LMFDB - Embedded newform 2475.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +O(q^{10})$	$q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +11.0000 q^{11} +40.8769 q^{13} +16.0000 q^{14} +11.4233 q^{16} -98.7083 q^{17} -39.6458 q^{19} +17.1771 q^{22} +61.6932 q^{23} +63.8314 q^{26} -56.9848 q^{28} +149.093 q^{29} +54.7386 q^{31} +187.255 q^{32} -154.138 q^{34} -44.8939 q^{37} -61.9091 q^{38} -336.479 q^{41} +2.36745 q^{43} -61.1771 q^{44} +96.3371 q^{46} -333.295 q^{47} -238.015 q^{49} -227.339 q^{52} +640.064 q^{53} -216.985 q^{56} +232.816 q^{58} +370.773 q^{59} -714.405 q^{61} +85.4773 q^{62} +201.022 q^{64} +404.985 q^{67} +548.972 q^{68} -939.292 q^{71} +362.570 q^{73} -70.1042 q^{74} +220.492 q^{76} +112.708 q^{77} +951.835 q^{79} -525.430 q^{82} +735.221 q^{83} +3.69690 q^{86} -232.948 q^{88} -385.879 q^{89} +418.833 q^{91} -343.110 q^{92} -520.458 q^{94} +966.345 q^{97} -371.673 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8}+O(q^{10}) $ 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8	$ 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8} + 22 q^{11} + 90 q^{13} + 32 q^{14} - 39 q^{16} - 16 q^{17} - 170 q^{19} - 11 q^{22} - 124 q^{23} - 62 q^{26} - 48 q^{28} + 158 q^{29} + 60 q^{31} + 123 q^{32} - 366 q^{34} + 372 q^{37} + 272 q^{38} - 38 q^{41} + 516 q^{43} - 77 q^{44} + 572 q^{46} + 224 q^{47} - 542 q^{49} - 298 q^{52} + 472 q^{53} - 368 q^{56} + 210 q^{58} - 248 q^{59} + 72 q^{61} + 72 q^{62} + 769 q^{64} + 744 q^{67} + 430 q^{68} - 2060 q^{71} + 486 q^{73} - 1138 q^{74} + 408 q^{76} + 44 q^{77} + 642 q^{79} - 1290 q^{82} - 286 q^{83} - 1312 q^{86} + 33 q^{88} - 244 q^{89} + 112 q^{91} - 76 q^{92} - 1948 q^{94} + 168 q^{97} + 407 q^{98}+O(q^{100}) $ 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8 + 22 * q^11 + 90 * q^13 + 32 * q^14 - 39 * q^16 - 16 * q^17 - 170 * q^19 - 11 * q^22 - 124 * q^23 - 62 * q^26 - 48 * q^28 + 158 * q^29 + 60 * q^31 + 123 * q^32 - 366 * q^34 + 372 * q^37 + 272 * q^38 - 38 * q^41 + 516 * q^43 - 77 * q^44 + 572 * q^46 + 224 * q^47 - 542 * q^49 - 298 * q^52 + 472 * q^53 - 368 * q^56 + 210 * q^58 - 248 * q^59 + 72 * q^61 + 72 * q^62 + 769 * q^64 + 744 * q^67 + 430 * q^68 - 2060 * q^71 + 486 * q^73 - 1138 * q^74 + 408 * q^76 + 44 * q^77 + 642 * q^79 - 1290 * q^82 - 286 * q^83 - 1312 * q^86 + 33 * q^88 - 244 * q^89 + 112 * q^91 - 76 * q^92 - 1948 * q^94 + 168 * q^97 + 407 * 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 $

$2$	1.56155	0.552092	0.276046	−	0.961144i	$-0.410976\pi$
$2$	1.56155	0.552092	0.276046	+	0.961144i	$0.410976\pi$
$3$	0	0
$3$	0	0
$4$	−5.56155	−0.695194
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.2462	0.553243	0.276622	−	0.960979i	$-0.410785\pi$
$7$	10.2462	0.553243	0.276622	+	0.960979i	$0.410785\pi$
$8$	−21.1771	−0.935904
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	40.8769	0.872093	0.436047	−	0.899924i	$-0.356378\pi$
$13$	40.8769	0.872093	0.436047	+	0.899924i	$0.356378\pi$
$14$	16.0000	0.305441
$15$	0	0
$16$	11.4233	0.178489
$17$	−98.7083	−1.40825	−0.704126	−	0.710075i	$-0.748661\pi$
$17$	−98.7083	−1.40825	−0.704126	+	0.710075i	$0.748661\pi$
$18$	0	0
$19$	−39.6458	−0.478704	−0.239352	−	0.970933i	$-0.576935\pi$
$19$	−39.6458	−0.478704	−0.239352	+	0.970933i	$0.576935\pi$
$20$	0	0
$21$	0	0
$22$	17.1771	0.166462
$23$	61.6932	0.559301	0.279650	−	0.960102i	$-0.409781\pi$
$23$	61.6932	0.559301	0.279650	+	0.960102i	$0.409781\pi$
$24$	0	0
$25$	0	0
$26$	63.8314	0.481476
$27$	0	0
$28$	−56.9848	−0.384612
$29$	149.093	0.954684	0.477342	−	0.878718i	$-0.341600\pi$
$29$	149.093	0.954684	0.477342	+	0.878718i	$0.341600\pi$
$30$	0	0
$31$	54.7386	0.317140	0.158570	−	0.987348i	$-0.449312\pi$
$31$	54.7386	0.317140	0.158570	+	0.987348i	$0.449312\pi$
$32$	187.255	1.03445
$33$	0	0
$34$	−154.138	−0.777485
$35$	0	0
$36$	0	0
$37$	−44.8939	−0.199473	−0.0997367	−	0.995014i	$-0.531800\pi$
$37$	−44.8939	−0.199473	−0.0997367	+	0.995014i	$0.531800\pi$
$38$	−61.9091	−0.264289
$39$	0	0
$40$	0	0
$41$	−336.479	−1.28169	−0.640844	−	0.767671i	$-0.721415\pi$
$41$	−336.479	−1.28169	−0.640844	+	0.767671i	$0.721415\pi$
$42$	0	0
$43$	2.36745	0.00839611	0.00419806	−	0.999991i	$-0.498664\pi$
$43$	2.36745	0.00839611	0.00419806	+	0.999991i	$0.498664\pi$
$44$	−61.1771	−0.209609
$45$	0	0
$46$	96.3371	0.308786
$47$	−333.295	−1.03439	−0.517193	−	0.855869i	$-0.673023\pi$
$47$	−333.295	−1.03439	−0.517193	+	0.855869i	$0.673023\pi$
$48$	0	0
$49$	−238.015	−0.693922
$50$	0	0
$51$	0	0
$52$	−227.339	−0.606274
$53$	640.064	1.65886	0.829430	−	0.558610i	$-0.188665\pi$
$53$	640.064	1.65886	0.829430	+	0.558610i	$0.188665\pi$
$54$	0	0
$55$	0	0
$56$	−216.985	−0.517782
$57$	0	0
$58$	232.816	0.527074
$59$	370.773	0.818144	0.409072	−	0.912502i	$-0.365853\pi$
$59$	370.773	0.818144	0.409072	+	0.912502i	$0.365853\pi$
$60$	0	0
$61$	−714.405	−1.49951	−0.749756	−	0.661715i	$-0.769829\pi$
$61$	−714.405	−1.49951	−0.749756	+	0.661715i	$0.769829\pi$
$62$	85.4773	0.175091
$63$	0	0
$64$	201.022	0.392621
$65$	0	0
$66$	0	0
$67$	404.985	0.738459	0.369230	−	0.929338i	$-0.379622\pi$
$67$	404.985	0.738459	0.369230	+	0.929338i	$0.379622\pi$
$68$	548.972	0.979009
$69$	0	0
$70$	0	0
$71$	−939.292	−1.57005	−0.785024	−	0.619465i	$-0.787349\pi$
$71$	−939.292	−1.57005	−0.785024	+	0.619465i	$0.787349\pi$
$72$	0	0
$73$	362.570	0.581310	0.290655	−	0.956828i	$-0.406127\pi$
$73$	362.570	0.581310	0.290655	+	0.956828i	$0.406127\pi$
$74$	−70.1042	−0.110128
$75$	0	0
$76$	220.492	0.332792
$77$	112.708	0.166809
$78$	0	0
$79$	951.835	1.35557	0.677784	−	0.735261i	$-0.262941\pi$
$79$	951.835	1.35557	0.677784	+	0.735261i	$0.262941\pi$
$80$	0	0
$81$	0	0
$82$	−525.430	−0.707610
$83$	735.221	0.972302	0.486151	−	0.873875i	$-0.338401\pi$
$83$	735.221	0.972302	0.486151	+	0.873875i	$0.338401\pi$
$84$	0	0
$85$	0	0
$86$	3.69690	0.00463543
$87$	0	0
$88$	−232.948	−0.282186
$89$	−385.879	−0.459585	−0.229793	−	0.973240i	$-0.573805\pi$
$89$	−385.879	−0.459585	−0.229793	+	0.973240i	$0.573805\pi$
$90$	0	0
$91$	418.833	0.482480
$92$	−343.110	−0.388823
$93$	0	0
$94$	−520.458	−0.571076
$95$	0	0
$96$	0	0
$97$	966.345	1.01152	0.505760	−	0.862674i	$-0.331212\pi$
$97$	966.345	1.01152	0.505760	+	0.862674i	$0.331212\pi$
$98$	−371.673	−0.383109
$99$	0	0
$100$	0	0
$101$	−348.600	−0.343436	−0.171718	−	0.985146i	$-0.554932\pi$
$101$	−348.600	−0.343436	−0.171718	+	0.985146i	$0.554932\pi$
$102$	0	0
$103$	1536.38	1.46975	0.734873	−	0.678204i	$-0.237242\pi$
$103$	1536.38	1.46975	0.734873	+	0.678204i	$0.237242\pi$
$104$	−865.653	−0.816195
$105$	0	0
$106$	999.494	0.915844
$107$	−779.180	−0.703983	−0.351991	−	0.936003i	$-0.614495\pi$
$107$	−779.180	−0.703983	−0.351991	+	0.936003i	$0.614495\pi$
$108$	0	0
$109$	−1501.79	−1.31968	−0.659842	−	0.751404i	$-0.729377\pi$
$109$	−1501.79	−1.31968	−0.659842	+	0.751404i	$0.729377\pi$
$110$	0	0
$111$	0	0
$112$	117.045	0.0987478
$113$	170.000	0.141524	0.0707622	−	0.997493i	$-0.477457\pi$
$113$	170.000	0.141524	0.0707622	+	0.997493i	$0.477457\pi$
$114$	0	0
$115$	0	0
$116$	−829.187	−0.663691
$117$	0	0
$118$	578.981	0.451691
$119$	−1011.39	−0.779106
$120$	0	0
$121$	121.000	0.0909091
$122$	−1115.58	−0.827869
$123$	0	0
$124$	−304.432	−0.220474
$125$	0	0
$126$	0	0
$127$	1739.82	1.21562	0.607811	−	0.794082i	$-0.292048\pi$
$127$	1739.82	1.21562	0.607811	+	0.794082i	$0.292048\pi$
$128$	−1184.13	−0.817683
$129$	0	0
$130$	0	0
$131$	−312.837	−0.208647	−0.104323	−	0.994543i	$-0.533268\pi$
$131$	−312.837	−0.208647	−0.104323	+	0.994543i	$0.533268\pi$
$132$	0	0
$133$	−406.220	−0.264840
$134$	632.405	0.407698
$135$	0	0
$136$	2090.35	1.31799
$137$	−716.928	−0.447090	−0.223545	−	0.974694i	$-0.571763\pi$
$137$	−716.928	−0.447090	−0.223545	+	0.974694i	$0.571763\pi$
$138$	0	0
$139$	−876.483	−0.534837	−0.267418	−	0.963581i	$-0.586171\pi$
$139$	−876.483	−0.534837	−0.267418	+	0.963581i	$0.586171\pi$
$140$	0	0
$141$	0	0
$142$	−1466.75	−0.866811
$143$	449.646	0.262946
$144$	0	0
$145$	0	0
$146$	566.172	0.320937
$147$	0	0
$148$	249.680	0.138673
$149$	2376.36	1.30657	0.653285	−	0.757112i	$-0.273390\pi$
$149$	2376.36	1.30657	0.653285	+	0.757112i	$0.273390\pi$
$150$	0	0
$151$	−92.8466	−0.0500381	−0.0250190	−	0.999687i	$-0.507965\pi$
$151$	−92.8466	−0.0500381	−0.0250190	+	0.999687i	$0.507965\pi$
$152$	839.583	0.448021
$153$	0	0
$154$	176.000	0.0920941
$155$	0	0
$156$	0	0
$157$	1881.24	0.956301	0.478150	−	0.878278i	$-0.341307\pi$
$157$	1881.24	0.956301	0.478150	+	0.878278i	$0.341307\pi$
$158$	1486.34	0.748398
$159$	0	0
$160$	0	0
$161$	632.121	0.309429
$162$	0	0
$163$	2465.49	1.18474	0.592369	−	0.805667i	$-0.298193\pi$
$163$	2465.49	1.18474	0.592369	+	0.805667i	$0.298193\pi$
$164$	1871.35	0.891022
$165$	0	0
$166$	1148.09	0.536800
$167$	−1254.30	−0.581200	−0.290600	−	0.956845i	$-0.593855\pi$
$167$	−1254.30	−0.581200	−0.290600	+	0.956845i	$0.593855\pi$
$168$	0	0
$169$	−526.080	−0.239454
$170$	0	0
$171$	0	0
$172$	−13.1667	−0.00583693
$173$	−1206.71	−0.530314	−0.265157	−	0.964205i	$-0.585424\pi$
$173$	−1206.71	−0.530314	−0.265157	+	0.964205i	$0.585424\pi$
$174$	0	0
$175$	0	0
$176$	125.656	0.0538164
$177$	0	0
$178$	−602.570	−0.253733
$179$	1442.29	0.602244	0.301122	−	0.953586i	$-0.402639\pi$
$179$	1442.29	0.602244	0.301122	+	0.953586i	$0.402639\pi$
$180$	0	0
$181$	4261.81	1.75015	0.875076	−	0.483985i	$-0.160811\pi$
$181$	4261.81	1.75015	0.875076	+	0.483985i	$0.160811\pi$
$182$	654.030	0.266373
$183$	0	0
$184$	−1306.48	−0.523451
$185$	0	0
$186$	0	0
$187$	−1085.79	−0.424604
$188$	1853.64	0.719099
$189$	0	0
$190$	0	0
$191$	852.223	0.322852	0.161426	−	0.986885i	$-0.448391\pi$
$191$	852.223	0.322852	0.161426	+	0.986885i	$0.448391\pi$
$192$	0	0
$193$	2459.95	0.917468	0.458734	−	0.888574i	$-0.348303\pi$
$193$	2459.95	0.917468	0.458734	+	0.888574i	$0.348303\pi$
$194$	1509.00	0.558452
$195$	0	0
$196$	1323.73	0.482410
$197$	3477.06	1.25751	0.628756	−	0.777602i	$-0.283564\pi$
$197$	3477.06	1.25751	0.628756	+	0.777602i	$0.283564\pi$
$198$	0	0
$199$	3995.04	1.42312	0.711560	−	0.702626i	$-0.247989\pi$
$199$	3995.04	1.42312	0.711560	+	0.702626i	$0.247989\pi$
$200$	0	0
$201$	0	0
$202$	−544.358	−0.189608
$203$	1527.64	0.528173
$204$	0	0
$205$	0	0
$206$	2399.14	0.811436
$207$	0	0
$208$	466.949	0.155659
$209$	−436.104	−0.144335
$210$	0	0
$211$	1046.13	0.341319	0.170660	−	0.985330i	$-0.445410\pi$
$211$	1046.13	0.341319	0.170660	+	0.985330i	$0.445410\pi$
$212$	−3559.75	−1.15323
$213$	0	0
$214$	−1216.73	−0.388664
$215$	0	0
$216$	0	0
$217$	560.864	0.175456
$218$	−2345.13	−0.728587
$219$	0	0
$220$	0	0
$221$	−4034.89	−1.22813
$222$	0	0
$223$	506.265	0.152027	0.0760135	−	0.997107i	$-0.475781\pi$
$223$	506.265	0.152027	0.0760135	+	0.997107i	$0.475781\pi$
$224$	1918.65	0.572300
$225$	0	0
$226$	265.464	0.0781345
$227$	−4286.29	−1.25326	−0.626632	−	0.779315i	$-0.715567\pi$
$227$	−4286.29	−1.25326	−0.626632	+	0.779315i	$0.715567\pi$
$228$	0	0
$229$	5709.37	1.64754	0.823769	−	0.566926i	$-0.191867\pi$
$229$	5709.37	1.64754	0.823769	+	0.566926i	$0.191867\pi$
$230$	0	0
$231$	0	0
$232$	−3157.35	−0.893492
$233$	2946.09	0.828348	0.414174	−	0.910198i	$-0.364071\pi$
$233$	2946.09	0.828348	0.414174	+	0.910198i	$0.364071\pi$
$234$	0	0
$235$	0	0
$236$	−2062.07	−0.568769
$237$	0	0
$238$	−1579.33	−0.430139
$239$	2078.89	0.562646	0.281323	−	0.959613i	$-0.409227\pi$
$239$	2078.89	0.562646	0.281323	+	0.959613i	$0.409227\pi$
$240$	0	0
$241$	1853.37	0.495378	0.247689	−	0.968840i	$-0.420329\pi$
$241$	1853.37	0.495378	0.247689	+	0.968840i	$0.420329\pi$
$242$	188.948	0.0501902
$243$	0	0
$244$	3973.20	1.04245
$245$	0	0
$246$	0	0
$247$	−1620.60	−0.417475
$248$	−1159.20	−0.296813
$249$	0	0
$250$	0	0
$251$	2358.39	0.593068	0.296534	−	0.955022i	$-0.404169\pi$
$251$	2358.39	0.593068	0.296534	+	0.955022i	$0.404169\pi$
$252$	0	0
$253$	678.625	0.168635
$254$	2716.82	0.671135
$255$	0	0
$256$	−3457.26	−0.844057
$257$	5519.25	1.33962	0.669809	−	0.742534i	$-0.266376\pi$
$257$	5519.25	1.33962	0.669809	+	0.742534i	$0.266376\pi$
$258$	0	0
$259$	−459.993	−0.110357
$260$	0	0
$261$	0	0
$262$	−488.512	−0.115192
$263$	2259.65	0.529795	0.264898	−	0.964277i	$-0.414662\pi$
$263$	2259.65	0.529795	0.264898	+	0.964277i	$0.414662\pi$
$264$	0	0
$265$	0	0
$266$	−634.333	−0.146216
$267$	0	0
$268$	−2252.34	−0.513373
$269$	−7039.53	−1.59557	−0.797783	−	0.602944i	$-0.793994\pi$
$269$	−7039.53	−1.59557	−0.797783	+	0.602944i	$0.793994\pi$
$270$	0	0
$271$	5155.08	1.15553	0.577765	−	0.816203i	$-0.303925\pi$
$271$	5155.08	1.15553	0.577765	+	0.816203i	$0.303925\pi$
$272$	−1127.57	−0.251357
$273$	0	0
$274$	−1119.52	−0.246835
$275$	0	0
$276$	0	0
$277$	−9074.52	−1.96836	−0.984179	−	0.177175i	$-0.943304\pi$
$277$	−9074.52	−1.96836	−0.984179	+	0.177175i	$0.943304\pi$
$278$	−1368.67	−0.295279
$279$	0	0
$280$	0	0
$281$	3407.79	0.723459	0.361729	−	0.932283i	$-0.382186\pi$
$281$	3407.79	0.723459	0.361729	+	0.932283i	$0.382186\pi$
$282$	0	0
$283$	8827.73	1.85425	0.927127	−	0.374746i	$-0.122270\pi$
$283$	8827.73	1.85425	0.927127	+	0.374746i	$0.122270\pi$
$284$	5223.92	1.09149
$285$	0	0
$286$	702.146	0.145170
$287$	−3447.64	−0.709085
$288$	0	0
$289$	4830.33	0.983174
$290$	0	0
$291$	0	0
$292$	−2016.45	−0.404123
$293$	−4528.29	−0.902886	−0.451443	−	0.892300i	$-0.649091\pi$
$293$	−4528.29	−0.902886	−0.451443	+	0.892300i	$0.649091\pi$
$294$	0	0
$295$	0	0
$296$	950.722	0.186688
$297$	0	0
$298$	3710.81	0.721347
$299$	2521.83	0.487762
$300$	0	0
$301$	24.2574	0.00464509
$302$	−144.985	−0.0276256
$303$	0	0
$304$	−452.886	−0.0854434
$305$	0	0
$306$	0	0
$307$	−568.106	−0.105614	−0.0528071	−	0.998605i	$-0.516817\pi$
$307$	−568.106	−0.105614	−0.0528071	+	0.998605i	$0.516817\pi$
$308$	−626.833	−0.115965
$309$	0	0
$310$	0	0
$311$	6853.59	1.24962	0.624809	−	0.780778i	$-0.285177\pi$
$311$	6853.59	1.24962	0.624809	+	0.780778i	$0.285177\pi$
$312$	0	0
$313$	1138.92	0.205673	0.102837	−	0.994698i	$-0.467208\pi$
$313$	1138.92	0.205673	0.102837	+	0.994698i	$0.467208\pi$
$314$	2937.65	0.527966
$315$	0	0
$316$	−5293.68	−0.942382
$317$	3207.48	0.568297	0.284148	−	0.958780i	$-0.408289\pi$
$317$	3207.48	0.568297	0.284148	+	0.958780i	$0.408289\pi$
$318$	0	0
$319$	1640.02	0.287848
$320$	0	0
$321$	0	0
$322$	987.091	0.170834
$323$	3913.37	0.674136
$324$	0	0
$325$	0	0
$326$	3850.00	0.654085
$327$	0	0
$328$	7125.65	1.19954
$329$	−3415.02	−0.572267
$330$	0	0
$331$	−9135.12	−1.51695	−0.758477	−	0.651700i	$-0.774056\pi$
$331$	−9135.12	−1.51695	−0.758477	+	0.651700i	$0.774056\pi$
$332$	−4088.97	−0.675938
$333$	0	0
$334$	−1958.65	−0.320876
$335$	0	0
$336$	0	0
$337$	3470.05	0.560907	0.280453	−	0.959868i	$-0.409515\pi$
$337$	3470.05	0.560907	0.280453	+	0.959868i	$0.409515\pi$
$338$	−821.501	−0.132200
$339$	0	0
$340$	0	0
$341$	602.125	0.0956214
$342$	0	0
$343$	−5953.20	−0.937151
$344$	−50.1357	−0.00785795
$345$	0	0
$346$	−1884.34	−0.292782
$347$	−89.3315	−0.0138201	−0.00691004	−	0.999976i	$-0.502200\pi$
$347$	−89.3315	−0.0138201	−0.00691004	+	0.999976i	$0.502200\pi$
$348$	0	0
$349$	−149.375	−0.0229107	−0.0114554	−	0.999934i	$-0.503646\pi$
$349$	−149.375	−0.0229107	−0.0114554	+	0.999934i	$0.503646\pi$
$350$	0	0
$351$	0	0
$352$	2059.80	0.311897
$353$	7867.64	1.18627	0.593133	−	0.805104i	$-0.297891\pi$
$353$	7867.64	1.18627	0.593133	+	0.805104i	$0.297891\pi$
$354$	0	0
$355$	0	0
$356$	2146.09	0.319501
$357$	0	0
$358$	2252.21	0.332494
$359$	−4974.22	−0.731279	−0.365639	−	0.930757i	$-0.619150\pi$
$359$	−4974.22	−0.731279	−0.365639	+	0.930757i	$0.619150\pi$
$360$	0	0
$361$	−5287.21	−0.770842
$362$	6655.04	0.966246
$363$	0	0
$364$	−2329.36	−0.335417
$365$	0	0
$366$	0	0
$367$	13266.7	1.88696	0.943479	−	0.331433i	$-0.107532\pi$
$367$	13266.7	1.88696	0.943479	+	0.331433i	$0.107532\pi$
$368$	704.739	0.0998290
$369$	0	0
$370$	0	0
$371$	6558.23	0.917754
$372$	0	0
$373$	4632.77	0.643099	0.321549	−	0.946893i	$-0.395796\pi$
$373$	4632.77	0.643099	0.321549	+	0.946893i	$0.395796\pi$
$374$	−1695.52	−0.234421
$375$	0	0
$376$	7058.22	0.968085
$377$	6094.45	0.832573
$378$	0	0
$379$	6503.31	0.881406	0.440703	−	0.897653i	$-0.354729\pi$
$379$	6503.31	0.881406	0.440703	+	0.897653i	$0.354729\pi$
$380$	0	0
$381$	0	0
$382$	1330.79	0.178244
$383$	12734.5	1.69897	0.849484	−	0.527614i	$-0.176913\pi$
$383$	12734.5	1.69897	0.849484	+	0.527614i	$0.176913\pi$
$384$	0	0
$385$	0	0
$386$	3841.35	0.506527
$387$	0	0
$388$	−5374.38	−0.703203
$389$	−12024.6	−1.56728	−0.783639	−	0.621216i	$-0.786639\pi$
$389$	−12024.6	−1.56728	−0.783639	+	0.621216i	$0.786639\pi$
$390$	0	0
$391$	−6089.63	−0.787636
$392$	5040.47	0.649444
$393$	0	0
$394$	5429.61	0.694263
$395$	0	0
$396$	0	0
$397$	5223.65	0.660371	0.330186	−	0.943916i	$-0.392889\pi$
$397$	5223.65	0.660371	0.330186	+	0.943916i	$0.392889\pi$
$398$	6238.46	0.785693
$399$	0	0
$400$	0	0
$401$	−9648.18	−1.20151	−0.600757	−	0.799432i	$-0.705134\pi$
$401$	−9648.18	−1.20151	−0.600757	+	0.799432i	$0.705134\pi$
$402$	0	0
$403$	2237.55	0.276576
$404$	1938.76	0.238755
$405$	0	0
$406$	2385.48	0.291600
$407$	−493.833	−0.0601435
$408$	0	0
$409$	−2010.47	−0.243060	−0.121530	−	0.992588i	$-0.538780\pi$
$409$	−2010.47	−0.243060	−0.121530	+	0.992588i	$0.538780\pi$
$410$	0	0
$411$	0	0
$412$	−8544.65	−1.02176
$413$	3799.02	0.452633
$414$	0	0
$415$	0	0
$416$	7654.39	0.902133
$417$	0	0
$418$	−681.000	−0.0796861
$419$	−4435.27	−0.517129	−0.258565	−	0.965994i	$-0.583250\pi$
$419$	−4435.27	−0.517129	−0.258565	+	0.965994i	$0.583250\pi$
$420$	0	0
$421$	15217.9	1.76170	0.880852	−	0.473392i	$-0.156971\pi$
$421$	15217.9	1.76170	0.880852	+	0.473392i	$0.156971\pi$
$422$	1633.58	0.188440
$423$	0	0
$424$	−13554.7	−1.55253
$425$	0	0
$426$	0	0
$427$	−7319.95	−0.829595
$428$	4333.45	0.489405
$429$	0	0
$430$	0	0
$431$	5622.11	0.628324	0.314162	−	0.949369i	$-0.398277\pi$
$431$	5622.11	0.628324	0.314162	+	0.949369i	$0.398277\pi$
$432$	0	0
$433$	14306.3	1.58780	0.793898	−	0.608051i	$-0.208049\pi$
$433$	14306.3	1.58780	0.793898	+	0.608051i	$0.208049\pi$
$434$	875.818	0.0968678
$435$	0	0
$436$	8352.29	0.917436
$437$	−2445.88	−0.267740
$438$	0	0
$439$	4384.20	0.476643	0.238322	−	0.971186i	$-0.423403\pi$
$439$	4384.20	0.476643	0.238322	+	0.971186i	$0.423403\pi$
$440$	0	0
$441$	0	0
$442$	−6300.69	−0.678039
$443$	−10090.0	−1.08214	−0.541071	−	0.840977i	$-0.681981\pi$
$443$	−10090.0	−1.08214	−0.541071	+	0.840977i	$0.681981\pi$
$444$	0	0
$445$	0	0
$446$	790.560	0.0839330
$447$	0	0
$448$	2059.71	0.217215
$449$	−9582.52	−1.00719	−0.503594	−	0.863941i	$-0.667989\pi$
$449$	−9582.52	−1.00719	−0.503594	+	0.863941i	$0.667989\pi$
$450$	0	0
$451$	−3701.27	−0.386444
$452$	−945.464	−0.0983869
$453$	0	0
$454$	−6693.27	−0.691918
$455$	0	0
$456$	0	0
$457$	−9999.34	−1.02352	−0.511761	−	0.859128i	$-0.671007\pi$
$457$	−9999.34	−1.02352	−0.511761	+	0.859128i	$0.671007\pi$
$458$	8915.49	0.909593
$459$	0	0
$460$	0	0
$461$	11115.8	1.12302	0.561512	−	0.827468i	$-0.310220\pi$
$461$	11115.8	1.12302	0.561512	+	0.827468i	$0.310220\pi$
$462$	0	0
$463$	1567.16	0.157305	0.0786524	−	0.996902i	$-0.474938\pi$
$463$	1567.16	0.157305	0.0786524	+	0.996902i	$0.474938\pi$
$464$	1703.13	0.170401
$465$	0	0
$466$	4600.48	0.457325
$467$	12648.8	1.25335	0.626675	−	0.779281i	$-0.284415\pi$
$467$	12648.8	1.25335	0.626675	+	0.779281i	$0.284415\pi$
$468$	0	0
$469$	4149.56	0.408548
$470$	0	0
$471$	0	0
$472$	−7851.88	−0.765704
$473$	26.0420	0.00253152
$474$	0	0
$475$	0	0
$476$	5624.88	0.541630
$477$	0	0
$478$	3246.30	0.310633
$479$	10719.2	1.02249	0.511247	−	0.859434i	$-0.329184\pi$
$479$	10719.2	1.02249	0.511247	+	0.859434i	$0.329184\pi$
$480$	0	0
$481$	−1835.12	−0.173959
$482$	2894.14	0.273494
$483$	0	0
$484$	−672.948	−0.0631995
$485$	0	0
$486$	0	0
$487$	−7161.20	−0.666335	−0.333167	−	0.942868i	$-0.608117\pi$
$487$	−7161.20	−0.666335	−0.333167	+	0.942868i	$0.608117\pi$
$488$	15129.0	1.40340
$489$	0	0
$490$	0	0
$491$	−14567.3	−1.33893	−0.669463	−	0.742845i	$-0.733476\pi$
$491$	−14567.3	−1.33893	−0.669463	+	0.742845i	$0.733476\pi$
$492$	0	0
$493$	−14716.7	−1.34444
$494$	−2530.65	−0.230485
$495$	0	0
$496$	625.295	0.0566060
$497$	−9624.18	−0.868619
$498$	0	0
$499$	−4638.99	−0.416172	−0.208086	−	0.978111i	$-0.566723\pi$
$499$	−4638.99	−0.416172	−0.208086	+	0.978111i	$0.566723\pi$
$500$	0	0
$501$	0	0
$502$	3682.75	0.327428
$503$	−12206.3	−1.08201	−0.541006	−	0.841019i	$-0.681956\pi$
$503$	−12206.3	−1.08201	−0.541006	+	0.841019i	$0.681956\pi$
$504$	0	0
$505$	0	0
$506$	1059.71	0.0931024
$507$	0	0
$508$	−9676.09	−0.845093
$509$	−10018.6	−0.872427	−0.436214	−	0.899843i	$-0.643681\pi$
$509$	−10018.6	−0.872427	−0.436214	+	0.899843i	$0.643681\pi$
$510$	0	0
$511$	3714.97	0.321606
$512$	4074.36	0.351686
$513$	0	0
$514$	8618.61	0.739592
$515$	0	0
$516$	0	0
$517$	−3666.25	−0.311879
$518$	−718.303	−0.0609274
$519$	0	0
$520$	0	0
$521$	−1054.72	−0.0886916	−0.0443458	−	0.999016i	$-0.514120\pi$
$521$	−1054.72	−0.0886916	−0.0443458	+	0.999016i	$0.514120\pi$
$522$	0	0
$523$	16234.2	1.35730	0.678652	−	0.734460i	$-0.262564\pi$
$523$	16234.2	1.35730	0.678652	+	0.734460i	$0.262564\pi$
$524$	1739.86	0.145050
$525$	0	0
$526$	3528.57	0.292496
$527$	−5403.16	−0.446613
$528$	0	0
$529$	−8360.95	−0.687183
$530$	0	0
$531$	0	0
$532$	2259.21	0.184115
$533$	−13754.2	−1.11775
$534$	0	0
$535$	0	0
$536$	−8576.40	−0.691127
$537$	0	0
$538$	−10992.6	−0.880900
$539$	−2618.17	−0.209225
$540$	0	0
$541$	675.936	0.0537167	0.0268584	−	0.999639i	$-0.491450\pi$
$541$	675.936	0.0537167	0.0268584	+	0.999639i	$0.491450\pi$
$542$	8049.93	0.637960
$543$	0	0
$544$	−18483.6	−1.45676
$545$	0	0
$546$	0	0
$547$	−13058.2	−1.02071	−0.510355	−	0.859964i	$-0.670486\pi$
$547$	−13058.2	−1.02071	−0.510355	+	0.859964i	$0.670486\pi$
$548$	3987.23	0.310814
$549$	0	0
$550$	0	0
$551$	−5910.91	−0.457011
$552$	0	0
$553$	9752.70	0.749959
$554$	−14170.3	−1.08672
$555$	0	0
$556$	4874.61	0.371815
$557$	−6710.48	−0.510471	−0.255236	−	0.966879i	$-0.582153\pi$
$557$	−6710.48	−0.510471	−0.255236	+	0.966879i	$0.582153\pi$
$558$	0	0
$559$	96.7741	0.00732219
$560$	0	0
$561$	0	0
$562$	5321.45	0.399416
$563$	−20820.5	−1.55858	−0.779288	−	0.626666i	$-0.784419\pi$
$563$	−20820.5	−1.55858	−0.779288	+	0.626666i	$0.784419\pi$
$564$	0	0
$565$	0	0
$566$	13785.0	1.02372
$567$	0	0
$568$	19891.5	1.46941
$569$	−3251.08	−0.239530	−0.119765	−	0.992802i	$-0.538214\pi$
$569$	−3251.08	−0.239530	−0.119765	+	0.992802i	$0.538214\pi$
$570$	0	0
$571$	−4637.50	−0.339883	−0.169941	−	0.985454i	$-0.554358\pi$
$571$	−4637.50	−0.339883	−0.169941	+	0.985454i	$0.554358\pi$
$572$	−2500.73	−0.182798
$573$	0	0
$574$	−5383.67	−0.391481
$575$	0	0
$576$	0	0
$577$	−14462.4	−1.04346	−0.521730	−	0.853111i	$-0.674713\pi$
$577$	−14462.4	−1.04346	−0.521730	+	0.853111i	$0.674713\pi$
$578$	7542.82	0.542803
$579$	0	0
$580$	0	0
$581$	7533.23	0.537920
$582$	0	0
$583$	7040.71	0.500165
$584$	−7678.18	−0.544050
$585$	0	0
$586$	−7071.17	−0.498476
$587$	−22759.7	−1.60033	−0.800166	−	0.599779i	$-0.795255\pi$
$587$	−22759.7	−1.60033	−0.800166	+	0.599779i	$0.795255\pi$
$588$	0	0
$589$	−2170.16	−0.151816
$590$	0	0
$591$	0	0
$592$	−512.836	−0.0356038
$593$	−14956.4	−1.03573	−0.517864	−	0.855463i	$-0.673273\pi$
$593$	−14956.4	−1.03573	−0.517864	+	0.855463i	$0.673273\pi$
$594$	0	0
$595$	0	0
$596$	−13216.2	−0.908319
$597$	0	0
$598$	3937.96	0.269290
$599$	−2150.77	−0.146708	−0.0733539	−	0.997306i	$-0.523370\pi$
$599$	−2150.77	−0.146708	−0.0733539	+	0.997306i	$0.523370\pi$
$600$	0	0
$601$	27759.8	1.88410	0.942050	−	0.335472i	$-0.108896\pi$
$601$	27759.8	1.88410	0.942050	+	0.335472i	$0.108896\pi$
$602$	37.8792	0.00256452
$603$	0	0
$604$	516.371	0.0347862
$605$	0	0
$606$	0	0
$607$	10991.5	0.734974	0.367487	−	0.930029i	$-0.380218\pi$
$607$	10991.5	0.734974	0.367487	+	0.930029i	$0.380218\pi$
$608$	−7423.87	−0.495194
$609$	0	0
$610$	0	0
$611$	−13624.1	−0.902081
$612$	0	0
$613$	−10646.1	−0.701457	−0.350728	−	0.936477i	$-0.614066\pi$
$613$	−10646.1	−0.701457	−0.350728	+	0.936477i	$0.614066\pi$
$614$	−887.128	−0.0583088
$615$	0	0
$616$	−2386.83	−0.156117
$617$	7199.92	0.469786	0.234893	−	0.972021i	$-0.424526\pi$
$617$	7199.92	0.469786	0.234893	+	0.972021i	$0.424526\pi$
$618$	0	0
$619$	12186.9	0.791332	0.395666	−	0.918395i	$-0.370514\pi$
$619$	12186.9	0.791332	0.395666	+	0.918395i	$0.370514\pi$
$620$	0	0
$621$	0	0
$622$	10702.2	0.689905
$623$	−3953.80	−0.254262
$624$	0	0
$625$	0	0
$626$	1778.49	0.113551
$627$	0	0
$628$	−10462.6	−0.664814
$629$	4431.40	0.280909
$630$	0	0
$631$	7370.64	0.465009	0.232505	−	0.972595i	$-0.425308\pi$
$631$	7370.64	0.465009	0.232505	+	0.972595i	$0.425308\pi$
$632$	−20157.1	−1.26868
$633$	0	0
$634$	5008.65	0.313752
$635$	0	0
$636$	0	0
$637$	−9729.32	−0.605164
$638$	2560.98	0.158919
$639$	0	0
$640$	0	0
$641$	25014.9	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$641$	25014.9	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$642$	0	0
$643$	21668.2	1.32894	0.664472	−	0.747313i	$-0.268657\pi$
$643$	21668.2	1.32894	0.664472	+	0.747313i	$0.268657\pi$
$644$	−3515.58	−0.215113
$645$	0	0
$646$	6110.94	0.372185
$647$	−27625.3	−1.67861	−0.839305	−	0.543661i	$-0.817038\pi$
$647$	−27625.3	−1.67861	−0.839305	+	0.543661i	$0.817038\pi$
$648$	0	0
$649$	4078.50	0.246680
$650$	0	0
$651$	0	0
$652$	−13712.0	−0.823623
$653$	−14314.0	−0.857810	−0.428905	−	0.903350i	$-0.641101\pi$
$653$	−14314.0	−0.857810	−0.428905	+	0.903350i	$0.641101\pi$
$654$	0	0
$655$	0	0
$656$	−3843.70	−0.228767
$657$	0	0
$658$	−5332.73	−0.315944
$659$	28327.8	1.67450	0.837249	−	0.546822i	$-0.184163\pi$
$659$	28327.8	1.67450	0.837249	+	0.546822i	$0.184163\pi$
$660$	0	0
$661$	−32190.9	−1.89422	−0.947112	−	0.320905i	$-0.896013\pi$
$661$	−32190.9	−1.89422	−0.947112	+	0.320905i	$0.896013\pi$
$662$	−14265.0	−0.837498
$663$	0	0
$664$	−15569.8	−0.909981
$665$	0	0
$666$	0	0
$667$	9198.01	0.533955
$668$	6975.84	0.404047
$669$	0	0
$670$	0	0
$671$	−7858.46	−0.452120
$672$	0	0
$673$	6207.38	0.355538	0.177769	−	0.984072i	$-0.443112\pi$
$673$	6207.38	0.355538	0.177769	+	0.984072i	$0.443112\pi$
$674$	5418.66	0.309672
$675$	0	0
$676$	2925.82	0.166467
$677$	28831.1	1.63674	0.818368	−	0.574695i	$-0.194879\pi$
$677$	28831.1	1.63674	0.818368	+	0.574695i	$0.194879\pi$
$678$	0	0
$679$	9901.37	0.559617
$680$	0	0
$681$	0	0
$682$	940.250	0.0527918
$683$	3193.10	0.178888	0.0894441	−	0.995992i	$-0.471491\pi$
$683$	3193.10	0.178888	0.0894441	+	0.995992i	$0.471491\pi$
$684$	0	0
$685$	0	0
$686$	−9296.24	−0.517394
$687$	0	0
$688$	27.0441	0.00149861
$689$	26163.8	1.44668
$690$	0	0
$691$	7682.49	0.422946	0.211473	−	0.977384i	$-0.432174\pi$
$691$	7682.49	0.422946	0.211473	+	0.977384i	$0.432174\pi$
$692$	6711.17	0.368671
$693$	0	0
$694$	−139.496	−0.00762996
$695$	0	0
$696$	0	0
$697$	33213.3	1.80494
$698$	−233.256	−0.0126488
$699$	0	0
$700$	0	0
$701$	−26551.6	−1.43058	−0.715292	−	0.698825i	$-0.753707\pi$
$701$	−26551.6	−1.43058	−0.715292	+	0.698825i	$0.753707\pi$
$702$	0	0
$703$	1779.86	0.0954887
$704$	2211.24	0.118380
$705$	0	0
$706$	12285.7	0.654928
$707$	−3571.83	−0.190004
$708$	0	0
$709$	−16304.6	−0.863655	−0.431828	−	0.901956i	$-0.642131\pi$
$709$	−16304.6	−0.863655	−0.431828	+	0.901956i	$0.642131\pi$
$710$	0	0
$711$	0	0
$712$	8171.79	0.430127
$713$	3377.00	0.177377
$714$	0	0
$715$	0	0
$716$	−8021.36	−0.418676
$717$	0	0
$718$	−7767.50	−0.403733
$719$	3973.62	0.206107	0.103053	−	0.994676i	$-0.467139\pi$
$719$	3973.62	0.206107	0.103053	+	0.994676i	$0.467139\pi$
$720$	0	0
$721$	15742.1	0.813128
$722$	−8256.25	−0.425576
$723$	0	0
$724$	−23702.3	−1.21670
$725$	0	0
$726$	0	0
$727$	10780.4	0.549961	0.274980	−	0.961450i	$-0.411329\pi$
$727$	10780.4	0.549961	0.274980	+	0.961450i	$0.411329\pi$
$728$	−8869.67	−0.451555
$729$	0	0
$730$	0	0
$731$	−233.687	−0.0118238
$732$	0	0
$733$	9211.46	0.464165	0.232083	−	0.972696i	$-0.425446\pi$
$733$	9211.46	0.464165	0.232083	+	0.972696i	$0.425446\pi$
$734$	20716.6	1.04177
$735$	0	0
$736$	11552.3	0.578566
$737$	4454.83	0.222654
$738$	0	0
$739$	11084.7	0.551768	0.275884	−	0.961191i	$-0.411029\pi$
$739$	11084.7	0.551768	0.275884	+	0.961191i	$0.411029\pi$
$740$	0	0
$741$	0	0
$742$	10241.0	0.506685
$743$	−27420.4	−1.35391	−0.676955	−	0.736024i	$-0.736701\pi$
$743$	−27420.4	−1.35391	−0.676955	+	0.736024i	$0.736701\pi$
$744$	0	0
$745$	0	0
$746$	7234.32	0.355050
$747$	0	0
$748$	6038.69	0.295182
$749$	−7983.64	−0.389474
$750$	0	0
$751$	−11290.8	−0.548614	−0.274307	−	0.961642i	$-0.588448\pi$
$751$	−11290.8	−0.548614	−0.274307	+	0.961642i	$0.588448\pi$
$752$	−3807.33	−0.184626
$753$	0	0
$754$	9516.81	0.459657
$755$	0	0
$756$	0	0
$757$	−3739.19	−0.179528	−0.0897642	−	0.995963i	$-0.528611\pi$
$757$	−3739.19	−0.179528	−0.0897642	+	0.995963i	$0.528611\pi$
$758$	10155.3	0.486617
$759$	0	0
$760$	0	0
$761$	15621.1	0.744107	0.372053	−	0.928211i	$-0.378654\pi$
$761$	15621.1	0.744107	0.372053	+	0.928211i	$0.378654\pi$
$762$	0	0
$763$	−15387.7	−0.730106
$764$	−4739.69	−0.224445
$765$	0	0
$766$	19885.7	0.937987
$767$	15156.0	0.713498
$768$	0	0
$769$	40241.7	1.88706	0.943531	−	0.331284i	$-0.107482\pi$
$769$	40241.7	1.88706	0.943531	+	0.331284i	$0.107482\pi$
$770$	0	0
$771$	0	0
$772$	−13681.2	−0.637818
$773$	22821.4	1.06187	0.530936	−	0.847412i	$-0.321840\pi$
$773$	22821.4	1.06187	0.530936	+	0.847412i	$0.321840\pi$
$774$	0	0
$775$	0	0
$776$	−20464.4	−0.946685
$777$	0	0
$778$	−18777.0	−0.865282
$779$	13340.0	0.613549
$780$	0	0
$781$	−10332.2	−0.473387
$782$	−9509.28	−0.434848
$783$	0	0
$784$	−2718.92	−0.123857
$785$	0	0
$786$	0	0
$787$	29454.3	1.33410	0.667048	−	0.745015i	$-0.267558\pi$
$787$	29454.3	1.33410	0.667048	+	0.745015i	$0.267558\pi$
$788$	−19337.8	−0.874216
$789$	0	0
$790$	0	0
$791$	1741.86	0.0782974
$792$	0	0
$793$	−29202.7	−1.30771
$794$	8157.00	0.364586
$795$	0	0
$796$	−22218.6	−0.989344
$797$	−27440.3	−1.21955	−0.609777	−	0.792573i	$-0.708741\pi$
$797$	−27440.3	−1.21955	−0.609777	+	0.792573i	$0.708741\pi$
$798$	0	0
$799$	32899.0	1.45668
$800$	0	0
$801$	0	0
$802$	−15066.1	−0.663346
$803$	3988.27	0.175272
$804$	0	0
$805$	0	0
$806$	3494.05	0.152695
$807$	0	0
$808$	7382.34	0.321423
$809$	5060.18	0.219909	0.109954	−	0.993937i	$-0.464930\pi$
$809$	5060.18	0.219909	0.109954	+	0.993937i	$0.464930\pi$
$810$	0	0
$811$	30480.1	1.31973	0.659865	−	0.751384i	$-0.270613\pi$
$811$	30480.1	1.31973	0.659865	+	0.751384i	$0.270613\pi$
$812$	−8496.03	−0.367183
$813$	0	0
$814$	−771.146	−0.0332048
$815$	0	0
$816$	0	0
$817$	−93.8596	−0.00401925
$818$	−3139.46	−0.134192
$819$	0	0
$820$	0	0
$821$	−37909.0	−1.61149	−0.805745	−	0.592263i	$-0.798235\pi$
$821$	−37909.0	−1.61149	−0.805745	+	0.592263i	$0.798235\pi$
$822$	0	0
$823$	−23636.0	−1.00109	−0.500546	−	0.865710i	$-0.666867\pi$
$823$	−23636.0	−1.00109	−0.500546	+	0.865710i	$0.666867\pi$
$824$	−32536.0	−1.37554
$825$	0	0
$826$	5932.36	0.249895
$827$	−42634.3	−1.79267	−0.896336	−	0.443376i	$-0.853781\pi$
$827$	−42634.3	−1.79267	−0.896336	+	0.443376i	$0.853781\pi$
$828$	0	0
$829$	−45152.5	−1.89169	−0.945845	−	0.324619i	$-0.894764\pi$
$829$	−45152.5	−1.89169	−0.945845	+	0.324619i	$0.894764\pi$
$830$	0	0
$831$	0	0
$832$	8217.15	0.342402
$833$	23494.1	0.977217
$834$	0	0
$835$	0	0
$836$	2425.42	0.100341
$837$	0	0
$838$	−6925.91	−0.285503
$839$	−30431.5	−1.25222	−0.626110	−	0.779734i	$-0.715354\pi$
$839$	−30431.5	−1.25222	−0.626110	+	0.779734i	$0.715354\pi$
$840$	0	0
$841$	−2160.34	−0.0885784
$842$	23763.6	0.972623
$843$	0	0
$844$	−5818.09	−0.237283
$845$	0	0
$846$	0	0
$847$	1239.79	0.0502949
$848$	7311.64	0.296088
$849$	0	0
$850$	0	0
$851$	−2769.65	−0.111566
$852$	0	0
$853$	−10367.2	−0.416139	−0.208070	−	0.978114i	$-0.566718\pi$
$853$	−10367.2	−0.416139	−0.208070	+	0.978114i	$0.566718\pi$
$854$	−11430.5	−0.458013
$855$	0	0
$856$	16500.8	0.658860
$857$	12947.1	0.516063	0.258032	−	0.966136i	$-0.416926\pi$
$857$	12947.1	0.516063	0.258032	+	0.966136i	$0.416926\pi$
$858$	0	0
$859$	−20383.5	−0.809636	−0.404818	−	0.914397i	$-0.632665\pi$
$859$	−20383.5	−0.809636	−0.404818	+	0.914397i	$0.632665\pi$
$860$	0	0
$861$	0	0
$862$	8779.22	0.346893
$863$	9056.42	0.357224	0.178612	−	0.983920i	$-0.442839\pi$
$863$	9056.42	0.357224	0.178612	+	0.983920i	$0.442839\pi$
$864$	0	0
$865$	0	0
$866$	22340.0	0.876610
$867$	0	0
$868$	−3119.27	−0.121976
$869$	10470.2	0.408719
$870$	0	0
$871$	16554.5	0.644005
$872$	31803.6	1.23510
$873$	0	0
$874$	−3819.37	−0.147817
$875$	0	0
$876$	0	0
$877$	2867.88	0.110424	0.0552118	−	0.998475i	$-0.482417\pi$
$877$	2867.88	0.110424	0.0552118	+	0.998475i	$0.482417\pi$
$878$	6846.16	0.263151
$879$	0	0
$880$	0	0
$881$	11862.5	0.453640	0.226820	−	0.973937i	$-0.427167\pi$
$881$	11862.5	0.453640	0.226820	+	0.973937i	$0.427167\pi$
$882$	0	0
$883$	−33463.8	−1.27537	−0.637683	−	0.770299i	$-0.720107\pi$
$883$	−33463.8	−1.27537	−0.637683	+	0.770299i	$0.720107\pi$
$884$	22440.3	0.853787
$885$	0	0
$886$	−15756.0	−0.597442
$887$	2420.75	0.0916357	0.0458178	−	0.998950i	$-0.485411\pi$
$887$	2420.75	0.0916357	0.0458178	+	0.998950i	$0.485411\pi$
$888$	0	0
$889$	17826.5	0.672534
$890$	0	0
$891$	0	0
$892$	−2815.62	−0.105688
$893$	13213.8	0.495165
$894$	0	0
$895$	0	0
$896$	−12132.9	−0.452378
$897$	0	0
$898$	−14963.6	−0.556060
$899$	8161.14	0.302769
$900$	0	0
$901$	−63179.7	−2.33609
$902$	−5779.73	−0.213352
$903$	0	0
$904$	−3600.10	−0.132453
$905$	0	0
$906$	0	0
$907$	38154.1	1.39679	0.698393	−	0.715714i	$-0.253899\pi$
$907$	38154.1	1.39679	0.698393	+	0.715714i	$0.253899\pi$
$908$	23838.4	0.871262
$909$	0	0
$910$	0	0
$911$	35758.0	1.30045	0.650227	−	0.759740i	$-0.274674\pi$
$911$	35758.0	1.30045	0.650227	+	0.759740i	$0.274674\pi$
$912$	0	0
$913$	8087.44	0.293160
$914$	−15614.5	−0.565079
$915$	0	0
$916$	−31753.0	−1.14536
$917$	−3205.39	−0.115432
$918$	0	0
$919$	17387.5	0.624115	0.312058	−	0.950063i	$-0.398982\pi$
$919$	17387.5	0.624115	0.312058	+	0.950063i	$0.398982\pi$
$920$	0	0
$921$	0	0
$922$	17357.9	0.620013
$923$	−38395.3	−1.36923
$924$	0	0
$925$	0	0
$926$	2447.20	0.0868467
$927$	0	0
$928$	27918.3	0.987569
$929$	−6955.93	−0.245658	−0.122829	−	0.992428i	$-0.539197\pi$
$929$	−6955.93	−0.245658	−0.122829	+	0.992428i	$0.539197\pi$
$930$	0	0
$931$	9436.31	0.332183
$932$	−16384.9	−0.575863
$933$	0	0
$934$	19751.7	0.691965
$935$	0	0
$936$	0	0
$937$	16074.5	0.560438	0.280219	−	0.959936i	$-0.409593\pi$
$937$	16074.5	0.560438	0.280219	+	0.959936i	$0.409593\pi$
$938$	6479.76	0.225556
$939$	0	0
$940$	0	0
$941$	687.126	0.0238041	0.0119021	−	0.999929i	$-0.496211\pi$
$941$	687.126	0.0238041	0.0119021	+	0.999929i	$0.496211\pi$
$942$	0	0
$943$	−20758.5	−0.716849
$944$	4235.44	0.146030
$945$	0	0
$946$	40.6659	0.00139763
$947$	35352.0	1.21308	0.606540	−	0.795053i	$-0.292557\pi$
$947$	35352.0	1.21308	0.606540	+	0.795053i	$0.292557\pi$
$948$	0	0
$949$	14820.7	0.506956
$950$	0	0
$951$	0	0
$952$	21418.2	0.729168
$953$	−19390.7	−0.659103	−0.329552	−	0.944138i	$-0.606898\pi$
$953$	−19390.7	−0.659103	−0.329552	+	0.944138i	$0.606898\pi$
$954$	0	0
$955$	0	0
$956$	−11561.9	−0.391148
$957$	0	0
$958$	16738.7	0.564511
$959$	−7345.80	−0.247349
$960$	0	0
$961$	−26794.7	−0.899422
$962$	−2865.64	−0.0960416
$963$	0	0
$964$	−10307.6	−0.344384
$965$	0	0
$966$	0	0
$967$	−28643.6	−0.952551	−0.476275	−	0.879296i	$-0.658013\pi$
$967$	−28643.6	−0.952551	−0.476275	+	0.879296i	$0.658013\pi$
$968$	−2562.43	−0.0850821
$969$	0	0
$970$	0	0
$971$	19574.8	0.646946	0.323473	−	0.946237i	$-0.395149\pi$
$971$	19574.8	0.646946	0.323473	+	0.946237i	$0.395149\pi$
$972$	0	0
$973$	−8980.63	−0.295895
$974$	−11182.6	−0.367878
$975$	0	0
$976$	−8160.86	−0.267646
$977$	50095.5	1.64043	0.820213	−	0.572058i	$-0.193855\pi$
$977$	50095.5	1.64043	0.820213	+	0.572058i	$0.193855\pi$
$978$	0	0
$979$	−4244.67	−0.138570
$980$	0	0
$981$	0	0
$982$	−22747.6	−0.739211
$983$	14445.4	0.468706	0.234353	−	0.972152i	$-0.424703\pi$
$983$	14445.4	0.468706	0.234353	+	0.972152i	$0.424703\pi$
$984$	0	0
$985$	0	0
$986$	−22980.9	−0.742253
$987$	0	0
$988$	9013.05	0.290226
$989$	146.056	0.00469595
$990$	0	0
$991$	29120.1	0.933430	0.466715	−	0.884408i	$-0.345437\pi$
$991$	29120.1	0.933430	0.466715	+	0.884408i	$0.345437\pi$
$992$	10250.1	0.328064
$993$	0	0
$994$	−15028.7	−0.479558
$995$	0	0
$996$	0	0
$997$	9137.45	0.290257	0.145128	−	0.989413i	$-0.453640\pi$
$997$	9137.45	0.290257	0.145128	+	0.989413i	$0.453640\pi$
$998$	−7244.03	−0.229765
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.n.1.2		2
3.2	odd	2		825.4.a.m.1.1		2
5.4	even	2		495.4.a.d.1.1		2
15.2	even	4		825.4.c.j.199.2		4
15.8	even	4		825.4.c.j.199.3		4
15.14	odd	2		165.4.a.c.1.2	✓	2
165.164	even	2		1815.4.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.c.1.2	✓	2	15.14	odd	2
495.4.a.d.1.1		2	5.4	even	2
825.4.a.m.1.1		2	3.2	odd	2
825.4.c.j.199.2		4	15.2	even	4
825.4.c.j.199.3		4	15.8	even	4
1815.4.a.n.1.1		2	165.164	even	2
2475.4.a.n.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +O(q^{10})\)	\(q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +11.0000 q^{11} +40.8769 q^{13} +16.0000 q^{14} +11.4233 q^{16} -98.7083 q^{17} -39.6458 q^{19} +17.1771 q^{22} +61.6932 q^{23} +63.8314 q^{26} -56.9848 q^{28} +149.093 q^{29} +54.7386 q^{31} +187.255 q^{32} -154.138 q^{34} -44.8939 q^{37} -61.9091 q^{38} -336.479 q^{41} +2.36745 q^{43} -61.1771 q^{44} +96.3371 q^{46} -333.295 q^{47} -238.015 q^{49} -227.339 q^{52} +640.064 q^{53} -216.985 q^{56} +232.816 q^{58} +370.773 q^{59} -714.405 q^{61} +85.4773 q^{62} +201.022 q^{64} +404.985 q^{67} +548.972 q^{68} -939.292 q^{71} +362.570 q^{73} -70.1042 q^{74} +220.492 q^{76} +112.708 q^{77} +951.835 q^{79} -525.430 q^{82} +735.221 q^{83} +3.69690 q^{86} -232.948 q^{88} -385.879 q^{89} +418.833 q^{91} -343.110 q^{92} -520.458 q^{94} +966.345 q^{97} -371.673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8	\( 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8} + 22 q^{11} + 90 q^{13} + 32 q^{14} - 39 q^{16} - 16 q^{17} - 170 q^{19} - 11 q^{22} - 124 q^{23} - 62 q^{26} - 48 q^{28} + 158 q^{29} + 60 q^{31} + 123 q^{32} - 366 q^{34} + 372 q^{37} + 272 q^{38} - 38 q^{41} + 516 q^{43} - 77 q^{44} + 572 q^{46} + 224 q^{47} - 542 q^{49} - 298 q^{52} + 472 q^{53} - 368 q^{56} + 210 q^{58} - 248 q^{59} + 72 q^{61} + 72 q^{62} + 769 q^{64} + 744 q^{67} + 430 q^{68} - 2060 q^{71} + 486 q^{73} - 1138 q^{74} + 408 q^{76} + 44 q^{77} + 642 q^{79} - 1290 q^{82} - 286 q^{83} - 1312 q^{86} + 33 q^{88} - 244 q^{89} + 112 q^{91} - 76 q^{92} - 1948 q^{94} + 168 q^{97} + 407 q^{98}+O(q^{100}) \) 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8 + 22 * q^11 + 90 * q^13 + 32 * q^14 - 39 * q^16 - 16 * q^17 - 170 * q^19 - 11 * q^22 - 124 * q^23 - 62 * q^26 - 48 * q^28 + 158 * q^29 + 60 * q^31 + 123 * q^32 - 366 * q^34 + 372 * q^37 + 272 * q^38 - 38 * q^41 + 516 * q^43 - 77 * q^44 + 572 * q^46 + 224 * q^47 - 542 * q^49 - 298 * q^52 + 472 * q^53 - 368 * q^56 + 210 * q^58 - 248 * q^59 + 72 * q^61 + 72 * q^62 + 769 * q^64 + 744 * q^67 + 430 * q^68 - 2060 * q^71 + 486 * q^73 - 1138 * q^74 + 408 * q^76 + 44 * q^77 + 642 * q^79 - 1290 * q^82 - 286 * q^83 - 1312 * q^86 + 33 * q^88 - 244 * q^89 + 112 * q^91 - 76 * q^92 - 1948 * q^94 + 168 * q^97 + 407 * q^98

Introduction

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