0
$706$	32027.1	1.70730
$707$	3323.97	0.176818
$708$	0	0
$709$	−13306.8	−0.704860	−0.352430	−	0.935838i	$-0.614645\pi$
$709$	−13306.8	−0.704860	−0.352430	+	0.935838i	$0.614645\pi$
$710$	0	0
$711$	0	0
$712$	−22020.6	−1.15907
$713$	−36575.6	−1.92113
$714$	0	0
$715$	0	0
$716$	55602.0	2.90216
$717$	0	0
$718$	50839.9	2.64252
$719$	−10701.2	−0.555062	−0.277531	−	0.960717i	$-0.589516\pi$
$719$	−10701.2	−0.555062	−0.277531	+	0.960717i	$0.589516\pi$
$720$	0	0
$721$	−13997.1	−0.722996
$722$	18068.0	0.931333
$723$	0	0
$724$	61576.5	3.16088
$725$	0	0
$726$	0	0
$727$	2121.14	0.108210	0.0541051	−	0.998535i	$-0.482769\pi$
$727$	2121.14	0.108210	0.0541051	+	0.998535i	$0.482769\pi$
$728$	16285.6	0.829098
$729$	0	0
$730$	0	0
$731$	71.7808	0.00363189
$732$	0	0
$733$	21584.0	1.08762	0.543809	−	0.839209i	$-0.316981\pi$
$733$	21584.0	1.08762	0.543809	+	0.839209i	$0.316981\pi$
$734$	2757.33	0.138658
$735$	0	0
$736$	50135.0	2.51087
$737$	16738.4	0.836588
$738$	0	0
$739$	−9945.21	−0.495048	−0.247524	−	0.968882i	$-0.579617\pi$
$739$	−9945.21	−0.495048	−0.247524	+	0.968882i	$0.579617\pi$
$740$	0	0
$741$	0	0
$742$	−19824.7	−0.980848
$743$	2867.01	0.141562	0.0707808	−	0.997492i	$-0.477451\pi$
$743$	2867.01	0.141562	0.0707808	+	0.997492i	$0.477451\pi$
$744$	0	0
$745$	0	0
$746$	17037.1	0.836158
$747$	0	0
$748$	7267.71	0.355259
$749$	−8167.15	−0.398426
$750$	0	0
$751$	−10824.1	−0.525934	−0.262967	−	0.964805i	$-0.584701\pi$
$751$	−10824.1	−0.525934	−0.262967	+	0.964805i	$0.584701\pi$
$752$	42724.0	2.07179
$753$	0	0
$754$	−6965.31	−0.336421
$755$	0	0
$756$	0	0
$757$	14512.0	0.696761	0.348381	−	0.937353i	$-0.386732\pi$
$757$	14512.0	0.696761	0.348381	+	0.937353i	$0.386732\pi$
$758$	−35023.9	−1.67827
$759$	0	0
$760$	0	0
$761$	33075.8	1.57556	0.787778	−	0.615959i	$-0.211231\pi$
$761$	33075.8	1.57556	0.787778	+	0.615959i	$0.211231\pi$
$762$	0	0
$763$	9360.23	0.444120
$764$	−5193.13	−0.245917
$765$	0	0
$766$	75032.8	3.53922
$767$	1338.68	0.0630206
$768$	0	0
$769$	6728.44	0.315518	0.157759	−	0.987478i	$-0.449573\pi$
$769$	6728.44	0.315518	0.157759	+	0.987478i	$0.449573\pi$
$770$	0	0
$771$	0	0
$772$	−80311.5	−3.74414
$773$	−24233.3	−1.12757	−0.563784	−	0.825922i	$-0.690655\pi$
$773$	−24233.3	−1.12757	−0.563784	+	0.825922i	$0.690655\pi$
$774$	0	0
$775$	0	0
$776$	90186.0	4.17202
$777$	0	0
$778$	15412.4	0.710231
$779$	−19820.6	−0.911615
$780$	0	0
$781$	12640.1	0.579127
$782$	−6975.51	−0.318982
$783$	0	0
$784$	8350.72	0.380408
$785$	0	0
$786$	0	0
$787$	17200.4	0.779069	0.389535	−	0.921012i	$-0.372636\pi$
$787$	17200.4	0.779069	0.389535	+	0.921012i	$0.372636\pi$
$788$	−56992.7	−2.57650
$789$	0	0
$790$	0	0
$791$	−6346.05	−0.285258
$792$	0	0
$793$	2155.66	0.0965318
$794$	4282.92	0.191430
$795$	0	0
$796$	−60801.9	−2.70737
$797$	−4208.87	−0.187059	−0.0935295	−	0.995617i	$-0.529815\pi$
$797$	−4208.87	−0.187059	−0.0935295	+	0.995617i	$0.529815\pi$
$798$	0	0
$799$	−2644.78	−0.117104
$800$	0	0
$801$	0	0
$802$	12336.1	0.543146
$803$	20199.7	0.887709
$804$	0	0
$805$	0	0
$806$	−57410.2	−2.50892
$807$	0	0
$808$	29618.7	1.28958
$809$	23632.1	1.02702	0.513511	−	0.858083i	$-0.328344\pi$
$809$	23632.1	1.02702	0.513511	+	0.858083i	$0.328344\pi$
$810$	0	0
$811$	28425.1	1.23075	0.615377	−	0.788233i	$-0.289004\pi$
$811$	28425.1	1.23075	0.615377	+	0.788233i	$0.289004\pi$
$812$	−4912.85	−0.212324
$813$	0	0
$814$	47638.9	2.05128
$815$	0	0
$816$	0	0
$817$	398.699	0.0170731
$818$	14387.7	0.614981
$819$	0	0
$820$	0	0
$821$	−39409.6	−1.67528	−0.837640	−	0.546223i	$-0.816065\pi$
$821$	−39409.6	−1.67528	−0.837640	+	0.546223i	$0.816065\pi$
$822$	0	0
$823$	−16346.6	−0.692352	−0.346176	−	0.938170i	$-0.612520\pi$
$823$	−16346.6	−0.692352	−0.346176	+	0.938170i	$0.612520\pi$
$824$	−124723.	−5.27300
$825$	0	0
$826$	1325.23	0.0558241
$827$	−3738.87	−0.157211	−0.0786054	−	0.996906i	$-0.525047\pi$
$827$	−3738.87	−0.157211	−0.0786054	+	0.996906i	$0.525047\pi$
$828$	0	0
$829$	−45196.2	−1.89352	−0.946761	−	0.321937i	$-0.895666\pi$
$829$	−45196.2	−1.89352	−0.946761	+	0.321937i	$0.895666\pi$
$830$	0	0
$831$	0	0
$832$	27840.8	1.16010
$833$	−516.942	−0.0215018
$834$	0	0
$835$	0	0
$836$	40367.8	1.67004
$837$	0	0
$838$	70189.5	2.89338
$839$	−15899.7	−0.654254	−0.327127	−	0.944980i	$-0.606080\pi$
$839$	−15899.7	−0.654254	−0.327127	+	0.944980i	$0.606080\pi$
$840$	0	0
$841$	−23135.7	−0.948612
$842$	58063.4	2.37648
$843$	0	0
$844$	11804.1	0.481414
$845$	0	0
$846$	0	0
$847$	864.465	0.0350689
$848$	−91500.0	−3.70534
$849$	0	0
$850$	0	0
$851$	−32577.4	−1.31227
$852$	0	0
$853$	−33926.7	−1.36182	−0.680908	−	0.732369i	$-0.738415\pi$
$853$	−33926.7	−1.36182	−0.680908	+	0.732369i	$0.738415\pi$
$854$	2134.01	0.0855086
$855$	0	0
$856$	−72774.7	−2.90583
$857$	−35432.4	−1.41231	−0.706154	−	0.708058i	$-0.749572\pi$
$857$	−35432.4	−1.41231	−0.706154	+	0.708058i	$0.749572\pi$
$858$	0	0
$859$	−6780.17	−0.269309	−0.134655	−	0.990893i	$-0.542992\pi$
$859$	−6780.17	−0.269309	−0.134655	+	0.990893i	$0.542992\pi$
$860$	0	0
$861$	0	0
$862$	−34422.1	−1.36012
$863$	−30675.1	−1.20995	−0.604977	−	0.796243i	$-0.706818\pi$
$863$	−30675.1	−1.20995	−0.604977	+	0.796243i	$0.706818\pi$
$864$	0	0
$865$	0	0
$866$	−61802.4	−2.42509
$867$	0	0
$868$	−40493.2	−1.58344
$869$	24105.2	0.940981
$870$	0	0
$871$	−17966.6	−0.698938
$872$	83405.8	3.23908
$873$	0	0
$874$	−38744.8	−1.49950
$875$	0	0
$876$	0	0
$877$	−40861.3	−1.57330	−0.786652	−	0.617397i	$-0.788187\pi$
$877$	−40861.3	−1.57330	−0.786652	+	0.617397i	$0.788187\pi$
$878$	77073.9	2.96255
$879$	0	0
$880$	0	0
$881$	43839.0	1.67647	0.838236	−	0.545308i	$-0.183587\pi$
$881$	43839.0	1.67647	0.838236	+	0.545308i	$0.183587\pi$
$882$	0	0
$883$	−44625.1	−1.70074	−0.850371	−	0.526183i	$-0.823623\pi$
$883$	−44625.1	−1.70074	−0.850371	+	0.526183i	$0.823623\pi$
$884$	−7801.01	−0.296806
$885$	0	0
$886$	80388.6	3.04820
$887$	−43967.5	−1.66436	−0.832178	−	0.554509i	$-0.812906\pi$
$887$	−43967.5	−1.66436	−0.832178	+	0.554509i	$0.812906\pi$
$888$	0	0
$889$	−12004.2	−0.452878
$890$	0	0
$891$	0	0
$892$	74933.5	2.81273
$893$	−14690.2	−0.550491
$894$	0	0
$895$	0	0
$896$	5162.95	0.192502
$897$	0	0
$898$	56329.7	2.09326
$899$	10330.1	0.383234
$900$	0	0
$901$	5664.21	0.209436
$902$	62000.4	2.28868
$903$	0	0
$904$	−56547.4	−2.08046
$905$	0	0
$906$	0	0
$907$	13584.3	0.497309	0.248654	−	0.968592i	$-0.420012\pi$
$907$	13584.3	0.497309	0.248654	+	0.968592i	$0.420012\pi$
$908$	36232.1	1.32423
$909$	0	0
$910$	0	0
$911$	16421.6	0.597226	0.298613	−	0.954374i	$-0.403476\pi$
$911$	16421.6	0.597226	0.298613	+	0.954374i	$0.403476\pi$
$912$	0	0
$913$	−46369.0	−1.68082
$914$	22303.6	0.807152
$915$	0	0
$916$	−16856.0	−0.608010
$917$	3294.28	0.118633
$918$	0	0
$919$	−29487.3	−1.05843	−0.529214	−	0.848488i	$-0.677513\pi$
$919$	−29487.3	−1.05843	−0.529214	+	0.848488i	$0.677513\pi$
$920$	0	0
$921$	0	0
$922$	4800.81	0.171482
$923$	−13567.6	−0.483839
$924$	0	0
$925$	0	0
$926$	23505.9	0.834181
$927$	0	0
$928$	−14159.7	−0.500878
$929$	−3441.85	−0.121554	−0.0607769	−	0.998151i	$-0.519358\pi$
$929$	−3441.85	−0.121554	−0.0607769	+	0.998151i	$0.519358\pi$
$930$	0	0
$931$	−2871.30	−0.101077
$932$	−130667.	−4.59242
$933$	0	0
$934$	23364.8	0.818545
$935$	0	0
$936$	0	0
$937$	−5646.60	−0.196869	−0.0984346	−	0.995144i	$-0.531384\pi$
$937$	−5646.60	−0.196869	−0.0984346	+	0.995144i	$0.531384\pi$
$938$	−17786.2	−0.619125
$939$	0	0
$940$	0	0
$941$	44680.1	1.54785	0.773927	−	0.633275i	$-0.218290\pi$
$941$	44680.1	1.54785	0.773927	+	0.633275i	$0.218290\pi$
$942$	0	0
$943$	−42398.4	−1.46414
$944$	6116.54	0.210886
$945$	0	0
$946$	−1247.16	−0.0428633
$947$	−48924.6	−1.67881	−0.839406	−	0.543505i	$-0.817097\pi$
$947$	−48924.6	−1.67881	−0.839406	+	0.543505i	$0.817097\pi$
$948$	0	0
$949$	−21681.9	−0.741647
$950$	0	0
$951$	0	0
$952$	−4606.29	−0.156818
$953$	52014.3	1.76801	0.884003	−	0.467482i	$-0.154839\pi$
$953$	52014.3	1.76801	0.884003	+	0.467482i	$0.154839\pi$
$954$	0	0
$955$	0	0
$956$	3619.14	0.122439
$957$	0	0
$958$	14521.1	0.489723
$959$	3107.37	0.104632
$960$	0	0
$961$	55352.8	1.85804
$962$	−51134.5	−1.71377
$963$	0	0
$964$	30210.9	1.00936
$965$	0	0
$966$	0	0
$967$	47117.7	1.56691	0.783456	−	0.621448i	$-0.213455\pi$
$967$	47117.7	1.56691	0.783456	+	0.621448i	$0.213455\pi$
$968$	7702.95	0.255767
$969$	0	0
$970$	0	0
$971$	8195.04	0.270846	0.135423	−	0.990788i	$-0.456761\pi$
$971$	8195.04	0.270846	0.135423	+	0.990788i	$0.456761\pi$
$972$	0	0
$973$	−11689.9	−0.385159
$974$	−3537.35	−0.116370
$975$	0	0
$976$	9849.42	0.323025
$977$	4643.51	0.152056	0.0760282	−	0.997106i	$-0.475776\pi$
$977$	4643.51	0.152056	0.0760282	+	0.997106i	$0.475776\pi$
$978$	0	0
$979$	−12267.8	−0.400490
$980$	0	0
$981$	0	0
$982$	−43490.1	−1.41326
$983$	43986.5	1.42721	0.713607	−	0.700546i	$-0.247060\pi$
$983$	43986.5	1.42721	0.713607	+	0.700546i	$0.247060\pi$
$984$	0	0
$985$	0	0
$986$	1970.10	0.0636317
$987$	0	0
$988$	−43329.9	−1.39525
$989$	852.859	0.0274210
$990$	0	0
$991$	1595.21	0.0511337	0.0255668	−	0.999673i	$-0.491861\pi$
$991$	1595.21	0.0511337	0.0255668	+	0.999673i	$0.491861\pi$
$992$	−116709.	−3.73539
$993$	0	0
$994$	−13431.3	−0.428588
$995$	0	0
$996$	0	0
$997$	−21501.2	−0.682998	−0.341499	−	0.939882i	$-0.610935\pi$
$997$	−21501.2	−0.682998	−0.341499	+	0.939882i	$0.610935\pi$
$998$	−43069.2	−1.36606
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.p.1.1		2
3.2	odd	2		525.4.a.n.1.2		2
5.4	even	2		63.4.a.e.1.2		2
15.2	even	4		525.4.d.g.274.4		4
15.8	even	4		525.4.d.g.274.1		4
15.14	odd	2		21.4.a.c.1.1	✓	2
20.19	odd	2		1008.4.a.ba.1.1		2
35.4	even	6		441.4.e.q.226.1		4
35.9	even	6		441.4.e.q.361.1		4
35.19	odd	6		441.4.e.p.361.1		4
35.24	odd	6		441.4.e.p.226.1		4
35.34	odd	2		441.4.a.r.1.2		2
60.59	even	2		336.4.a.m.1.2		2
105.44	odd	6		147.4.e.l.67.2		4
105.59	even	6		147.4.e.m.79.2		4
105.74	odd	6		147.4.e.l.79.2		4
105.89	even	6		147.4.e.m.67.2		4
105.104	even	2		147.4.a.i.1.1		2
120.29	odd	2		1344.4.a.bg.1.1		2
120.59	even	2		1344.4.a.bo.1.1		2
420.419	odd	2		2352.4.a.bz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.1	✓	2	15.14	odd	2
63.4.a.e.1.2		2	5.4	even	2
147.4.a.i.1.1		2	105.104	even	2
147.4.e.l.67.2		4	105.44	odd	6
147.4.e.l.79.2		4	105.74	odd	6
147.4.e.m.67.2		4	105.89	even	6
147.4.e.m.79.2		4	105.59	even	6
336.4.a.m.1.2		2	60.59	even	2
441.4.a.r.1.2		2	35.34	odd	2
441.4.e.p.226.1		4	35.24	odd	6
441.4.e.p.361.1		4	35.19	odd	6
441.4.e.q.226.1		4	35.4	even	6
441.4.e.q.361.1		4	35.9	even	6
525.4.a.n.1.2		2	3.2	odd	2
525.4.d.g.274.1		4	15.8	even	4
525.4.d.g.274.4		4	15.2	even	4
1008.4.a.ba.1.1		2	20.19	odd	2
1344.4.a.bg.1.1		2	120.29	odd	2
1344.4.a.bo.1.1		2	120.59	even	2
1575.4.a.p.1.1		2	1.1	even	1	trivial
2352.4.a.bz.1.1		2	420.419	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.27492 q^{2} +19.8248 q^{4} -7.00000 q^{7} -62.3746 q^{8} +O(q^{10})\)	\(q-5.27492 q^{2} +19.8248 q^{4} -7.00000 q^{7} -62.3746 q^{8} -34.7492 q^{11} +37.2990 q^{13} +36.9244 q^{14} +170.423 q^{16} -10.5498 q^{17} -58.5980 q^{19} +183.299 q^{22} -125.347 q^{23} -196.749 q^{26} -138.773 q^{28} +35.4020 q^{29} +291.794 q^{31} -399.969 q^{32} +55.6495 q^{34} +259.897 q^{37} +309.100 q^{38} +338.248 q^{41} -6.80397 q^{43} -688.894 q^{44} +661.196 q^{46} +250.694 q^{47} +49.0000 q^{49} +739.444 q^{52} -536.900 q^{53} +436.622 q^{56} -186.743 q^{58} +35.8904 q^{59} +57.7940 q^{61} -1539.19 q^{62} +746.423 q^{64} -481.691 q^{67} -209.148 q^{68} -363.752 q^{71} -581.299 q^{73} -1370.94 q^{74} -1161.69 q^{76} +243.244 q^{77} -693.691 q^{79} -1784.23 q^{82} +1334.39 q^{83} +35.8904 q^{86} +2167.47 q^{88} +353.038 q^{89} -261.093 q^{91} -2484.98 q^{92} -1322.39 q^{94} -1445.88 q^{97} -258.471 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8	\( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8 + 6 * q^11 - 16 * q^13 + 21 * q^14 + 137 * q^16 - 6 * q^17 + 64 * q^19 + 276 * q^22 + 6 * q^23 - 318 * q^26 - 119 * q^28 + 252 * q^29 + 40 * q^31 - 279 * q^32 + 66 * q^34 + 248 * q^37 + 588 * q^38 + 450 * q^41 - 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 98 * q^49 + 890 * q^52 - 1104 * q^53 + 609 * q^56 + 306 * q^58 - 804 * q^59 - 428 * q^61 - 2112 * q^62 + 1289 * q^64 - 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 - 1508 * q^76 - 42 * q^77 - 572 * q^79 - 1530 * q^82 + 1944 * q^83 - 804 * q^86 + 1164 * q^88 - 366 * q^89 + 112 * q^91 - 2856 * q^92 - 1920 * q^94 - 808 * q^97 - 147 * q^98

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