LMFDB - Embedded newform 2475.4.a.bg.1.5

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 24x^{3} + 31x^{2} + 108x - 84 $ x^5 - 2x^4 - 24x^3 + 31x^2 + 108x - 84
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.95823$ 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)

$f(q)$	$=$	$q+3.95823 q^{2} +7.66762 q^{4} +27.0299 q^{7} -1.31563 q^{8} +O(q^{10})$	$q+3.95823 q^{2} +7.66762 q^{4} +27.0299 q^{7} -1.31563 q^{8} -11.0000 q^{11} +82.9282 q^{13} +106.991 q^{14} -66.5486 q^{16} +100.512 q^{17} +38.6071 q^{19} -43.5406 q^{22} -19.5133 q^{23} +328.249 q^{26} +207.255 q^{28} +100.230 q^{29} +121.874 q^{31} -252.890 q^{32} +397.851 q^{34} -347.319 q^{37} +152.816 q^{38} -108.386 q^{41} +268.039 q^{43} -84.3438 q^{44} -77.2381 q^{46} -568.074 q^{47} +387.616 q^{49} +635.862 q^{52} -603.970 q^{53} -35.5614 q^{56} +396.734 q^{58} -32.7793 q^{59} +156.323 q^{61} +482.407 q^{62} -468.608 q^{64} +745.522 q^{67} +770.689 q^{68} +264.725 q^{71} +334.805 q^{73} -1374.77 q^{74} +296.025 q^{76} -297.329 q^{77} +1287.57 q^{79} -429.018 q^{82} +127.129 q^{83} +1060.96 q^{86} +14.4720 q^{88} +615.817 q^{89} +2241.54 q^{91} -149.620 q^{92} -2248.57 q^{94} -579.630 q^{97} +1534.28 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 12 q^{4} + 24 q^{7} - 27 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 12 * q^4 + 24 * q^7 - 27 * q^8	$ 5 q - 2 q^{2} + 12 q^{4} + 24 q^{7} - 27 q^{8} - 55 q^{11} + 111 q^{13} - 47 q^{14} - 56 q^{16} - 40 q^{17} + 205 q^{19} + 22 q^{22} - 287 q^{23} + 354 q^{26} + 460 q^{28} - 251 q^{29} - 289 q^{31} - 248 q^{32} + 522 q^{34} + 224 q^{37} - 540 q^{38} + 462 q^{41} + 593 q^{43} - 132 q^{44} - 972 q^{46} - 766 q^{47} + 75 q^{49} - 696 q^{53} + 527 q^{56} + 1461 q^{58} + 22 q^{59} + 720 q^{61} + 998 q^{62} - 317 q^{64} + 1230 q^{67} + 109 q^{68} - 951 q^{71} + 666 q^{73} - 873 q^{74} + 290 q^{76} - 264 q^{77} - 588 q^{79} - 1807 q^{82} + 867 q^{83} + 411 q^{86} + 297 q^{88} + 51 q^{89} + 2172 q^{91} + 4137 q^{92} - 865 q^{94} + 2849 q^{97} + 4104 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 12 * q^4 + 24 * q^7 - 27 * q^8 - 55 * q^11 + 111 * q^13 - 47 * q^14 - 56 * q^16 - 40 * q^17 + 205 * q^19 + 22 * q^22 - 287 * q^23 + 354 * q^26 + 460 * q^28 - 251 * q^29 - 289 * q^31 - 248 * q^32 + 522 * q^34 + 224 * q^37 - 540 * q^38 + 462 * q^41 + 593 * q^43 - 132 * q^44 - 972 * q^46 - 766 * q^47 + 75 * q^49 - 696 * q^53 + 527 * q^56 + 1461 * q^58 + 22 * q^59 + 720 * q^61 + 998 * q^62 - 317 * q^64 + 1230 * q^67 + 109 * q^68 - 951 * q^71 + 666 * q^73 - 873 * q^74 + 290 * q^76 - 264 * q^77 - 588 * q^79 - 1807 * q^82 + 867 * q^83 + 411 * q^86 + 297 * q^88 + 51 * q^89 + 2172 * q^91 + 4137 * q^92 - 865 * q^94 + 2849 * q^97 + 4104 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.95823	1.39945	0.699724	−	0.714414i	$-0.253306\pi$
$2$	3.95823	1.39945	0.699724	+	0.714414i	$0.253306\pi$
$3$	0	0
$3$	0	0
$4$	7.66762	0.958453
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27.0299	1.45948	0.729739	−	0.683726i	$-0.239642\pi$
$7$	27.0299	1.45948	0.729739	+	0.683726i	$0.239642\pi$
$8$	−1.31563	−0.0581433
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	82.9282	1.76924	0.884621	−	0.466310i	$-0.154417\pi$
$13$	82.9282	1.76924	0.884621	+	0.466310i	$0.154417\pi$
$14$	106.991	2.04246
$15$	0	0
$16$	−66.5486	−1.03982
$17$	100.512	1.43399	0.716993	−	0.697080i	$-0.245518\pi$
$17$	100.512	1.43399	0.716993	+	0.697080i	$0.245518\pi$
$18$	0	0
$19$	38.6071	0.466162	0.233081	−	0.972457i	$-0.425119\pi$
$19$	38.6071	0.466162	0.233081	+	0.972457i	$0.425119\pi$
$20$	0	0
$21$	0	0
$22$	−43.5406	−0.421949
$23$	−19.5133	−0.176904	−0.0884522	−	0.996080i	$-0.528192\pi$
$23$	−19.5133	−0.176904	−0.0884522	+	0.996080i	$0.528192\pi$
$24$	0	0
$25$	0	0
$26$	328.249	2.47596
$27$	0	0
$28$	207.255	1.39884
$29$	100.230	0.641801	0.320900	−	0.947113i	$-0.396014\pi$
$29$	100.230	0.641801	0.320900	+	0.947113i	$0.396014\pi$
$30$	0	0
$31$	121.874	0.706105	0.353053	−	0.935603i	$-0.385144\pi$
$31$	121.874	0.706105	0.353053	+	0.935603i	$0.385144\pi$
$32$	−252.890	−1.39703
$33$	0	0
$34$	397.851	2.00679
$35$	0	0
$36$	0	0
$37$	−347.319	−1.54321	−0.771606	−	0.636101i	$-0.780546\pi$
$37$	−347.319	−1.54321	−0.771606	+	0.636101i	$0.780546\pi$
$38$	152.816	0.652370
$39$	0	0
$40$	0	0
$41$	−108.386	−0.412856	−0.206428	−	0.978462i	$-0.566184\pi$
$41$	−108.386	−0.412856	−0.206428	+	0.978462i	$0.566184\pi$
$42$	0	0
$43$	268.039	0.950594	0.475297	−	0.879825i	$-0.342341\pi$
$43$	268.039	0.950594	0.475297	+	0.879825i	$0.342341\pi$
$44$	−84.3438	−0.288984
$45$	0	0
$46$	−77.2381	−0.247568
$47$	−568.074	−1.76302	−0.881512	−	0.472162i	$-0.843474\pi$
$47$	−568.074	−1.76302	−0.881512	+	0.472162i	$0.843474\pi$
$48$	0	0
$49$	387.616	1.13008
$50$	0	0
$51$	0	0
$52$	635.862	1.69574
$53$	−603.970	−1.56531	−0.782657	−	0.622453i	$-0.786136\pi$
$53$	−603.970	−1.56531	−0.782657	+	0.622453i	$0.786136\pi$
$54$	0	0
$55$	0	0
$56$	−35.5614	−0.0848588
$57$	0	0
$58$	396.734	0.898167
$59$	−32.7793	−0.0723304	−0.0361652	−	0.999346i	$-0.511514\pi$
$59$	−32.7793	−0.0723304	−0.0361652	+	0.999346i	$0.511514\pi$
$60$	0	0
$61$	156.323	0.328117	0.164059	−	0.986451i	$-0.447541\pi$
$61$	156.323	0.328117	0.164059	+	0.986451i	$0.447541\pi$
$62$	482.407	0.988157
$63$	0	0
$64$	−468.608	−0.915251
$65$	0	0
$66$	0	0
$67$	745.522	1.35940	0.679702	−	0.733489i	$-0.262109\pi$
$67$	745.522	1.35940	0.679702	+	0.733489i	$0.262109\pi$
$68$	770.689	1.37441
$69$	0	0
$70$	0	0
$71$	264.725	0.442494	0.221247	−	0.975218i	$-0.428987\pi$
$71$	264.725	0.442494	0.221247	+	0.975218i	$0.428987\pi$
$72$	0	0
$73$	334.805	0.536794	0.268397	−	0.963308i	$-0.413506\pi$
$73$	334.805	0.536794	0.268397	+	0.963308i	$0.413506\pi$
$74$	−1374.77	−2.15964
$75$	0	0
$76$	296.025	0.446795
$77$	−297.329	−0.440049
$78$	0	0
$79$	1287.57	1.83370	0.916851	−	0.399230i	$-0.130723\pi$
$79$	1287.57	1.83370	0.916851	+	0.399230i	$0.130723\pi$
$80$	0	0
$81$	0	0
$82$	−429.018	−0.577770
$83$	127.129	0.168123	0.0840617	−	0.996461i	$-0.473211\pi$
$83$	127.129	0.168123	0.0840617	+	0.996461i	$0.473211\pi$
$84$	0	0
$85$	0	0
$86$	1060.96	1.33031
$87$	0	0
$88$	14.4720	0.0175309
$89$	615.817	0.733443	0.366722	−	0.930331i	$-0.380480\pi$
$89$	615.817	0.733443	0.366722	+	0.930331i	$0.380480\pi$
$90$	0	0
$91$	2241.54	2.58217
$92$	−149.620	−0.169554
$93$	0	0
$94$	−2248.57	−2.46726
$95$	0	0
$96$	0	0
$97$	−579.630	−0.606727	−0.303363	−	0.952875i	$-0.598110\pi$
$97$	−579.630	−0.606727	−0.303363	+	0.952875i	$0.598110\pi$
$98$	1534.28	1.58148
$99$	0	0
$100$	0	0
$101$	599.271	0.590393	0.295197	−	0.955437i	$-0.404615\pi$
$101$	599.271	0.590393	0.295197	+	0.955437i	$0.404615\pi$
$102$	0	0
$103$	991.632	0.948625	0.474313	−	0.880356i	$-0.342697\pi$
$103$	991.632	0.948625	0.474313	+	0.880356i	$0.342697\pi$
$104$	−109.103	−0.102870
$105$	0	0
$106$	−2390.65	−2.19057
$107$	867.633	0.783899	0.391950	−	0.919987i	$-0.371801\pi$
$107$	867.633	0.783899	0.391950	+	0.919987i	$0.371801\pi$
$108$	0	0
$109$	−1312.40	−1.15326	−0.576630	−	0.817006i	$-0.695632\pi$
$109$	−1312.40	−1.15326	−0.576630	+	0.817006i	$0.695632\pi$
$110$	0	0
$111$	0	0
$112$	−1798.80	−1.51760
$113$	909.383	0.757058	0.378529	−	0.925589i	$-0.376430\pi$
$113$	909.383	0.757058	0.378529	+	0.925589i	$0.376430\pi$
$114$	0	0
$115$	0	0
$116$	768.525	0.615136
$117$	0	0
$118$	−129.748	−0.101223
$119$	2716.83	2.09287
$120$	0	0
$121$	121.000	0.0909091
$122$	618.764	0.459183
$123$	0	0
$124$	934.486	0.676768
$125$	0	0
$126$	0	0
$127$	−339.579	−0.237266	−0.118633	−	0.992938i	$-0.537851\pi$
$127$	−339.579	−0.237266	−0.118633	+	0.992938i	$0.537851\pi$
$128$	168.256	0.116186
$129$	0	0
$130$	0	0
$131$	−1502.96	−1.00240	−0.501199	−	0.865332i	$-0.667108\pi$
$131$	−1502.96	−1.00240	−0.501199	+	0.865332i	$0.667108\pi$
$132$	0	0
$133$	1043.55	0.680354
$134$	2950.95	1.90241
$135$	0	0
$136$	−132.237	−0.0833767
$137$	−2164.22	−1.34965	−0.674825	−	0.737978i	$-0.735781\pi$
$137$	−2164.22	−1.34965	−0.674825	+	0.737978i	$0.735781\pi$
$138$	0	0
$139$	−390.133	−0.238062	−0.119031	−	0.992891i	$-0.537979\pi$
$139$	−390.133	−0.238062	−0.119031	+	0.992891i	$0.537979\pi$
$140$	0	0
$141$	0	0
$142$	1047.84	0.619246
$143$	−912.211	−0.533447
$144$	0	0
$145$	0	0
$146$	1325.24	0.751214
$147$	0	0
$148$	−2663.11	−1.47910
$149$	2735.95	1.50428	0.752141	−	0.659003i	$-0.229021\pi$
$149$	2735.95	1.50428	0.752141	+	0.659003i	$0.229021\pi$
$150$	0	0
$151$	1831.35	0.986974	0.493487	−	0.869753i	$-0.335722\pi$
$151$	1831.35	0.986974	0.493487	+	0.869753i	$0.335722\pi$
$152$	−50.7928	−0.0271042
$153$	0	0
$154$	−1176.90	−0.615826
$155$	0	0
$156$	0	0
$157$	3816.59	1.94011	0.970055	−	0.242884i	$-0.0780933\pi$
$157$	3816.59	1.94011	0.970055	+	0.242884i	$0.0780933\pi$
$158$	5096.48	2.56617
$159$	0	0
$160$	0	0
$161$	−527.442	−0.258188
$162$	0	0
$163$	−4.62217	−0.00222108	−0.00111054	−	0.999999i	$-0.500353\pi$
$163$	−4.62217	−0.00222108	−0.00111054	+	0.999999i	$0.500353\pi$
$164$	−831.065	−0.395703
$165$	0	0
$166$	503.207	0.235280
$167$	−2057.83	−0.953533	−0.476767	−	0.879030i	$-0.658191\pi$
$167$	−2057.83	−0.953533	−0.476767	+	0.879030i	$0.658191\pi$
$168$	0	0
$169$	4680.09	2.13022
$170$	0	0
$171$	0	0
$172$	2055.22	0.911100
$173$	4045.58	1.77792	0.888959	−	0.457987i	$-0.151429\pi$
$173$	4045.58	1.77792	0.888959	+	0.457987i	$0.151429\pi$
$174$	0	0
$175$	0	0
$176$	732.034	0.313518
$177$	0	0
$178$	2437.55	1.02642
$179$	−3607.09	−1.50618	−0.753092	−	0.657916i	$-0.771438\pi$
$179$	−3607.09	−1.50618	−0.753092	+	0.657916i	$0.771438\pi$
$180$	0	0
$181$	−664.649	−0.272945	−0.136472	−	0.990644i	$-0.543576\pi$
$181$	−664.649	−0.272945	−0.136472	+	0.990644i	$0.543576\pi$
$182$	8872.55	3.61361
$183$	0	0
$184$	25.6723	0.0102858
$185$	0	0
$186$	0	0
$187$	−1105.63	−0.432363
$188$	−4355.78	−1.68977
$189$	0	0
$190$	0	0
$191$	−89.6468	−0.0339613	−0.0169807	−	0.999856i	$-0.505405\pi$
$191$	−89.6468	−0.0339613	−0.0169807	+	0.999856i	$0.505405\pi$
$192$	0	0
$193$	2119.54	0.790505	0.395253	−	0.918572i	$-0.370657\pi$
$193$	2119.54	0.790505	0.395253	+	0.918572i	$0.370657\pi$
$194$	−2294.31	−0.849082
$195$	0	0
$196$	2972.09	1.08312
$197$	−2736.33	−0.989623	−0.494812	−	0.869000i	$-0.664763\pi$
$197$	−2736.33	−0.989623	−0.494812	+	0.869000i	$0.664763\pi$
$198$	0	0
$199$	−1249.59	−0.445130	−0.222565	−	0.974918i	$-0.571443\pi$
$199$	−1249.59	−0.445130	−0.222565	+	0.974918i	$0.571443\pi$
$200$	0	0
$201$	0	0
$202$	2372.06	0.826224
$203$	2709.21	0.936694
$204$	0	0
$205$	0	0
$206$	3925.11	1.32755
$207$	0	0
$208$	−5518.75	−1.83970
$209$	−424.678	−0.140553
$210$	0	0
$211$	944.864	0.308280	0.154140	−	0.988049i	$-0.450739\pi$
$211$	944.864	0.308280	0.154140	+	0.988049i	$0.450739\pi$
$212$	−4631.01	−1.50028
$213$	0	0
$214$	3434.29	1.09703
$215$	0	0
$216$	0	0
$217$	3294.25	1.03054
$218$	−5194.79	−1.61393
$219$	0	0
$220$	0	0
$221$	8335.29	2.53707
$222$	0	0
$223$	−5252.19	−1.57719	−0.788593	−	0.614916i	$-0.789190\pi$
$223$	−5252.19	−1.57719	−0.788593	+	0.614916i	$0.789190\pi$
$224$	−6835.59	−2.03894
$225$	0	0
$226$	3599.55	1.05946
$227$	−576.634	−0.168602	−0.0843008	−	0.996440i	$-0.526866\pi$
$227$	−576.634	−0.168602	−0.0843008	+	0.996440i	$0.526866\pi$
$228$	0	0
$229$	−4617.17	−1.33236	−0.666182	−	0.745789i	$-0.732073\pi$
$229$	−4617.17	−1.33236	−0.666182	+	0.745789i	$0.732073\pi$
$230$	0	0
$231$	0	0
$232$	−131.866	−0.0373164
$233$	−2916.94	−0.820150	−0.410075	−	0.912052i	$-0.634498\pi$
$233$	−2916.94	−0.820150	−0.410075	+	0.912052i	$0.634498\pi$
$234$	0	0
$235$	0	0
$236$	−251.339	−0.0693253
$237$	0	0
$238$	10753.9	2.92886
$239$	649.560	0.175802	0.0879008	−	0.996129i	$-0.471984\pi$
$239$	649.560	0.175802	0.0879008	+	0.996129i	$0.471984\pi$
$240$	0	0
$241$	1206.97	0.322605	0.161302	−	0.986905i	$-0.448431\pi$
$241$	1206.97	0.322605	0.161302	+	0.986905i	$0.448431\pi$
$242$	478.946	0.127222
$243$	0	0
$244$	1198.63	0.314485
$245$	0	0
$246$	0	0
$247$	3201.62	0.824754
$248$	−160.342	−0.0410553
$249$	0	0
$250$	0	0
$251$	2729.80	0.686468	0.343234	−	0.939250i	$-0.388478\pi$
$251$	2729.80	0.686468	0.343234	+	0.939250i	$0.388478\pi$
$252$	0	0
$253$	214.646	0.0533387
$254$	−1344.13	−0.332041
$255$	0	0
$256$	4414.86	1.07785
$257$	−3924.17	−0.952464	−0.476232	−	0.879320i	$-0.657998\pi$
$257$	−3924.17	−0.952464	−0.476232	+	0.879320i	$0.657998\pi$
$258$	0	0
$259$	−9387.99	−2.25228
$260$	0	0
$261$	0	0
$262$	−5949.07	−1.40280
$263$	1503.06	0.352406	0.176203	−	0.984354i	$-0.443618\pi$
$263$	1503.06	0.352406	0.176203	+	0.984354i	$0.443618\pi$
$264$	0	0
$265$	0	0
$266$	4130.61	0.952119
$267$	0	0
$268$	5716.38	1.30292
$269$	2116.10	0.479631	0.239815	−	0.970819i	$-0.422913\pi$
$269$	2116.10	0.479631	0.239815	+	0.970819i	$0.422913\pi$
$270$	0	0
$271$	−6259.37	−1.40306	−0.701531	−	0.712639i	$-0.747500\pi$
$271$	−6259.37	−1.40306	−0.701531	+	0.712639i	$0.747500\pi$
$272$	−6688.94	−1.49109
$273$	0	0
$274$	−8566.50	−1.88876
$275$	0	0
$276$	0	0
$277$	−2873.23	−0.623232	−0.311616	−	0.950208i	$-0.600870\pi$
$277$	−2873.23	−0.623232	−0.311616	+	0.950208i	$0.600870\pi$
$278$	−1544.24	−0.333155
$279$	0	0
$280$	0	0
$281$	766.809	0.162790	0.0813949	−	0.996682i	$-0.474062\pi$
$281$	766.809	0.162790	0.0813949	+	0.996682i	$0.474062\pi$
$282$	0	0
$283$	5160.63	1.08399	0.541993	−	0.840383i	$-0.317670\pi$
$283$	5160.63	1.08399	0.541993	+	0.840383i	$0.317670\pi$
$284$	2029.81	0.424109
$285$	0	0
$286$	−3610.74	−0.746531
$287$	−2929.67	−0.602554
$288$	0	0
$289$	5189.69	1.05632
$290$	0	0
$291$	0	0
$292$	2567.16	0.514491
$293$	−3915.69	−0.780740	−0.390370	−	0.920658i	$-0.627653\pi$
$293$	−3915.69	−0.780740	−0.390370	+	0.920658i	$0.627653\pi$
$294$	0	0
$295$	0	0
$296$	456.944	0.0897274
$297$	0	0
$298$	10829.5	2.10516
$299$	−1618.20	−0.312987
$300$	0	0
$301$	7245.07	1.38737
$302$	7248.91	1.38122
$303$	0	0
$304$	−2569.25	−0.484725
$305$	0	0
$306$	0	0
$307$	−8255.96	−1.53483	−0.767415	−	0.641151i	$-0.778457\pi$
$307$	−8255.96	−1.53483	−0.767415	+	0.641151i	$0.778457\pi$
$308$	−2279.81	−0.421766
$309$	0	0
$310$	0	0
$311$	9181.60	1.67409	0.837043	−	0.547138i	$-0.184283\pi$
$311$	9181.60	1.67409	0.837043	+	0.547138i	$0.184283\pi$
$312$	0	0
$313$	−7061.96	−1.27529	−0.637645	−	0.770330i	$-0.720091\pi$
$313$	−7061.96	−1.27529	−0.637645	+	0.770330i	$0.720091\pi$
$314$	15107.0	2.71508
$315$	0	0
$316$	9872.56	1.75752
$317$	6122.11	1.08471	0.542353	−	0.840151i	$-0.317533\pi$
$317$	6122.11	1.08471	0.542353	+	0.840151i	$0.317533\pi$
$318$	0	0
$319$	−1102.53	−0.193510
$320$	0	0
$321$	0	0
$322$	−2087.74	−0.361321
$323$	3880.48	0.668470
$324$	0	0
$325$	0	0
$326$	−18.2956	−0.00310828
$327$	0	0
$328$	142.596	0.0240048
$329$	−15355.0	−2.57309
$330$	0	0
$331$	4715.92	0.783113	0.391557	−	0.920154i	$-0.371937\pi$
$331$	4715.92	0.783113	0.391557	+	0.920154i	$0.371937\pi$
$332$	974.778	0.161138
$333$	0	0
$334$	−8145.39	−1.33442
$335$	0	0
$336$	0	0
$337$	9490.05	1.53399	0.766997	−	0.641651i	$-0.221750\pi$
$337$	9490.05	1.53399	0.766997	+	0.641651i	$0.221750\pi$
$338$	18524.9	2.98113
$339$	0	0
$340$	0	0
$341$	−1340.62	−0.212899
$342$	0	0
$343$	1205.97	0.189843
$344$	−352.641	−0.0552707
$345$	0	0
$346$	16013.4	2.48810
$347$	6489.79	1.00401	0.502003	−	0.864866i	$-0.332597\pi$
$347$	6489.79	1.00401	0.502003	+	0.864866i	$0.332597\pi$
$348$	0	0
$349$	7155.44	1.09748	0.548742	−	0.835992i	$-0.315107\pi$
$349$	7155.44	1.09748	0.548742	+	0.835992i	$0.315107\pi$
$350$	0	0
$351$	0	0
$352$	2781.79	0.421221
$353$	−929.258	−0.140112	−0.0700558	−	0.997543i	$-0.522318\pi$
$353$	−929.258	−0.140112	−0.0700558	+	0.997543i	$0.522318\pi$
$354$	0	0
$355$	0	0
$356$	4721.85	0.702971
$357$	0	0
$358$	−14277.7	−2.10782
$359$	8426.85	1.23886	0.619432	−	0.785050i	$-0.287363\pi$
$359$	8426.85	1.23886	0.619432	+	0.785050i	$0.287363\pi$
$360$	0	0
$361$	−5368.49	−0.782693
$362$	−2630.84	−0.381972
$363$	0	0
$364$	17187.3	2.47489
$365$	0	0
$366$	0	0
$367$	5108.50	0.726599	0.363299	−	0.931672i	$-0.381650\pi$
$367$	5108.50	0.726599	0.363299	+	0.931672i	$0.381650\pi$
$368$	1298.58	0.183949
$369$	0	0
$370$	0	0
$371$	−16325.3	−2.28454
$372$	0	0
$373$	−12264.2	−1.70246	−0.851231	−	0.524791i	$-0.824144\pi$
$373$	−12264.2	−1.70246	−0.851231	+	0.524791i	$0.824144\pi$
$374$	−4376.36	−0.605069
$375$	0	0
$376$	747.376	0.102508
$377$	8311.89	1.13550
$378$	0	0
$379$	−2724.96	−0.369318	−0.184659	−	0.982803i	$-0.559118\pi$
$379$	−2724.96	−0.369318	−0.184659	+	0.982803i	$0.559118\pi$
$380$	0	0
$381$	0	0
$382$	−354.843	−0.0475271
$383$	11606.0	1.54841	0.774205	−	0.632935i	$-0.218150\pi$
$383$	11606.0	1.54841	0.774205	+	0.632935i	$0.218150\pi$
$384$	0	0
$385$	0	0
$386$	8389.62	1.10627
$387$	0	0
$388$	−4444.38	−0.581519
$389$	−2655.45	−0.346110	−0.173055	−	0.984912i	$-0.555364\pi$
$389$	−2655.45	−0.346110	−0.173055	+	0.984912i	$0.555364\pi$
$390$	0	0
$391$	−1961.32	−0.253678
$392$	−509.960	−0.0657063
$393$	0	0
$394$	−10831.1	−1.38493
$395$	0	0
$396$	0	0
$397$	10994.8	1.38996	0.694980	−	0.719029i	$-0.255413\pi$
$397$	10994.8	1.38996	0.694980	+	0.719029i	$0.255413\pi$
$398$	−4946.16	−0.622936
$399$	0	0
$400$	0	0
$401$	4626.29	0.576124	0.288062	−	0.957612i	$-0.406989\pi$
$401$	4626.29	0.576124	0.288062	+	0.957612i	$0.406989\pi$
$402$	0	0
$403$	10106.8	1.24927
$404$	4594.98	0.565864
$405$	0	0
$406$	10723.7	1.31085
$407$	3820.51	0.465296
$408$	0	0
$409$	−7991.43	−0.966139	−0.483069	−	0.875582i	$-0.660478\pi$
$409$	−7991.43	−0.966139	−0.483069	+	0.875582i	$0.660478\pi$
$410$	0	0
$411$	0	0
$412$	7603.46	0.909212
$413$	−886.021	−0.105565
$414$	0	0
$415$	0	0
$416$	−20971.7	−2.47169
$417$	0	0
$418$	−1680.98	−0.196697
$419$	−14032.5	−1.63612	−0.818059	−	0.575134i	$-0.804950\pi$
$419$	−14032.5	−1.63612	−0.818059	+	0.575134i	$0.804950\pi$
$420$	0	0
$421$	5305.98	0.614247	0.307123	−	0.951670i	$-0.400634\pi$
$421$	5305.98	0.614247	0.307123	+	0.951670i	$0.400634\pi$
$422$	3739.99	0.431422
$423$	0	0
$424$	794.602	0.0910125
$425$	0	0
$426$	0	0
$427$	4225.40	0.478880
$428$	6652.68	0.751330
$429$	0	0
$430$	0	0
$431$	−16459.2	−1.83947	−0.919737	−	0.392536i	$-0.871598\pi$
$431$	−16459.2	−1.83947	−0.919737	+	0.392536i	$0.871598\pi$
$432$	0	0
$433$	6138.61	0.681300	0.340650	−	0.940190i	$-0.389353\pi$
$433$	6138.61	0.681300	0.340650	+	0.940190i	$0.389353\pi$
$434$	13039.4	1.44219
$435$	0	0
$436$	−10063.0	−1.10534
$437$	−753.352	−0.0824661
$438$	0	0
$439$	−13563.5	−1.47461	−0.737303	−	0.675562i	$-0.763901\pi$
$439$	−13563.5	−1.47461	−0.737303	+	0.675562i	$0.763901\pi$
$440$	0	0
$441$	0	0
$442$	32993.0	3.55050
$443$	2667.69	0.286108	0.143054	−	0.989715i	$-0.454308\pi$
$443$	2667.69	0.286108	0.143054	+	0.989715i	$0.454308\pi$
$444$	0	0
$445$	0	0
$446$	−20789.4	−2.20719
$447$	0	0
$448$	−12666.4	−1.33579
$449$	−5625.14	−0.591240	−0.295620	−	0.955306i	$-0.595526\pi$
$449$	−5625.14	−0.591240	−0.295620	+	0.955306i	$0.595526\pi$
$450$	0	0
$451$	1192.25	0.124481
$452$	6972.81	0.725604
$453$	0	0
$454$	−2282.45	−0.235949
$455$	0	0
$456$	0	0
$457$	−10988.9	−1.12481	−0.562404	−	0.826863i	$-0.690123\pi$
$457$	−10988.9	−1.12481	−0.562404	+	0.826863i	$0.690123\pi$
$458$	−18275.9	−1.86457
$459$	0	0
$460$	0	0
$461$	−1796.10	−0.181459	−0.0907295	−	0.995876i	$-0.528920\pi$
$461$	−1796.10	−0.181459	−0.0907295	+	0.995876i	$0.528920\pi$
$462$	0	0
$463$	720.304	0.0723010	0.0361505	−	0.999346i	$-0.488490\pi$
$463$	720.304	0.0723010	0.0361505	+	0.999346i	$0.488490\pi$
$464$	−6670.16	−0.667358
$465$	0	0
$466$	−11545.9	−1.14776
$467$	−131.849	−0.0130647	−0.00653236	−	0.999979i	$-0.502079\pi$
$467$	−131.849	−0.0130647	−0.00653236	+	0.999979i	$0.502079\pi$
$468$	0	0
$469$	20151.4	1.98402
$470$	0	0
$471$	0	0
$472$	43.1254	0.00420553
$473$	−2948.43	−0.286615
$474$	0	0
$475$	0	0
$476$	20831.7	2.00592
$477$	0	0
$478$	2571.11	0.246025
$479$	−15015.2	−1.43228	−0.716141	−	0.697956i	$-0.754093\pi$
$479$	−15015.2	−1.43228	−0.716141	+	0.697956i	$0.754093\pi$
$480$	0	0
$481$	−28802.5	−2.73032
$482$	4777.47	0.451468
$483$	0	0
$484$	927.782	0.0871321
$485$	0	0
$486$	0	0
$487$	−10065.4	−0.936561	−0.468281	−	0.883580i	$-0.655126\pi$
$487$	−10065.4	−0.936561	−0.468281	+	0.883580i	$0.655126\pi$
$488$	−205.664	−0.0190778
$489$	0	0
$490$	0	0
$491$	−8049.15	−0.739823	−0.369912	−	0.929067i	$-0.620612\pi$
$491$	−8049.15	−0.739823	−0.369912	+	0.929067i	$0.620612\pi$
$492$	0	0
$493$	10074.3	0.920334
$494$	12672.8	1.15420
$495$	0	0
$496$	−8110.55	−0.734223
$497$	7155.48	0.645810
$498$	0	0
$499$	−7688.18	−0.689720	−0.344860	−	0.938654i	$-0.612074\pi$
$499$	−7688.18	−0.689720	−0.344860	+	0.938654i	$0.612074\pi$
$500$	0	0
$501$	0	0
$502$	10805.2	0.960676
$503$	3158.78	0.280006	0.140003	−	0.990151i	$-0.455289\pi$
$503$	3158.78	0.280006	0.140003	+	0.990151i	$0.455289\pi$
$504$	0	0
$505$	0	0
$506$	849.620	0.0746447
$507$	0	0
$508$	−2603.76	−0.227408
$509$	5722.71	0.498340	0.249170	−	0.968460i	$-0.419842\pi$
$509$	5722.71	0.498340	0.249170	+	0.968460i	$0.419842\pi$
$510$	0	0
$511$	9049.74	0.783439
$512$	16129.0	1.39220
$513$	0	0
$514$	−15532.8	−1.33292
$515$	0	0
$516$	0	0
$517$	6248.81	0.531571
$518$	−37159.9	−3.15195
$519$	0	0
$520$	0	0
$521$	3228.04	0.271445	0.135723	−	0.990747i	$-0.456664\pi$
$521$	3228.04	0.271445	0.135723	+	0.990747i	$0.456664\pi$
$522$	0	0
$523$	−8397.13	−0.702067	−0.351033	−	0.936363i	$-0.614170\pi$
$523$	−8397.13	−0.702067	−0.351033	+	0.936363i	$0.614170\pi$
$524$	−11524.1	−0.960751
$525$	0	0
$526$	5949.48	0.493174
$527$	12249.8	1.01255
$528$	0	0
$529$	−11786.2	−0.968705
$530$	0	0
$531$	0	0
$532$	8001.53	0.652087
$533$	−8988.28	−0.730442
$534$	0	0
$535$	0	0
$536$	−980.833	−0.0790402
$537$	0	0
$538$	8376.01	0.671218
$539$	−4263.78	−0.340731
$540$	0	0
$541$	636.671	0.0505964	0.0252982	−	0.999680i	$-0.491946\pi$
$541$	636.671	0.0505964	0.0252982	+	0.999680i	$0.491946\pi$
$542$	−24776.1	−1.96351
$543$	0	0
$544$	−25418.5	−2.00332
$545$	0	0
$546$	0	0
$547$	−16014.7	−1.25181	−0.625904	−	0.779900i	$-0.715270\pi$
$547$	−16014.7	−1.25181	−0.625904	+	0.779900i	$0.715270\pi$
$548$	−16594.4	−1.29358
$549$	0	0
$550$	0	0
$551$	3869.59	0.299183
$552$	0	0
$553$	34802.8	2.67625
$554$	−11372.9	−0.872181
$555$	0	0
$556$	−2991.39	−0.228171
$557$	17221.3	1.31004	0.655019	−	0.755612i	$-0.272660\pi$
$557$	17221.3	1.31004	0.655019	+	0.755612i	$0.272660\pi$
$558$	0	0
$559$	22228.0	1.68183
$560$	0	0
$561$	0	0
$562$	3035.21	0.227816
$563$	2137.21	0.159987	0.0799934	−	0.996795i	$-0.474510\pi$
$563$	2137.21	0.159987	0.0799934	+	0.996795i	$0.474510\pi$
$564$	0	0
$565$	0	0
$566$	20427.0	1.51698
$567$	0	0
$568$	−348.280	−0.0257280
$569$	18820.1	1.38661	0.693304	−	0.720645i	$-0.256154\pi$
$569$	18820.1	1.38661	0.693304	+	0.720645i	$0.256154\pi$
$570$	0	0
$571$	−12458.4	−0.913081	−0.456540	−	0.889703i	$-0.650912\pi$
$571$	−12458.4	−0.913081	−0.456540	+	0.889703i	$0.650912\pi$
$572$	−6994.49	−0.511283
$573$	0	0
$574$	−11596.3	−0.843243
$575$	0	0
$576$	0	0
$577$	−7596.79	−0.548108	−0.274054	−	0.961714i	$-0.588365\pi$
$577$	−7596.79	−0.548108	−0.274054	+	0.961714i	$0.588365\pi$
$578$	20542.0	1.47826
$579$	0	0
$580$	0	0
$581$	3436.29	0.245372
$582$	0	0
$583$	6643.67	0.471960
$584$	−440.480	−0.0312109
$585$	0	0
$586$	−15499.2	−1.09260
$587$	−14595.8	−1.02629	−0.513146	−	0.858302i	$-0.671520\pi$
$587$	−14595.8	−1.02629	−0.513146	+	0.858302i	$0.671520\pi$
$588$	0	0
$589$	4705.21	0.329160
$590$	0	0
$591$	0	0
$592$	23113.6	1.60466
$593$	6816.78	0.472060	0.236030	−	0.971746i	$-0.424154\pi$
$593$	6816.78	0.472060	0.236030	+	0.971746i	$0.424154\pi$
$594$	0	0
$595$	0	0
$596$	20978.2	1.44178
$597$	0	0
$598$	−6405.22	−0.438008
$599$	2023.88	0.138053	0.0690263	−	0.997615i	$-0.478011\pi$
$599$	2023.88	0.138053	0.0690263	+	0.997615i	$0.478011\pi$
$600$	0	0
$601$	−14564.3	−0.988503	−0.494252	−	0.869319i	$-0.664558\pi$
$601$	−14564.3	−0.988503	−0.494252	+	0.869319i	$0.664558\pi$
$602$	28677.7	1.94155
$603$	0	0
$604$	14042.1	0.945968
$605$	0	0
$606$	0	0
$607$	−29493.8	−1.97218	−0.986092	−	0.166199i	$-0.946851\pi$
$607$	−29493.8	−1.97218	−0.986092	+	0.166199i	$0.946851\pi$
$608$	−9763.35	−0.651243
$609$	0	0
$610$	0	0
$611$	−47109.4	−3.11922
$612$	0	0
$613$	11895.7	0.783788	0.391894	−	0.920010i	$-0.371820\pi$
$613$	11895.7	0.783788	0.391894	+	0.920010i	$0.371820\pi$
$614$	−32679.0	−2.14791
$615$	0	0
$616$	391.176	0.0255859
$617$	−4875.99	−0.318152	−0.159076	−	0.987266i	$-0.550852\pi$
$617$	−4875.99	−0.318152	−0.159076	+	0.987266i	$0.550852\pi$
$618$	0	0
$619$	3651.22	0.237084	0.118542	−	0.992949i	$-0.462178\pi$
$619$	3651.22	0.237084	0.118542	+	0.992949i	$0.462178\pi$
$620$	0	0
$621$	0	0
$622$	36342.9	2.34279
$623$	16645.5	1.07044
$624$	0	0
$625$	0	0
$626$	−27952.9	−1.78470
$627$	0	0
$628$	29264.2	1.85950
$629$	−34909.7	−2.21294
$630$	0	0
$631$	−10343.2	−0.652546	−0.326273	−	0.945276i	$-0.605793\pi$
$631$	−10343.2	−0.652546	−0.326273	+	0.945276i	$0.605793\pi$
$632$	−1693.96	−0.106617
$633$	0	0
$634$	24232.8	1.51799
$635$	0	0
$636$	0	0
$637$	32144.3	1.99938
$638$	−4364.07	−0.270807
$639$	0	0
$640$	0	0
$641$	−28447.8	−1.75292	−0.876459	−	0.481477i	$-0.840101\pi$
$641$	−28447.8	−1.75292	−0.876459	+	0.481477i	$0.840101\pi$
$642$	0	0
$643$	11854.4	0.727048	0.363524	−	0.931585i	$-0.381573\pi$
$643$	11854.4	0.727048	0.363524	+	0.931585i	$0.381573\pi$
$644$	−4044.23	−0.247461
$645$	0	0
$646$	15359.9	0.935489
$647$	−16073.9	−0.976706	−0.488353	−	0.872646i	$-0.662402\pi$
$647$	−16073.9	−0.976706	−0.488353	+	0.872646i	$0.662402\pi$
$648$	0	0
$649$	360.572	0.0218084
$650$	0	0
$651$	0	0
$652$	−35.4410	−0.00212880
$653$	−25950.1	−1.55514	−0.777569	−	0.628798i	$-0.783547\pi$
$653$	−25950.1	−1.55514	−0.777569	+	0.628798i	$0.783547\pi$
$654$	0	0
$655$	0	0
$656$	7212.95	0.429296
$657$	0	0
$658$	−60778.6	−3.60091
$659$	23160.6	1.36906	0.684529	−	0.728986i	$-0.260008\pi$
$659$	23160.6	1.36906	0.684529	+	0.728986i	$0.260008\pi$
$660$	0	0
$661$	−24160.6	−1.42169	−0.710847	−	0.703346i	$-0.751688\pi$
$661$	−24160.6	−1.42169	−0.710847	+	0.703346i	$0.751688\pi$
$662$	18666.7	1.09593
$663$	0	0
$664$	−167.255	−0.00977524
$665$	0	0
$666$	0	0
$667$	−1955.81	−0.113537
$668$	−15778.7	−0.913916
$669$	0	0
$670$	0	0
$671$	−1719.56	−0.0989310
$672$	0	0
$673$	7079.68	0.405500	0.202750	−	0.979231i	$-0.435012\pi$
$673$	7079.68	0.405500	0.202750	+	0.979231i	$0.435012\pi$
$674$	37563.8	2.14674
$675$	0	0
$676$	35885.2	2.04171
$677$	−26190.6	−1.48684	−0.743418	−	0.668827i	$-0.766797\pi$
$677$	−26190.6	−1.48684	−0.743418	+	0.668827i	$0.766797\pi$
$678$	0	0
$679$	−15667.3	−0.885505
$680$	0	0
$681$	0	0
$682$	−5306.47	−0.297941
$683$	24378.6	1.36577	0.682887	−	0.730524i	$-0.260724\pi$
$683$	24378.6	1.36577	0.682887	+	0.730524i	$0.260724\pi$
$684$	0	0
$685$	0	0
$686$	4773.51	0.265676
$687$	0	0
$688$	−17837.6	−0.988448
$689$	−50086.2	−2.76942
$690$	0	0
$691$	−20884.1	−1.14974	−0.574869	−	0.818245i	$-0.694947\pi$
$691$	−20884.1	−1.14974	−0.574869	+	0.818245i	$0.694947\pi$
$692$	31020.0	1.70405
$693$	0	0
$694$	25688.1	1.40505
$695$	0	0
$696$	0	0
$697$	−10894.1	−0.592030
$698$	28322.9	1.53587
$699$	0	0
$700$	0	0
$701$	5279.66	0.284465	0.142232	−	0.989833i	$-0.454572\pi$
$701$	5279.66	0.284465	0.142232	+	0.989833i	$0.454572\pi$
$702$	0	0
$703$	−13409.0	−0.719387
$704$	5154.69	0.275959
$705$	0	0
$706$	−3678.22	−0.196079
$707$	16198.2	0.861666
$708$	0	0
$709$	9859.71	0.522270	0.261135	−	0.965302i	$-0.415903\pi$
$709$	9859.71	0.522270	0.261135	+	0.965302i	$0.415903\pi$
$710$	0	0
$711$	0	0
$712$	−810.188	−0.0426448
$713$	−2378.17	−0.124913
$714$	0	0
$715$	0	0
$716$	−27657.8	−1.44361
$717$	0	0
$718$	33355.4	1.73372
$719$	11209.6	0.581431	0.290715	−	0.956810i	$-0.406107\pi$
$719$	11209.6	0.581431	0.290715	+	0.956810i	$0.406107\pi$
$720$	0	0
$721$	26803.7	1.38450
$722$	−21249.7	−1.09534
$723$	0	0
$724$	−5096.28	−0.261605
$725$	0	0
$726$	0	0
$727$	6297.20	0.321252	0.160626	−	0.987015i	$-0.448649\pi$
$727$	6297.20	0.321252	0.160626	+	0.987015i	$0.448649\pi$
$728$	−2949.05	−0.150136
$729$	0	0
$730$	0	0
$731$	26941.2	1.36314
$732$	0	0
$733$	−11939.8	−0.601644	−0.300822	−	0.953680i	$-0.597261\pi$
$733$	−11939.8	−0.601644	−0.300822	+	0.953680i	$0.597261\pi$
$734$	20220.7	1.01684
$735$	0	0
$736$	4934.71	0.247141
$737$	−8200.75	−0.409876
$738$	0	0
$739$	24617.0	1.22537	0.612687	−	0.790325i	$-0.290089\pi$
$739$	24617.0	1.22537	0.612687	+	0.790325i	$0.290089\pi$
$740$	0	0
$741$	0	0
$742$	−64619.2	−3.19710
$743$	34374.3	1.69727	0.848634	−	0.528980i	$-0.177425\pi$
$743$	34374.3	1.69727	0.848634	+	0.528980i	$0.177425\pi$
$744$	0	0
$745$	0	0
$746$	−48544.8	−2.38251
$747$	0	0
$748$	−8477.58	−0.414400
$749$	23452.0	1.14408
$750$	0	0
$751$	23103.7	1.12259	0.561296	−	0.827615i	$-0.310303\pi$
$751$	23103.7	1.12259	0.561296	+	0.827615i	$0.310303\pi$
$752$	37804.5	1.83323
$753$	0	0
$754$	32900.4	1.58907
$755$	0	0
$756$	0	0
$757$	19530.2	0.937696	0.468848	−	0.883279i	$-0.344669\pi$
$757$	19530.2	0.937696	0.468848	+	0.883279i	$0.344669\pi$
$758$	−10786.0	−0.516841
$759$	0	0
$760$	0	0
$761$	20257.0	0.964934	0.482467	−	0.875914i	$-0.339741\pi$
$761$	20257.0	0.964934	0.482467	+	0.875914i	$0.339741\pi$
$762$	0	0
$763$	−35474.1	−1.68316
$764$	−687.378	−0.0325503
$765$	0	0
$766$	45939.4	2.16692
$767$	−2718.33	−0.127970
$768$	0	0
$769$	−19356.1	−0.907669	−0.453834	−	0.891086i	$-0.649944\pi$
$769$	−19356.1	−0.907669	−0.453834	+	0.891086i	$0.649944\pi$
$770$	0	0
$771$	0	0
$772$	16251.8	0.757662
$773$	4799.04	0.223298	0.111649	−	0.993748i	$-0.464387\pi$
$773$	4799.04	0.223298	0.111649	+	0.993748i	$0.464387\pi$
$774$	0	0
$775$	0	0
$776$	762.580	0.0352771
$777$	0	0
$778$	−10510.9	−0.484363
$779$	−4184.48	−0.192458
$780$	0	0
$781$	−2911.97	−0.133417
$782$	−7763.37	−0.355010
$783$	0	0
$784$	−25795.3	−1.17508
$785$	0	0
$786$	0	0
$787$	−22195.1	−1.00530	−0.502648	−	0.864491i	$-0.667641\pi$
$787$	−22195.1	−1.00530	−0.502648	+	0.864491i	$0.667641\pi$
$788$	−20981.2	−0.948507
$789$	0	0
$790$	0	0
$791$	24580.5	1.10491
$792$	0	0
$793$	12963.6	0.580519
$794$	43520.0	1.94517
$795$	0	0
$796$	−9581.36	−0.426636
$797$	−22630.2	−1.00578	−0.502888	−	0.864352i	$-0.667729\pi$
$797$	−22630.2	−1.00578	−0.502888	+	0.864352i	$0.667729\pi$
$798$	0	0
$799$	−57098.3	−2.52815
$800$	0	0
$801$	0	0
$802$	18311.9	0.806255
$803$	−3682.85	−0.161849
$804$	0	0
$805$	0	0
$806$	40005.1	1.74829
$807$	0	0
$808$	−788.420	−0.0343274
$809$	−17563.6	−0.763293	−0.381647	−	0.924308i	$-0.624643\pi$
$809$	−17563.6	−0.763293	−0.381647	+	0.924308i	$0.624643\pi$
$810$	0	0
$811$	6370.17	0.275816	0.137908	−	0.990445i	$-0.455962\pi$
$811$	6370.17	0.275816	0.137908	+	0.990445i	$0.455962\pi$
$812$	20773.2	0.897777
$813$	0	0
$814$	15122.5	0.651157
$815$	0	0
$816$	0	0
$817$	10348.2	0.443131
$818$	−31632.0	−1.35206
$819$	0	0
$820$	0	0
$821$	−20761.6	−0.882562	−0.441281	−	0.897369i	$-0.645476\pi$
$821$	−20761.6	−0.882562	−0.441281	+	0.897369i	$0.645476\pi$
$822$	0	0
$823$	−666.345	−0.0282228	−0.0141114	−	0.999900i	$-0.504492\pi$
$823$	−666.345	−0.0282228	−0.0141114	+	0.999900i	$0.504492\pi$
$824$	−1304.62	−0.0551562
$825$	0	0
$826$	−3507.08	−0.147732
$827$	−27888.5	−1.17265	−0.586323	−	0.810078i	$-0.699425\pi$
$827$	−27888.5	−1.17265	−0.586323	+	0.810078i	$0.699425\pi$
$828$	0	0
$829$	−24909.8	−1.04361	−0.521804	−	0.853065i	$-0.674741\pi$
$829$	−24909.8	−1.04361	−0.521804	+	0.853065i	$0.674741\pi$
$830$	0	0
$831$	0	0
$832$	−38860.9	−1.61930
$833$	38960.1	1.62051
$834$	0	0
$835$	0	0
$836$	−3256.27	−0.134714
$837$	0	0
$838$	−55544.0	−2.28966
$839$	7039.78	0.289679	0.144839	−	0.989455i	$-0.453733\pi$
$839$	7039.78	0.289679	0.144839	+	0.989455i	$0.453733\pi$
$840$	0	0
$841$	−14343.0	−0.588092
$842$	21002.3	0.859606
$843$	0	0
$844$	7244.86	0.295472
$845$	0	0
$846$	0	0
$847$	3270.62	0.132680
$848$	40193.3	1.62765
$849$	0	0
$850$	0	0
$851$	6777.33	0.273001
$852$	0	0
$853$	15202.7	0.610235	0.305118	−	0.952315i	$-0.401304\pi$
$853$	15202.7	0.610235	0.305118	+	0.952315i	$0.401304\pi$
$854$	16725.1	0.670167
$855$	0	0
$856$	−1141.49	−0.0455785
$857$	−6963.28	−0.277551	−0.138776	−	0.990324i	$-0.544317\pi$
$857$	−6963.28	−0.277551	−0.138776	+	0.990324i	$0.544317\pi$
$858$	0	0
$859$	−25295.1	−1.00472	−0.502362	−	0.864658i	$-0.667535\pi$
$859$	−25295.1	−1.00472	−0.502362	+	0.864658i	$0.667535\pi$
$860$	0	0
$861$	0	0
$862$	−65149.5	−2.57425
$863$	−40402.3	−1.59364	−0.796819	−	0.604219i	$-0.793485\pi$
$863$	−40402.3	−1.59364	−0.796819	+	0.604219i	$0.793485\pi$
$864$	0	0
$865$	0	0
$866$	24298.1	0.953444
$867$	0	0
$868$	25259.1	0.987729
$869$	−14163.2	−0.552882
$870$	0	0
$871$	61824.8	2.40511
$872$	1726.64	0.0670543
$873$	0	0
$874$	−2981.94	−0.115407
$875$	0	0
$876$	0	0
$877$	15316.6	0.589744	0.294872	−	0.955537i	$-0.404723\pi$
$877$	15316.6	0.589744	0.294872	+	0.955537i	$0.404723\pi$
$878$	−53687.7	−2.06363
$879$	0	0
$880$	0	0
$881$	19661.6	0.751892	0.375946	−	0.926642i	$-0.377318\pi$
$881$	19661.6	0.751892	0.375946	+	0.926642i	$0.377318\pi$
$882$	0	0
$883$	44446.5	1.69393	0.846966	−	0.531646i	$-0.178426\pi$
$883$	44446.5	1.69393	0.846966	+	0.531646i	$0.178426\pi$
$884$	63911.9	2.43166
$885$	0	0
$886$	10559.3	0.400392
$887$	−8495.24	−0.321581	−0.160790	−	0.986989i	$-0.551404\pi$
$887$	−8495.24	−0.321581	−0.160790	+	0.986989i	$0.551404\pi$
$888$	0	0
$889$	−9178.78	−0.346284
$890$	0	0
$891$	0	0
$892$	−40271.8	−1.51166
$893$	−21931.7	−0.821855
$894$	0	0
$895$	0	0
$896$	4547.93	0.169571
$897$	0	0
$898$	−22265.6	−0.827409
$899$	12215.4	0.453179
$900$	0	0
$901$	−60706.3	−2.24464
$902$	4719.20	0.174204
$903$	0	0
$904$	−1196.41	−0.0440178
$905$	0	0
$906$	0	0
$907$	−27175.1	−0.994857	−0.497428	−	0.867505i	$-0.665722\pi$
$907$	−27175.1	−0.994857	−0.497428	+	0.867505i	$0.665722\pi$
$908$	−4421.41	−0.161597
$909$	0	0
$910$	0	0
$911$	15258.1	0.554911	0.277455	−	0.960738i	$-0.410509\pi$
$911$	15258.1	0.554911	0.277455	+	0.960738i	$0.410509\pi$
$912$	0	0
$913$	−1398.42	−0.0506911
$914$	−43496.5	−1.57411
$915$	0	0
$916$	−35402.7	−1.27701
$917$	−40624.9	−1.46298
$918$	0	0
$919$	1227.88	0.0440739	0.0220370	−	0.999757i	$-0.492985\pi$
$919$	1227.88	0.0440739	0.0220370	+	0.999757i	$0.492985\pi$
$920$	0	0
$921$	0	0
$922$	−7109.37	−0.253942
$923$	21953.2	0.782878
$924$	0	0
$925$	0	0
$926$	2851.13	0.101181
$927$	0	0
$928$	−25347.1	−0.896616
$929$	−27542.1	−0.972687	−0.486344	−	0.873768i	$-0.661670\pi$
$929$	−27542.1	−0.972687	−0.486344	+	0.873768i	$0.661670\pi$
$930$	0	0
$931$	14964.7	0.526799
$932$	−22366.0	−0.786075
$933$	0	0
$934$	−521.888	−0.0182834
$935$	0	0
$936$	0	0
$937$	−57180.2	−1.99359	−0.996796	−	0.0799827i	$-0.974513\pi$
$937$	−57180.2	−1.99359	−0.996796	+	0.0799827i	$0.974513\pi$
$938$	79764.0	2.77653
$939$	0	0
$940$	0	0
$941$	−23474.8	−0.813237	−0.406618	−	0.913598i	$-0.633292\pi$
$941$	−23474.8	−0.813237	−0.406618	+	0.913598i	$0.633292\pi$
$942$	0	0
$943$	2114.97	0.0730360
$944$	2181.41	0.0752107
$945$	0	0
$946$	−11670.6	−0.401103
$947$	−13294.2	−0.456180	−0.228090	−	0.973640i	$-0.573248\pi$
$947$	−13294.2	−0.456180	−0.228090	+	0.973640i	$0.573248\pi$
$948$	0	0
$949$	27764.8	0.949718
$950$	0	0
$951$	0	0
$952$	−3574.35	−0.121686
$953$	16727.6	0.568582	0.284291	−	0.958738i	$-0.408242\pi$
$953$	16727.6	0.568582	0.284291	+	0.958738i	$0.408242\pi$
$954$	0	0
$955$	0	0
$956$	4980.58	0.168497
$957$	0	0
$958$	−59433.8	−2.00440
$959$	−58498.8	−1.96978
$960$	0	0
$961$	−14937.7	−0.501416
$962$	−114007.	−3.82093
$963$	0	0
$964$	9254.59	0.309201
$965$	0	0
$966$	0	0
$967$	−17648.1	−0.586893	−0.293446	−	0.955976i	$-0.594802\pi$
$967$	−17648.1	−0.586893	−0.293446	+	0.955976i	$0.594802\pi$
$968$	−159.191	−0.00528575
$969$	0	0
$970$	0	0
$971$	1627.65	0.0537939	0.0268970	−	0.999638i	$-0.491437\pi$
$971$	1627.65	0.0537939	0.0268970	+	0.999638i	$0.491437\pi$
$972$	0	0
$973$	−10545.3	−0.347447
$974$	−39841.1	−1.31067
$975$	0	0
$976$	−10403.1	−0.341183
$977$	10385.4	0.340080	0.170040	−	0.985437i	$-0.445610\pi$
$977$	10385.4	0.340080	0.170040	+	0.985437i	$0.445610\pi$
$978$	0	0
$979$	−6773.98	−0.221141
$980$	0	0
$981$	0	0
$982$	−31860.4	−1.03534
$983$	−26696.5	−0.866211	−0.433105	−	0.901343i	$-0.642582\pi$
$983$	−26696.5	−0.866211	−0.433105	+	0.901343i	$0.642582\pi$
$984$	0	0
$985$	0	0
$986$	39876.5	1.28796
$987$	0	0
$988$	24548.8	0.790488
$989$	−5230.32	−0.168164
$990$	0	0
$991$	56889.6	1.82357	0.911784	−	0.410670i	$-0.134705\pi$
$991$	56889.6	1.82357	0.911784	+	0.410670i	$0.134705\pi$
$992$	−30820.7	−0.986451
$993$	0	0
$994$	28323.1	0.903776
$995$	0	0
$996$	0	0
$997$	34587.2	1.09868	0.549341	−	0.835598i	$-0.314879\pi$
$997$	34587.2	1.09868	0.549341	+	0.835598i	$0.314879\pi$
$998$	−30431.6	−0.965227
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bg.1.5		5
3.2	odd	2		275.4.a.i.1.1	yes	5
5.4	even	2		2475.4.a.bk.1.1		5
15.2	even	4		275.4.b.g.199.2		10
15.8	even	4		275.4.b.g.199.9		10
15.14	odd	2		275.4.a.f.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.f.1.5	✓	5	15.14	odd	2
275.4.a.i.1.1	yes	5	3.2	odd	2
275.4.b.g.199.2		10	15.2	even	4
275.4.b.g.199.9		10	15.8	even	4
2475.4.a.bg.1.5		5	1.1	even	1	trivial
2475.4.a.bk.1.1		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.95823 q^{2} +7.66762 q^{4} +27.0299 q^{7} -1.31563 q^{8} +O(q^{10})\)	\(q+3.95823 q^{2} +7.66762 q^{4} +27.0299 q^{7} -1.31563 q^{8} -11.0000 q^{11} +82.9282 q^{13} +106.991 q^{14} -66.5486 q^{16} +100.512 q^{17} +38.6071 q^{19} -43.5406 q^{22} -19.5133 q^{23} +328.249 q^{26} +207.255 q^{28} +100.230 q^{29} +121.874 q^{31} -252.890 q^{32} +397.851 q^{34} -347.319 q^{37} +152.816 q^{38} -108.386 q^{41} +268.039 q^{43} -84.3438 q^{44} -77.2381 q^{46} -568.074 q^{47} +387.616 q^{49} +635.862 q^{52} -603.970 q^{53} -35.5614 q^{56} +396.734 q^{58} -32.7793 q^{59} +156.323 q^{61} +482.407 q^{62} -468.608 q^{64} +745.522 q^{67} +770.689 q^{68} +264.725 q^{71} +334.805 q^{73} -1374.77 q^{74} +296.025 q^{76} -297.329 q^{77} +1287.57 q^{79} -429.018 q^{82} +127.129 q^{83} +1060.96 q^{86} +14.4720 q^{88} +615.817 q^{89} +2241.54 q^{91} -149.620 q^{92} -2248.57 q^{94} -579.630 q^{97} +1534.28 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 12 q^{4} + 24 q^{7} - 27 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 12 * q^4 + 24 * q^7 - 27 * q^8	\( 5 q - 2 q^{2} + 12 q^{4} + 24 q^{7} - 27 q^{8} - 55 q^{11} + 111 q^{13} - 47 q^{14} - 56 q^{16} - 40 q^{17} + 205 q^{19} + 22 q^{22} - 287 q^{23} + 354 q^{26} + 460 q^{28} - 251 q^{29} - 289 q^{31} - 248 q^{32} + 522 q^{34} + 224 q^{37} - 540 q^{38} + 462 q^{41} + 593 q^{43} - 132 q^{44} - 972 q^{46} - 766 q^{47} + 75 q^{49} - 696 q^{53} + 527 q^{56} + 1461 q^{58} + 22 q^{59} + 720 q^{61} + 998 q^{62} - 317 q^{64} + 1230 q^{67} + 109 q^{68} - 951 q^{71} + 666 q^{73} - 873 q^{74} + 290 q^{76} - 264 q^{77} - 588 q^{79} - 1807 q^{82} + 867 q^{83} + 411 q^{86} + 297 q^{88} + 51 q^{89} + 2172 q^{91} + 4137 q^{92} - 865 q^{94} + 2849 q^{97} + 4104 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 12 * q^4 + 24 * q^7 - 27 * q^8 - 55 * q^11 + 111 * q^13 - 47 * q^14 - 56 * q^16 - 40 * q^17 + 205 * q^19 + 22 * q^22 - 287 * q^23 + 354 * q^26 + 460 * q^28 - 251 * q^29 - 289 * q^31 - 248 * q^32 + 522 * q^34 + 224 * q^37 - 540 * q^38 + 462 * q^41 + 593 * q^43 - 132 * q^44 - 972 * q^46 - 766 * q^47 + 75 * q^49 - 696 * q^53 + 527 * q^56 + 1461 * q^58 + 22 * q^59 + 720 * q^61 + 998 * q^62 - 317 * q^64 + 1230 * q^67 + 109 * q^68 - 951 * q^71 + 666 * q^73 - 873 * q^74 + 290 * q^76 - 264 * q^77 - 588 * q^79 - 1807 * q^82 + 867 * q^83 + 411 * q^86 + 297 * q^88 + 51 * q^89 + 2172 * q^91 + 4137 * q^92 - 865 * q^94 + 2849 * q^97 + 4104 * q^98

Introduction

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