LMFDB - Embedded newform 2475.4.a.bt.1.4

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 $ x^7 - 2x^6 - 41x^5 + 40x^4 + 424x^3 - 168x^2 - 1042x - 388
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.434062$ 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)

$f(q)$	$=$	$q+1.43406 q^{2} -5.94347 q^{4} +23.1010 q^{7} -19.9958 q^{8} +O(q^{10})$	$q+1.43406 q^{2} -5.94347 q^{4} +23.1010 q^{7} -19.9958 q^{8} -11.0000 q^{11} +85.3360 q^{13} +33.1283 q^{14} +18.8725 q^{16} +109.083 q^{17} +121.498 q^{19} -15.7747 q^{22} +171.169 q^{23} +122.377 q^{26} -137.300 q^{28} -80.1854 q^{29} -255.029 q^{31} +187.031 q^{32} +156.432 q^{34} -103.684 q^{37} +174.236 q^{38} +194.671 q^{41} -434.049 q^{43} +65.3781 q^{44} +245.467 q^{46} -459.463 q^{47} +190.656 q^{49} -507.191 q^{52} +656.774 q^{53} -461.923 q^{56} -114.991 q^{58} +158.957 q^{59} +107.192 q^{61} -365.727 q^{62} +117.234 q^{64} -685.810 q^{67} -648.332 q^{68} -671.377 q^{71} +260.221 q^{73} -148.689 q^{74} -722.121 q^{76} -254.111 q^{77} +361.102 q^{79} +279.170 q^{82} +706.521 q^{83} -622.453 q^{86} +219.954 q^{88} +557.720 q^{89} +1971.35 q^{91} -1017.34 q^{92} -658.898 q^{94} -482.897 q^{97} +273.413 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8}+O(q^{10}) $ 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8	$ 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8} - 77 q^{11} - 38 q^{13} - 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} - 55 q^{22} + 334 q^{23} + 372 q^{26} - 812 q^{28} + 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} - 466 q^{37} - 494 q^{38} + 258 q^{41} - 308 q^{43} - 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} - 512 q^{52} + 110 q^{53} - 20 q^{56} - 1362 q^{58} + 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} - 2268 q^{67} - 1186 q^{68} + 166 q^{71} - 200 q^{73} + 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1006 q^{82} + 370 q^{83} - 106 q^{86} - 495 q^{88} + 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} - 3698 q^{97} + 697 q^{98}+O(q^{100}) $ 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8 - 77 * q^11 - 38 * q^13 - 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 - 55 * q^22 + 334 * q^23 + 372 * q^26 - 812 * q^28 + 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 - 466 * q^37 - 494 * q^38 + 258 * q^41 - 308 * q^43 - 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 - 512 * q^52 + 110 * q^53 - 20 * q^56 - 1362 * q^58 + 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 - 2268 * q^67 - 1186 * q^68 + 166 * q^71 - 200 * q^73 + 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1006 * q^82 + 370 * q^83 - 106 * q^86 - 495 * q^88 + 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 - 3698 * q^97 + 697 * 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 $

$2$	1.43406	0.507018	0.253509	−	0.967333i	$-0.418415\pi$
$2$	1.43406	0.507018	0.253509	+	0.967333i	$0.418415\pi$
$3$	0	0
$3$	0	0
$4$	−5.94347	−0.742933
$5$	0	0
$5$	0	0
$6$	0	0
$7$	23.1010	1.24734	0.623668	−	0.781689i	$-0.285642\pi$
$7$	23.1010	1.24734	0.623668	+	0.781689i	$0.285642\pi$
$8$	−19.9958	−0.883698
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	85.3360	1.82061	0.910305	−	0.413938i	$-0.135847\pi$
$13$	85.3360	1.82061	0.910305	+	0.413938i	$0.135847\pi$
$14$	33.1283	0.632422
$15$	0	0
$16$	18.8725	0.294883
$17$	109.083	1.55627	0.778134	−	0.628098i	$-0.216167\pi$
$17$	109.083	1.55627	0.778134	+	0.628098i	$0.216167\pi$
$18$	0	0
$19$	121.498	1.46703	0.733516	−	0.679672i	$-0.237878\pi$
$19$	121.498	1.46703	0.733516	+	0.679672i	$0.237878\pi$
$20$	0	0
$21$	0	0
$22$	−15.7747	−0.152872
$23$	171.169	1.55179	0.775896	−	0.630860i	$-0.217298\pi$
$23$	171.169	1.55179	0.775896	+	0.630860i	$0.217298\pi$
$24$	0	0
$25$	0	0
$26$	122.377	0.923082
$27$	0	0
$28$	−137.300	−0.926688
$29$	−80.1854	−0.513450	−0.256725	−	0.966485i	$-0.582643\pi$
$29$	−80.1854	−0.513450	−0.256725	+	0.966485i	$0.582643\pi$
$30$	0	0
$31$	−255.029	−1.47756	−0.738782	−	0.673944i	$-0.764599\pi$
$31$	−255.029	−1.47756	−0.738782	+	0.673944i	$0.764599\pi$
$32$	187.031	1.03321
$33$	0	0
$34$	156.432	0.789055
$35$	0	0
$36$	0	0
$37$	−103.684	−0.460689	−0.230345	−	0.973109i	$-0.573985\pi$
$37$	−103.684	−0.460689	−0.230345	+	0.973109i	$0.573985\pi$
$38$	174.236	0.743811
$39$	0	0
$40$	0	0
$41$	194.671	0.741523	0.370761	−	0.928728i	$-0.379097\pi$
$41$	194.671	0.741523	0.370761	+	0.928728i	$0.379097\pi$
$42$	0	0
$43$	−434.049	−1.53934	−0.769672	−	0.638439i	$-0.779580\pi$
$43$	−434.049	−1.53934	−0.769672	+	0.638439i	$0.779580\pi$
$44$	65.3781	0.224003
$45$	0	0
$46$	245.467	0.786786
$47$	−459.463	−1.42595	−0.712973	−	0.701191i	$-0.752652\pi$
$47$	−459.463	−1.42595	−0.712973	+	0.701191i	$0.752652\pi$
$48$	0	0
$49$	190.656	0.555849
$50$	0	0
$51$	0	0
$52$	−507.191	−1.35259
$53$	656.774	1.70217	0.851084	−	0.525029i	$-0.175946\pi$
$53$	656.774	1.70217	0.851084	+	0.525029i	$0.175946\pi$
$54$	0	0
$55$	0	0
$56$	−461.923	−1.10227
$57$	0	0
$58$	−114.991	−0.260328
$59$	158.957	0.350753	0.175377	−	0.984501i	$-0.443886\pi$
$59$	158.957	0.350753	0.175377	+	0.984501i	$0.443886\pi$
$60$	0	0
$61$	107.192	0.224992	0.112496	−	0.993652i	$-0.464115\pi$
$61$	107.192	0.224992	0.112496	+	0.993652i	$0.464115\pi$
$62$	−365.727	−0.749151
$63$	0	0
$64$	117.234	0.228972
$65$	0	0
$66$	0	0
$67$	−685.810	−1.25052	−0.625261	−	0.780416i	$-0.715008\pi$
$67$	−685.810	−1.25052	−0.625261	+	0.780416i	$0.715008\pi$
$68$	−648.332	−1.15620
$69$	0	0
$70$	0	0
$71$	−671.377	−1.12222	−0.561111	−	0.827741i	$-0.689626\pi$
$71$	−671.377	−1.12222	−0.561111	+	0.827741i	$0.689626\pi$
$72$	0	0
$73$	260.221	0.417213	0.208606	−	0.978000i	$-0.433107\pi$
$73$	260.221	0.417213	0.208606	+	0.978000i	$0.433107\pi$
$74$	−148.689	−0.233578
$75$	0	0
$76$	−722.121	−1.08991
$77$	−254.111	−0.376086
$78$	0	0
$79$	361.102	0.514267	0.257134	−	0.966376i	$-0.417222\pi$
$79$	361.102	0.514267	0.257134	+	0.966376i	$0.417222\pi$
$80$	0	0
$81$	0	0
$82$	279.170	0.375965
$83$	706.521	0.934346	0.467173	−	0.884166i	$-0.345273\pi$
$83$	706.521	0.934346	0.467173	+	0.884166i	$0.345273\pi$
$84$	0	0
$85$	0	0
$86$	−622.453	−0.780475
$87$	0	0
$88$	219.954	0.266445
$89$	557.720	0.664249	0.332124	−	0.943236i	$-0.392235\pi$
$89$	557.720	0.664249	0.332124	+	0.943236i	$0.392235\pi$
$90$	0	0
$91$	1971.35	2.27091
$92$	−1017.34	−1.15288
$93$	0	0
$94$	−658.898	−0.722980
$95$	0	0
$96$	0	0
$97$	−482.897	−0.505472	−0.252736	−	0.967535i	$-0.581330\pi$
$97$	−482.897	−0.505472	−0.252736	+	0.967535i	$0.581330\pi$
$98$	273.413	0.281825
$99$	0	0
$100$	0	0
$101$	900.150	0.886815	0.443408	−	0.896320i	$-0.353769\pi$
$101$	900.150	0.886815	0.443408	+	0.896320i	$0.353769\pi$
$102$	0	0
$103$	−757.943	−0.725072	−0.362536	−	0.931970i	$-0.618089\pi$
$103$	−757.943	−0.725072	−0.362536	+	0.931970i	$0.618089\pi$
$104$	−1706.36	−1.60887
$105$	0	0
$106$	941.856	0.863029
$107$	−793.166	−0.716619	−0.358310	−	0.933603i	$-0.616647\pi$
$107$	−793.166	−0.716619	−0.358310	+	0.933603i	$0.616647\pi$
$108$	0	0
$109$	1232.47	1.08302	0.541508	−	0.840696i	$-0.317854\pi$
$109$	1232.47	1.08302	0.541508	+	0.840696i	$0.317854\pi$
$110$	0	0
$111$	0	0
$112$	435.974	0.367818
$113$	1255.66	1.04533	0.522664	−	0.852539i	$-0.324938\pi$
$113$	1255.66	1.04533	0.522664	+	0.852539i	$0.324938\pi$
$114$	0	0
$115$	0	0
$116$	476.579	0.381459
$117$	0	0
$118$	227.954	0.177838
$119$	2519.93	1.94119
$120$	0	0
$121$	121.000	0.0909091
$122$	153.720	0.114075
$123$	0	0
$124$	1515.75	1.09773
$125$	0	0
$126$	0	0
$127$	2021.58	1.41249	0.706245	−	0.707968i	$-0.250388\pi$
$127$	2021.58	1.41249	0.706245	+	0.707968i	$0.250388\pi$
$128$	−1328.13	−0.917116
$129$	0	0
$130$	0	0
$131$	2782.03	1.85548	0.927738	−	0.373232i	$-0.121750\pi$
$131$	2782.03	1.85548	0.927738	+	0.373232i	$0.121750\pi$
$132$	0	0
$133$	2806.73	1.82988
$134$	−983.494	−0.634037
$135$	0	0
$136$	−2181.20	−1.37527
$137$	481.615	0.300344	0.150172	−	0.988660i	$-0.452017\pi$
$137$	481.615	0.300344	0.150172	+	0.988660i	$0.452017\pi$
$138$	0	0
$139$	−252.051	−0.153803	−0.0769017	−	0.997039i	$-0.524503\pi$
$139$	−252.051	−0.153803	−0.0769017	+	0.997039i	$0.524503\pi$
$140$	0	0
$141$	0	0
$142$	−962.796	−0.568986
$143$	−938.696	−0.548935
$144$	0	0
$145$	0	0
$146$	373.173	0.211534
$147$	0	0
$148$	616.241	0.342261
$149$	−2393.86	−1.31619	−0.658096	−	0.752934i	$-0.728638\pi$
$149$	−2393.86	−1.31619	−0.658096	+	0.752934i	$0.728638\pi$
$150$	0	0
$151$	−2799.55	−1.50877	−0.754384	−	0.656434i	$-0.772064\pi$
$151$	−2799.55	−1.50877	−0.754384	+	0.656434i	$0.772064\pi$
$152$	−2429.45	−1.29641
$153$	0	0
$154$	−364.411	−0.190682
$155$	0	0
$156$	0	0
$157$	811.381	0.412454	0.206227	−	0.978504i	$-0.433882\pi$
$157$	811.381	0.412454	0.206227	+	0.978504i	$0.433882\pi$
$158$	517.842	0.260743
$159$	0	0
$160$	0	0
$161$	3954.18	1.93561
$162$	0	0
$163$	−401.857	−0.193103	−0.0965517	−	0.995328i	$-0.530781\pi$
$163$	−401.857	−0.193103	−0.0965517	+	0.995328i	$0.530781\pi$
$164$	−1157.02	−0.550902
$165$	0	0
$166$	1013.19	0.473730
$167$	2483.79	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$167$	2483.79	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$168$	0	0
$169$	5085.23	2.31462
$170$	0	0
$171$	0	0
$172$	2579.75	1.14363
$173$	2068.32	0.908967	0.454483	−	0.890755i	$-0.349824\pi$
$173$	2068.32	0.908967	0.454483	+	0.890755i	$0.349824\pi$
$174$	0	0
$175$	0	0
$176$	−207.598	−0.0889105
$177$	0	0
$178$	799.805	0.336786
$179$	−2584.76	−1.07930	−0.539648	−	0.841891i	$-0.681443\pi$
$179$	−2584.76	−1.07930	−0.539648	+	0.841891i	$0.681443\pi$
$180$	0	0
$181$	2522.42	1.03586	0.517929	−	0.855424i	$-0.326703\pi$
$181$	2522.42	1.03586	0.517929	+	0.855424i	$0.326703\pi$
$182$	2827.03	1.15139
$183$	0	0
$184$	−3422.66	−1.37132
$185$	0	0
$186$	0	0
$187$	−1199.91	−0.469232
$188$	2730.80	1.05938
$189$	0	0
$190$	0	0
$191$	−2240.14	−0.848644	−0.424322	−	0.905511i	$-0.639487\pi$
$191$	−2240.14	−0.848644	−0.424322	+	0.905511i	$0.639487\pi$
$192$	0	0
$193$	−2692.31	−1.00413	−0.502064	−	0.864831i	$-0.667426\pi$
$193$	−2692.31	−1.00413	−0.502064	+	0.864831i	$0.667426\pi$
$194$	−692.504	−0.256283
$195$	0	0
$196$	−1133.16	−0.412959
$197$	−1843.20	−0.666612	−0.333306	−	0.942819i	$-0.608164\pi$
$197$	−1843.20	−0.666612	−0.333306	+	0.942819i	$0.608164\pi$
$198$	0	0
$199$	−3094.01	−1.10215	−0.551076	−	0.834455i	$-0.685783\pi$
$199$	−3094.01	−1.10215	−0.551076	+	0.834455i	$0.685783\pi$
$200$	0	0
$201$	0	0
$202$	1290.87	0.449631
$203$	−1852.36	−0.640445
$204$	0	0
$205$	0	0
$206$	−1086.94	−0.367624
$207$	0	0
$208$	1610.50	0.536867
$209$	−1336.48	−0.442327
$210$	0	0
$211$	−380.009	−0.123985	−0.0619927	−	0.998077i	$-0.519746\pi$
$211$	−380.009	−0.123985	−0.0619927	+	0.998077i	$0.519746\pi$
$212$	−3903.52	−1.26460
$213$	0	0
$214$	−1137.45	−0.363338
$215$	0	0
$216$	0	0
$217$	−5891.42	−1.84302
$218$	1767.43	0.549108
$219$	0	0
$220$	0	0
$221$	9308.72	2.83336
$222$	0	0
$223$	2016.16	0.605436	0.302718	−	0.953080i	$-0.402106\pi$
$223$	2016.16	0.605436	0.302718	+	0.953080i	$0.402106\pi$
$224$	4320.60	1.28876
$225$	0	0
$226$	1800.69	0.530000
$227$	−1813.60	−0.530278	−0.265139	−	0.964210i	$-0.585418\pi$
$227$	−1813.60	−0.530278	−0.265139	+	0.964210i	$0.585418\pi$
$228$	0	0
$229$	−895.092	−0.258294	−0.129147	−	0.991625i	$-0.541224\pi$
$229$	−895.092	−0.258294	−0.129147	+	0.991625i	$0.541224\pi$
$230$	0	0
$231$	0	0
$232$	1603.37	0.453735
$233$	4981.99	1.40078	0.700389	−	0.713761i	$-0.253010\pi$
$233$	4981.99	1.40078	0.700389	+	0.713761i	$0.253010\pi$
$234$	0	0
$235$	0	0
$236$	−944.756	−0.260586
$237$	0	0
$238$	3613.74	0.984217
$239$	2692.91	0.728829	0.364415	−	0.931237i	$-0.381269\pi$
$239$	2692.91	0.728829	0.364415	+	0.931237i	$0.381269\pi$
$240$	0	0
$241$	−6472.03	−1.72988	−0.864938	−	0.501878i	$-0.832643\pi$
$241$	−6472.03	−1.72988	−0.864938	+	0.501878i	$0.832643\pi$
$242$	173.522	0.0460925
$243$	0	0
$244$	−637.090	−0.167154
$245$	0	0
$246$	0	0
$247$	10368.2	2.67089
$248$	5099.50	1.30572
$249$	0	0
$250$	0	0
$251$	−5949.61	−1.49616	−0.748080	−	0.663608i	$-0.769024\pi$
$251$	−5949.61	−1.49616	−0.748080	+	0.663608i	$0.769024\pi$
$252$	0	0
$253$	−1882.86	−0.467883
$254$	2899.07	0.716157
$255$	0	0
$256$	−2842.48	−0.693966
$257$	4983.32	1.20954	0.604768	−	0.796401i	$-0.293266\pi$
$257$	4983.32	1.20954	0.604768	+	0.796401i	$0.293266\pi$
$258$	0	0
$259$	−2395.20	−0.574635
$260$	0	0
$261$	0	0
$262$	3989.61	0.940759
$263$	415.329	0.0973775	0.0486888	−	0.998814i	$-0.484496\pi$
$263$	415.329	0.0973775	0.0486888	+	0.998814i	$0.484496\pi$
$264$	0	0
$265$	0	0
$266$	4025.03	0.927783
$267$	0	0
$268$	4076.09	0.929054
$269$	959.649	0.217512	0.108756	−	0.994068i	$-0.465313\pi$
$269$	959.649	0.217512	0.108756	+	0.994068i	$0.465313\pi$
$270$	0	0
$271$	−997.792	−0.223659	−0.111829	−	0.993727i	$-0.535671\pi$
$271$	−997.792	−0.223659	−0.111829	+	0.993727i	$0.535671\pi$
$272$	2058.67	0.458917
$273$	0	0
$274$	690.666	0.152280
$275$	0	0
$276$	0	0
$277$	−7652.53	−1.65991	−0.829957	−	0.557828i	$-0.811635\pi$
$277$	−7652.53	−1.65991	−0.829957	+	0.557828i	$0.811635\pi$
$278$	−361.457	−0.0779811
$279$	0	0
$280$	0	0
$281$	−807.627	−0.171455	−0.0857277	−	0.996319i	$-0.527322\pi$
$281$	−807.627	−0.171455	−0.0857277	+	0.996319i	$0.527322\pi$
$282$	0	0
$283$	5795.36	1.21731	0.608654	−	0.793435i	$-0.291710\pi$
$283$	5795.36	1.21731	0.608654	+	0.793435i	$0.291710\pi$
$284$	3990.30	0.833736
$285$	0	0
$286$	−1346.15	−0.278320
$287$	4497.08	0.924929
$288$	0	0
$289$	6986.13	1.42197
$290$	0	0
$291$	0	0
$292$	−1546.61	−0.309961
$293$	−4286.00	−0.854576	−0.427288	−	0.904115i	$-0.640531\pi$
$293$	−4286.00	−0.854576	−0.427288	+	0.904115i	$0.640531\pi$
$294$	0	0
$295$	0	0
$296$	2073.24	0.407110
$297$	0	0
$298$	−3432.95	−0.667333
$299$	14606.9	2.82521
$300$	0	0
$301$	−10027.0	−1.92008
$302$	−4014.72	−0.764972
$303$	0	0
$304$	2292.98	0.432603
$305$	0	0
$306$	0	0
$307$	−8830.25	−1.64159	−0.820796	−	0.571221i	$-0.806470\pi$
$307$	−8830.25	−1.64159	−0.820796	+	0.571221i	$0.806470\pi$
$308$	1510.30	0.279407
$309$	0	0
$310$	0	0
$311$	3446.59	0.628419	0.314210	−	0.949354i	$-0.398261\pi$
$311$	3446.59	0.628419	0.314210	+	0.949354i	$0.398261\pi$
$312$	0	0
$313$	4359.69	0.787298	0.393649	−	0.919261i	$-0.371213\pi$
$313$	4359.69	0.787298	0.393649	+	0.919261i	$0.371213\pi$
$314$	1163.57	0.209121
$315$	0	0
$316$	−2146.20	−0.382066
$317$	−3954.65	−0.700679	−0.350340	−	0.936623i	$-0.613934\pi$
$317$	−3954.65	−0.700679	−0.350340	+	0.936623i	$0.613934\pi$
$318$	0	0
$319$	882.039	0.154811
$320$	0	0
$321$	0	0
$322$	5670.54	0.981387
$323$	13253.4	2.28309
$324$	0	0
$325$	0	0
$326$	−576.287	−0.0979068
$327$	0	0
$328$	−3892.59	−0.655282
$329$	−10614.0	−1.77864
$330$	0	0
$331$	2845.54	0.472523	0.236262	−	0.971689i	$-0.424078\pi$
$331$	2845.54	0.472523	0.236262	+	0.971689i	$0.424078\pi$
$332$	−4199.18	−0.694157
$333$	0	0
$334$	3561.92	0.583531
$335$	0	0
$336$	0	0
$337$	−5477.00	−0.885314	−0.442657	−	0.896691i	$-0.645964\pi$
$337$	−5477.00	−0.885314	−0.442657	+	0.896691i	$0.645964\pi$
$338$	7292.53	1.17355
$339$	0	0
$340$	0	0
$341$	2805.32	0.445503
$342$	0	0
$343$	−3519.29	−0.554006
$344$	8679.15	1.36032
$345$	0	0
$346$	2966.10	0.460862
$347$	−4121.77	−0.637660	−0.318830	−	0.947812i	$-0.603290\pi$
$347$	−4121.77	−0.637660	−0.318830	+	0.947812i	$0.603290\pi$
$348$	0	0
$349$	10299.6	1.57973	0.789867	−	0.613279i	$-0.210150\pi$
$349$	10299.6	1.57973	0.789867	+	0.613279i	$0.210150\pi$
$350$	0	0
$351$	0	0
$352$	−2057.34	−0.311524
$353$	6574.93	0.991355	0.495677	−	0.868507i	$-0.334920\pi$
$353$	6574.93	0.991355	0.495677	+	0.868507i	$0.334920\pi$
$354$	0	0
$355$	0	0
$356$	−3314.79	−0.493493
$357$	0	0
$358$	−3706.71	−0.547222
$359$	8334.97	1.22536	0.612678	−	0.790332i	$-0.290092\pi$
$359$	8334.97	1.22536	0.612678	+	0.790332i	$0.290092\pi$
$360$	0	0
$361$	7902.83	1.15218
$362$	3617.31	0.525198
$363$	0	0
$364$	−11716.6	−1.68714
$365$	0	0
$366$	0	0
$367$	−1298.89	−0.184746	−0.0923729	−	0.995724i	$-0.529445\pi$
$367$	−1298.89	−0.184746	−0.0923729	+	0.995724i	$0.529445\pi$
$368$	3230.39	0.457597
$369$	0	0
$370$	0	0
$371$	15172.1	2.12318
$372$	0	0
$373$	5715.86	0.793448	0.396724	−	0.917938i	$-0.370147\pi$
$373$	5715.86	0.793448	0.396724	+	0.917938i	$0.370147\pi$
$374$	−1720.75	−0.237909
$375$	0	0
$376$	9187.32	1.26011
$377$	−6842.69	−0.934792
$378$	0	0
$379$	3446.62	0.467126	0.233563	−	0.972342i	$-0.424961\pi$
$379$	3446.62	0.467126	0.233563	+	0.972342i	$0.424961\pi$
$380$	0	0
$381$	0	0
$382$	−3212.50	−0.430277
$383$	5307.81	0.708137	0.354069	−	0.935219i	$-0.384798\pi$
$383$	5307.81	0.708137	0.354069	+	0.935219i	$0.384798\pi$
$384$	0	0
$385$	0	0
$386$	−3860.94	−0.509110
$387$	0	0
$388$	2870.08	0.375532
$389$	−8148.22	−1.06203	−0.531017	−	0.847361i	$-0.678190\pi$
$389$	−8148.22	−1.06203	−0.531017	+	0.847361i	$0.678190\pi$
$390$	0	0
$391$	18671.7	2.41501
$392$	−3812.32	−0.491202
$393$	0	0
$394$	−2643.26	−0.337984
$395$	0	0
$396$	0	0
$397$	9697.26	1.22592	0.612962	−	0.790113i	$-0.289978\pi$
$397$	9697.26	1.22592	0.612962	+	0.790113i	$0.289978\pi$
$398$	−4437.00	−0.558811
$399$	0	0
$400$	0	0
$401$	15273.3	1.90202	0.951012	−	0.309153i	$-0.100046\pi$
$401$	15273.3	1.90202	0.951012	+	0.309153i	$0.100046\pi$
$402$	0	0
$403$	−21763.1	−2.69007
$404$	−5350.01	−0.658844
$405$	0	0
$406$	−2656.40	−0.324717
$407$	1140.52	0.138903
$408$	0	0
$409$	4890.84	0.591287	0.295644	−	0.955298i	$-0.404466\pi$
$409$	4890.84	0.591287	0.295644	+	0.955298i	$0.404466\pi$
$410$	0	0
$411$	0	0
$412$	4504.81	0.538680
$413$	3672.07	0.437508
$414$	0	0
$415$	0	0
$416$	15960.4	1.88107
$417$	0	0
$418$	−1916.60	−0.224267
$419$	−9513.26	−1.10920	−0.554598	−	0.832118i	$-0.687128\pi$
$419$	−9513.26	−1.10920	−0.554598	+	0.832118i	$0.687128\pi$
$420$	0	0
$421$	9213.00	1.06654	0.533271	−	0.845945i	$-0.320963\pi$
$421$	9213.00	1.06654	0.533271	+	0.845945i	$0.320963\pi$
$422$	−544.957	−0.0628627
$423$	0	0
$424$	−13132.7	−1.50420
$425$	0	0
$426$	0	0
$427$	2476.24	0.280640
$428$	4714.15	0.532400
$429$	0	0
$430$	0	0
$431$	9388.13	1.04921	0.524606	−	0.851345i	$-0.324213\pi$
$431$	9388.13	1.04921	0.524606	+	0.851345i	$0.324213\pi$
$432$	0	0
$433$	−12962.4	−1.43864	−0.719321	−	0.694678i	$-0.755547\pi$
$433$	−12962.4	−1.43864	−0.719321	+	0.694678i	$0.755547\pi$
$434$	−8448.66	−0.934444
$435$	0	0
$436$	−7325.11	−0.804609
$437$	20796.8	2.27653
$438$	0	0
$439$	−1076.83	−0.117071	−0.0585357	−	0.998285i	$-0.518643\pi$
$439$	−1076.83	−0.117071	−0.0585357	+	0.998285i	$0.518643\pi$
$440$	0	0
$441$	0	0
$442$	13349.3	1.43656
$443$	2508.84	0.269072	0.134536	−	0.990909i	$-0.457046\pi$
$443$	2508.84	0.269072	0.134536	+	0.990909i	$0.457046\pi$
$444$	0	0
$445$	0	0
$446$	2891.30	0.306967
$447$	0	0
$448$	2708.22	0.285605
$449$	976.211	0.102606	0.0513032	−	0.998683i	$-0.483663\pi$
$449$	976.211	0.102606	0.0513032	+	0.998683i	$0.483663\pi$
$450$	0	0
$451$	−2141.38	−0.223578
$452$	−7462.94	−0.776609
$453$	0	0
$454$	−2600.82	−0.268860
$455$	0	0
$456$	0	0
$457$	−9967.11	−1.02022	−0.510111	−	0.860108i	$-0.670396\pi$
$457$	−9967.11	−1.02022	−0.510111	+	0.860108i	$0.670396\pi$
$458$	−1283.62	−0.130960
$459$	0	0
$460$	0	0
$461$	2123.41	0.214527	0.107264	−	0.994231i	$-0.465791\pi$
$461$	2123.41	0.214527	0.107264	+	0.994231i	$0.465791\pi$
$462$	0	0
$463$	7343.50	0.737109	0.368555	−	0.929606i	$-0.379853\pi$
$463$	7343.50	0.737109	0.368555	+	0.929606i	$0.379853\pi$
$464$	−1513.30	−0.151408
$465$	0	0
$466$	7144.49	0.710219
$467$	7351.06	0.728408	0.364204	−	0.931319i	$-0.381341\pi$
$467$	7351.06	0.728408	0.364204	+	0.931319i	$0.381341\pi$
$468$	0	0
$469$	−15842.9	−1.55982
$470$	0	0
$471$	0	0
$472$	−3178.47	−0.309960
$473$	4774.54	0.464130
$474$	0	0
$475$	0	0
$476$	−14977.1	−1.44217
$477$	0	0
$478$	3861.81	0.369529
$479$	−327.620	−0.0312512	−0.0156256	−	0.999878i	$-0.504974\pi$
$479$	−327.620	−0.0312512	−0.0156256	+	0.999878i	$0.504974\pi$
$480$	0	0
$481$	−8847.95	−0.838736
$482$	−9281.30	−0.877078
$483$	0	0
$484$	−719.159	−0.0675394
$485$	0	0
$486$	0	0
$487$	−12591.3	−1.17159	−0.585797	−	0.810458i	$-0.699218\pi$
$487$	−12591.3	−1.17159	−0.585797	+	0.810458i	$0.699218\pi$
$488$	−2143.38	−0.198825
$489$	0	0
$490$	0	0
$491$	−2124.55	−0.195274	−0.0976369	−	0.995222i	$-0.531128\pi$
$491$	−2124.55	−0.195274	−0.0976369	+	0.995222i	$0.531128\pi$
$492$	0	0
$493$	−8746.87	−0.799065
$494$	14868.6	1.35419
$495$	0	0
$496$	−4813.03	−0.435708
$497$	−15509.5	−1.39979
$498$	0	0
$499$	8119.15	0.728383	0.364192	−	0.931324i	$-0.381345\pi$
$499$	8119.15	0.728383	0.364192	+	0.931324i	$0.381345\pi$
$500$	0	0
$501$	0	0
$502$	−8532.12	−0.758580
$503$	−8466.22	−0.750477	−0.375238	−	0.926928i	$-0.622439\pi$
$503$	−8466.22	−0.750477	−0.375238	+	0.926928i	$0.622439\pi$
$504$	0	0
$505$	0	0
$506$	−2700.14	−0.237225
$507$	0	0
$508$	−12015.2	−1.04939
$509$	3047.74	0.265400	0.132700	−	0.991156i	$-0.457635\pi$
$509$	3047.74	0.265400	0.132700	+	0.991156i	$0.457635\pi$
$510$	0	0
$511$	6011.36	0.520405
$512$	6548.70	0.565263
$513$	0	0
$514$	7146.39	0.613256
$515$	0	0
$516$	0	0
$517$	5054.09	0.429939
$518$	−3434.86	−0.291350
$519$	0	0
$520$	0	0
$521$	−18982.8	−1.59626	−0.798130	−	0.602486i	$-0.794177\pi$
$521$	−18982.8	−1.59626	−0.798130	+	0.602486i	$0.794177\pi$
$522$	0	0
$523$	−4340.00	−0.362859	−0.181429	−	0.983404i	$-0.558072\pi$
$523$	−4340.00	−0.362859	−0.181429	+	0.983404i	$0.558072\pi$
$524$	−16534.9	−1.37849
$525$	0	0
$526$	595.608	0.0493721
$527$	−27819.3	−2.29949
$528$	0	0
$529$	17131.9	1.40806
$530$	0	0
$531$	0	0
$532$	−16681.7	−1.35948
$533$	16612.4	1.35002
$534$	0	0
$535$	0	0
$536$	13713.3	1.10508
$537$	0	0
$538$	1376.20	0.110283
$539$	−2097.22	−0.167595
$540$	0	0
$541$	10372.3	0.824288	0.412144	−	0.911119i	$-0.364780\pi$
$541$	10372.3	0.824288	0.412144	+	0.911119i	$0.364780\pi$
$542$	−1430.90	−0.113399
$543$	0	0
$544$	20401.9	1.60795
$545$	0	0
$546$	0	0
$547$	5859.58	0.458021	0.229010	−	0.973424i	$-0.426451\pi$
$547$	5859.58	0.458021	0.229010	+	0.973424i	$0.426451\pi$
$548$	−2862.46	−0.223136
$549$	0	0
$550$	0	0
$551$	−9742.38	−0.753248
$552$	0	0
$553$	8341.81	0.641464
$554$	−10974.2	−0.841606
$555$	0	0
$556$	1498.06	0.114266
$557$	−16395.2	−1.24720	−0.623598	−	0.781745i	$-0.714330\pi$
$557$	−16395.2	−1.24720	−0.623598	+	0.781745i	$0.714330\pi$
$558$	0	0
$559$	−37040.0	−2.80255
$560$	0	0
$561$	0	0
$562$	−1158.19	−0.0869309
$563$	7660.01	0.573412	0.286706	−	0.958019i	$-0.407440\pi$
$563$	7660.01	0.573412	0.286706	+	0.958019i	$0.407440\pi$
$564$	0	0
$565$	0	0
$566$	8310.91	0.617197
$567$	0	0
$568$	13424.7	0.991705
$569$	−12844.6	−0.946353	−0.473177	−	0.880968i	$-0.656893\pi$
$569$	−12844.6	−0.946353	−0.473177	+	0.880968i	$0.656893\pi$
$570$	0	0
$571$	−19440.0	−1.42476	−0.712382	−	0.701792i	$-0.752384\pi$
$571$	−19440.0	−1.42476	−0.712382	+	0.701792i	$0.752384\pi$
$572$	5579.10	0.407822
$573$	0	0
$574$	6449.10	0.468955
$575$	0	0
$576$	0	0
$577$	−977.160	−0.0705021	−0.0352510	−	0.999378i	$-0.511223\pi$
$577$	−977.160	−0.0705021	−0.0352510	+	0.999378i	$0.511223\pi$
$578$	10018.6	0.720963
$579$	0	0
$580$	0	0
$581$	16321.3	1.16544
$582$	0	0
$583$	−7224.52	−0.513223
$584$	−5203.32	−0.368690
$585$	0	0
$586$	−6146.39	−0.433285
$587$	−7212.61	−0.507149	−0.253574	−	0.967316i	$-0.581606\pi$
$587$	−7212.61	−0.507149	−0.253574	+	0.967316i	$0.581606\pi$
$588$	0	0
$589$	−30985.5	−2.16763
$590$	0	0
$591$	0	0
$592$	−1956.77	−0.135849
$593$	−21076.0	−1.45950	−0.729752	−	0.683712i	$-0.760365\pi$
$593$	−21076.0	−1.45950	−0.729752	+	0.683712i	$0.760365\pi$
$594$	0	0
$595$	0	0
$596$	14227.8	0.977843
$597$	0	0
$598$	20947.2	1.43243
$599$	25691.7	1.75248	0.876238	−	0.481878i	$-0.160045\pi$
$599$	25691.7	1.75248	0.876238	+	0.481878i	$0.160045\pi$
$600$	0	0
$601$	8638.39	0.586302	0.293151	−	0.956066i	$-0.405296\pi$
$601$	8638.39	0.586302	0.293151	+	0.956066i	$0.405296\pi$
$602$	−14379.3	−0.973515
$603$	0	0
$604$	16639.0	1.12091
$605$	0	0
$606$	0	0
$607$	7794.30	0.521188	0.260594	−	0.965449i	$-0.416082\pi$
$607$	7794.30	0.521188	0.260594	+	0.965449i	$0.416082\pi$
$608$	22723.9	1.51575
$609$	0	0
$610$	0	0
$611$	−39208.7	−2.59609
$612$	0	0
$613$	−6321.35	−0.416504	−0.208252	−	0.978075i	$-0.566777\pi$
$613$	−6321.35	−0.416504	−0.208252	+	0.978075i	$0.566777\pi$
$614$	−12663.1	−0.832316
$615$	0	0
$616$	5081.15	0.332347
$617$	24939.9	1.62730	0.813650	−	0.581356i	$-0.197477\pi$
$617$	24939.9	1.62730	0.813650	+	0.581356i	$0.197477\pi$
$618$	0	0
$619$	5992.55	0.389113	0.194556	−	0.980891i	$-0.437673\pi$
$619$	5992.55	0.389113	0.194556	+	0.980891i	$0.437673\pi$
$620$	0	0
$621$	0	0
$622$	4942.63	0.318620
$623$	12883.9	0.828542
$624$	0	0
$625$	0	0
$626$	6252.07	0.399174
$627$	0	0
$628$	−4822.41	−0.306426
$629$	−11310.2	−0.716956
$630$	0	0
$631$	−7767.52	−0.490048	−0.245024	−	0.969517i	$-0.578796\pi$
$631$	−7767.52	−0.490048	−0.245024	+	0.969517i	$0.578796\pi$
$632$	−7220.52	−0.454457
$633$	0	0
$634$	−5671.22	−0.355257
$635$	0	0
$636$	0	0
$637$	16269.8	1.01198
$638$	1264.90	0.0784919
$639$	0	0
$640$	0	0
$641$	16348.1	1.00735	0.503674	−	0.863894i	$-0.331981\pi$
$641$	16348.1	1.00735	0.503674	+	0.863894i	$0.331981\pi$
$642$	0	0
$643$	−1423.25	−0.0872902	−0.0436451	−	0.999047i	$-0.513897\pi$
$643$	−1423.25	−0.0872902	−0.0436451	+	0.999047i	$0.513897\pi$
$644$	−23501.5	−1.43803
$645$	0	0
$646$	19006.2	1.15757
$647$	−2225.13	−0.135207	−0.0676035	−	0.997712i	$-0.521535\pi$
$647$	−2225.13	−0.135207	−0.0676035	+	0.997712i	$0.521535\pi$
$648$	0	0
$649$	−1748.53	−0.105756
$650$	0	0
$651$	0	0
$652$	2388.42	0.143463
$653$	8257.95	0.494883	0.247441	−	0.968903i	$-0.420410\pi$
$653$	8257.95	0.494883	0.247441	+	0.968903i	$0.420410\pi$
$654$	0	0
$655$	0	0
$656$	3673.92	0.218662
$657$	0	0
$658$	−15221.2	−0.901800
$659$	27126.3	1.60348	0.801739	−	0.597674i	$-0.203908\pi$
$659$	27126.3	1.60348	0.801739	+	0.597674i	$0.203908\pi$
$660$	0	0
$661$	2629.79	0.154746	0.0773728	−	0.997002i	$-0.475347\pi$
$661$	2629.79	0.154746	0.0773728	+	0.997002i	$0.475347\pi$
$662$	4080.68	0.239578
$663$	0	0
$664$	−14127.4	−0.825680
$665$	0	0
$666$	0	0
$667$	−13725.3	−0.796768
$668$	−14762.3	−0.855048
$669$	0	0
$670$	0	0
$671$	−1179.11	−0.0678376
$672$	0	0
$673$	3289.98	0.188439	0.0942196	−	0.995551i	$-0.469964\pi$
$673$	3289.98	0.188439	0.0942196	+	0.995551i	$0.469964\pi$
$674$	−7854.35	−0.448870
$675$	0	0
$676$	−30223.9	−1.71961
$677$	−4469.65	−0.253741	−0.126871	−	0.991919i	$-0.540493\pi$
$677$	−4469.65	−0.253741	−0.126871	+	0.991919i	$0.540493\pi$
$678$	0	0
$679$	−11155.4	−0.630493
$680$	0	0
$681$	0	0
$682$	4023.00	0.225878
$683$	27927.6	1.56460	0.782298	−	0.622905i	$-0.214048\pi$
$683$	27927.6	1.56460	0.782298	+	0.622905i	$0.214048\pi$
$684$	0	0
$685$	0	0
$686$	−5046.89	−0.280891
$687$	0	0
$688$	−8191.58	−0.453926
$689$	56046.5	3.09899
$690$	0	0
$691$	−30950.9	−1.70395	−0.851974	−	0.523584i	$-0.824595\pi$
$691$	−30950.9	−1.70395	−0.851974	+	0.523584i	$0.824595\pi$
$692$	−12293.0	−0.675301
$693$	0	0
$694$	−5910.87	−0.323305
$695$	0	0
$696$	0	0
$697$	21235.3	1.15401
$698$	14770.3	0.800953
$699$	0	0
$700$	0	0
$701$	9614.69	0.518034	0.259017	−	0.965873i	$-0.416601\pi$
$701$	9614.69	0.518034	0.259017	+	0.965873i	$0.416601\pi$
$702$	0	0
$703$	−12597.4	−0.675846
$704$	−1289.57	−0.0690377
$705$	0	0
$706$	9428.86	0.502634
$707$	20794.4	1.10616
$708$	0	0
$709$	9294.67	0.492340	0.246170	−	0.969227i	$-0.420828\pi$
$709$	9294.67	0.492340	0.246170	+	0.969227i	$0.420828\pi$
$710$	0	0
$711$	0	0
$712$	−11152.0	−0.586995
$713$	−43653.0	−2.29287
$714$	0	0
$715$	0	0
$716$	15362.4	0.801845
$717$	0	0
$718$	11952.9	0.621277
$719$	−12434.2	−0.644948	−0.322474	−	0.946578i	$-0.604514\pi$
$719$	−12434.2	−0.644948	−0.322474	+	0.946578i	$0.604514\pi$
$720$	0	0
$721$	−17509.2	−0.904408
$722$	11333.1	0.584177
$723$	0	0
$724$	−14991.9	−0.769573
$725$	0	0
$726$	0	0
$727$	2146.65	0.109511	0.0547556	−	0.998500i	$-0.482562\pi$
$727$	2146.65	0.109511	0.0547556	+	0.998500i	$0.482562\pi$
$728$	−39418.6	−2.00680
$729$	0	0
$730$	0	0
$731$	−47347.4	−2.39563
$732$	0	0
$733$	11617.8	0.585423	0.292711	−	0.956201i	$-0.405442\pi$
$733$	11617.8	0.585423	0.292711	+	0.956201i	$0.405442\pi$
$734$	−1862.69	−0.0936693
$735$	0	0
$736$	32013.9	1.60333
$737$	7543.91	0.377047
$738$	0	0
$739$	23974.7	1.19340	0.596702	−	0.802463i	$-0.296477\pi$
$739$	23974.7	1.19340	0.596702	+	0.802463i	$0.296477\pi$
$740$	0	0
$741$	0	0
$742$	21757.8	1.07649
$743$	13294.7	0.656443	0.328221	−	0.944601i	$-0.393551\pi$
$743$	13294.7	0.656443	0.328221	+	0.944601i	$0.393551\pi$
$744$	0	0
$745$	0	0
$746$	8196.91	0.402292
$747$	0	0
$748$	7131.65	0.348608
$749$	−18322.9	−0.893865
$750$	0	0
$751$	−25319.4	−1.23025	−0.615124	−	0.788430i	$-0.710894\pi$
$751$	−25319.4	−1.23025	−0.615124	+	0.788430i	$0.710894\pi$
$752$	−8671.21	−0.420487
$753$	0	0
$754$	−9812.85	−0.473956
$755$	0	0
$756$	0	0
$757$	−16978.3	−0.815173	−0.407586	−	0.913167i	$-0.633629\pi$
$757$	−16978.3	−0.815173	−0.407586	+	0.913167i	$0.633629\pi$
$758$	4942.66	0.236841
$759$	0	0
$760$	0	0
$761$	16484.5	0.785232	0.392616	−	0.919702i	$-0.371570\pi$
$761$	16484.5	0.785232	0.392616	+	0.919702i	$0.371570\pi$
$762$	0	0
$763$	28471.2	1.35089
$764$	13314.2	0.630485
$765$	0	0
$766$	7611.74	0.359038
$767$	13564.8	0.638585
$768$	0	0
$769$	−34803.2	−1.63204	−0.816019	−	0.578025i	$-0.803824\pi$
$769$	−34803.2	−1.63204	−0.816019	+	0.578025i	$0.803824\pi$
$770$	0	0
$771$	0	0
$772$	16001.6	0.745999
$773$	−33850.8	−1.57507	−0.787535	−	0.616270i	$-0.788643\pi$
$773$	−33850.8	−1.57507	−0.787535	+	0.616270i	$0.788643\pi$
$774$	0	0
$775$	0	0
$776$	9655.91	0.446684
$777$	0	0
$778$	−11685.1	−0.538470
$779$	23652.1	1.08784
$780$	0	0
$781$	7385.14	0.338363
$782$	26776.3	1.22445
$783$	0	0
$784$	3598.16	0.163910
$785$	0	0
$786$	0	0
$787$	−17983.2	−0.814526	−0.407263	−	0.913311i	$-0.633517\pi$
$787$	−17983.2	−0.814526	−0.407263	+	0.913311i	$0.633517\pi$
$788$	10955.0	0.495248
$789$	0	0
$790$	0	0
$791$	29006.9	1.30388
$792$	0	0
$793$	9147.31	0.409622
$794$	13906.5	0.621565
$795$	0	0
$796$	18389.1	0.818826
$797$	20935.7	0.930464	0.465232	−	0.885189i	$-0.345971\pi$
$797$	20935.7	0.930464	0.465232	+	0.885189i	$0.345971\pi$
$798$	0	0
$799$	−50119.6	−2.21915
$800$	0	0
$801$	0	0
$802$	21902.9	0.964360
$803$	−2862.43	−0.125794
$804$	0	0
$805$	0	0
$806$	−31209.7	−1.36391
$807$	0	0
$808$	−17999.2	−0.783676
$809$	37360.3	1.62363	0.811815	−	0.583914i	$-0.198480\pi$
$809$	37360.3	1.62363	0.811815	+	0.583914i	$0.198480\pi$
$810$	0	0
$811$	−32067.8	−1.38847	−0.694237	−	0.719747i	$-0.744258\pi$
$811$	−32067.8	−1.38847	−0.694237	+	0.719747i	$0.744258\pi$
$812$	11009.4	0.475808
$813$	0	0
$814$	1635.58	0.0704263
$815$	0	0
$816$	0	0
$817$	−52736.2	−2.25827
$818$	7013.77	0.299793
$819$	0	0
$820$	0	0
$821$	−3988.65	−0.169555	−0.0847777	−	0.996400i	$-0.527018\pi$
$821$	−3988.65	−0.169555	−0.0847777	+	0.996400i	$0.527018\pi$
$822$	0	0
$823$	7867.82	0.333238	0.166619	−	0.986021i	$-0.446715\pi$
$823$	7867.82	0.333238	0.166619	+	0.986021i	$0.446715\pi$
$824$	15155.7	0.640744
$825$	0	0
$826$	5265.97	0.221824
$827$	−9219.74	−0.387669	−0.193834	−	0.981034i	$-0.562092\pi$
$827$	−9219.74	−0.387669	−0.193834	+	0.981034i	$0.562092\pi$
$828$	0	0
$829$	35423.8	1.48410	0.742051	−	0.670344i	$-0.233853\pi$
$829$	35423.8	1.48410	0.742051	+	0.670344i	$0.233853\pi$
$830$	0	0
$831$	0	0
$832$	10004.3	0.416869
$833$	20797.4	0.865050
$834$	0	0
$835$	0	0
$836$	7943.33	0.328619
$837$	0	0
$838$	−13642.6	−0.562382
$839$	−159.682	−0.00657071	−0.00328536	−	0.999995i	$-0.501046\pi$
$839$	−159.682	−0.00657071	−0.00328536	+	0.999995i	$0.501046\pi$
$840$	0	0
$841$	−17959.3	−0.736369
$842$	13212.0	0.540755
$843$	0	0
$844$	2258.57	0.0921128
$845$	0	0
$846$	0	0
$847$	2795.22	0.113394
$848$	12395.0	0.501940
$849$	0	0
$850$	0	0
$851$	−17747.5	−0.714895
$852$	0	0
$853$	−26602.4	−1.06782	−0.533910	−	0.845541i	$-0.679278\pi$
$853$	−26602.4	−1.06782	−0.533910	+	0.845541i	$0.679278\pi$
$854$	3551.08	0.142290
$855$	0	0
$856$	15860.0	0.633275
$857$	36913.3	1.47133	0.735667	−	0.677343i	$-0.236869\pi$
$857$	36913.3	1.47133	0.735667	+	0.677343i	$0.236869\pi$
$858$	0	0
$859$	4372.37	0.173671	0.0868354	−	0.996223i	$-0.472325\pi$
$859$	4372.37	0.173671	0.0868354	+	0.996223i	$0.472325\pi$
$860$	0	0
$861$	0	0
$862$	13463.2	0.531969
$863$	4398.69	0.173503	0.0867515	−	0.996230i	$-0.472351\pi$
$863$	4398.69	0.173503	0.0867515	+	0.996230i	$0.472351\pi$
$864$	0	0
$865$	0	0
$866$	−18588.8	−0.729417
$867$	0	0
$868$	35015.4	1.36924
$869$	−3972.12	−0.155057
$870$	0	0
$871$	−58524.2	−2.27671
$872$	−24644.1	−0.957059
$873$	0	0
$874$	29823.8	1.15424
$875$	0	0
$876$	0	0
$877$	34751.4	1.33805	0.669025	−	0.743240i	$-0.266712\pi$
$877$	34751.4	1.33805	0.669025	+	0.743240i	$0.266712\pi$
$878$	−1544.24	−0.0593573
$879$	0	0
$880$	0	0
$881$	−31026.1	−1.18649	−0.593244	−	0.805023i	$-0.702153\pi$
$881$	−31026.1	−1.18649	−0.593244	+	0.805023i	$0.702153\pi$
$882$	0	0
$883$	46496.8	1.77207	0.886037	−	0.463615i	$-0.153448\pi$
$883$	46496.8	1.77207	0.886037	+	0.463615i	$0.153448\pi$
$884$	−55326.0	−2.10500
$885$	0	0
$886$	3597.84	0.136424
$887$	−27215.4	−1.03022	−0.515110	−	0.857124i	$-0.672249\pi$
$887$	−27215.4	−1.03022	−0.515110	+	0.857124i	$0.672249\pi$
$888$	0	0
$889$	46700.5	1.76185
$890$	0	0
$891$	0	0
$892$	−11983.0	−0.449799
$893$	−55823.9	−2.09191
$894$	0	0
$895$	0	0
$896$	−30681.0	−1.14395
$897$	0	0
$898$	1399.95	0.0520232
$899$	20449.6	0.758655
$900$	0	0
$901$	71643.0	2.64903
$902$	−3070.87	−0.113358
$903$	0	0
$904$	−25107.8	−0.923754
$905$	0	0
$906$	0	0
$907$	19319.3	0.707263	0.353632	−	0.935385i	$-0.384947\pi$
$907$	19319.3	0.707263	0.353632	+	0.935385i	$0.384947\pi$
$908$	10779.1	0.393961
$909$	0	0
$910$	0	0
$911$	−44063.1	−1.60250	−0.801249	−	0.598332i	$-0.795831\pi$
$911$	−44063.1	−1.60250	−0.801249	+	0.598332i	$0.795831\pi$
$912$	0	0
$913$	−7771.73	−0.281716
$914$	−14293.5	−0.517271
$915$	0	0
$916$	5319.95	0.191895
$917$	64267.7	2.31440
$918$	0	0
$919$	11766.0	0.422332	0.211166	−	0.977450i	$-0.432274\pi$
$919$	11766.0	0.422332	0.211166	+	0.977450i	$0.432274\pi$
$920$	0	0
$921$	0	0
$922$	3045.10	0.108769
$923$	−57292.6	−2.04313
$924$	0	0
$925$	0	0
$926$	10531.0	0.373727
$927$	0	0
$928$	−14997.1	−0.530501
$929$	−3733.33	−0.131848	−0.0659239	−	0.997825i	$-0.520999\pi$
$929$	−3733.33	−0.131848	−0.0659239	+	0.997825i	$0.520999\pi$
$930$	0	0
$931$	23164.4	0.815448
$932$	−29610.3	−1.04068
$933$	0	0
$934$	10541.9	0.369315
$935$	0	0
$936$	0	0
$937$	−4865.84	−0.169648	−0.0848239	−	0.996396i	$-0.527033\pi$
$937$	−4865.84	−0.169648	−0.0848239	+	0.996396i	$0.527033\pi$
$938$	−22719.7	−0.790857
$939$	0	0
$940$	0	0
$941$	−16216.8	−0.561797	−0.280898	−	0.959737i	$-0.590632\pi$
$941$	−16216.8	−0.561797	−0.280898	+	0.959737i	$0.590632\pi$
$942$	0	0
$943$	33321.6	1.15069
$944$	2999.92	0.103431
$945$	0	0
$946$	6846.98	0.235322
$947$	−1297.47	−0.0445216	−0.0222608	−	0.999752i	$-0.507086\pi$
$947$	−1297.47	−0.0445216	−0.0222608	+	0.999752i	$0.507086\pi$
$948$	0	0
$949$	22206.2	0.759582
$950$	0	0
$951$	0	0
$952$	−50388.0	−1.71543
$953$	−23938.9	−0.813701	−0.406851	−	0.913495i	$-0.633373\pi$
$953$	−23938.9	−0.813701	−0.406851	+	0.913495i	$0.633373\pi$
$954$	0	0
$955$	0	0
$956$	−16005.2	−0.541471
$957$	0	0
$958$	−469.827	−0.0158449
$959$	11125.8	0.374630
$960$	0	0
$961$	35248.6	1.18320
$962$	−12688.5	−0.425254
$963$	0	0
$964$	38466.3	1.28518
$965$	0	0
$966$	0	0
$967$	27087.0	0.900786	0.450393	−	0.892830i	$-0.351284\pi$
$967$	27087.0	0.900786	0.450393	+	0.892830i	$0.351284\pi$
$968$	−2419.49	−0.0803362
$969$	0	0
$970$	0	0
$971$	36157.4	1.19500	0.597500	−	0.801869i	$-0.296161\pi$
$971$	36157.4	1.19500	0.597500	+	0.801869i	$0.296161\pi$
$972$	0	0
$973$	−5822.63	−0.191845
$974$	−18056.7	−0.594019
$975$	0	0
$976$	2022.98	0.0663462
$977$	−34426.9	−1.12734	−0.563671	−	0.825999i	$-0.690611\pi$
$977$	−34426.9	−1.12734	−0.563671	+	0.825999i	$0.690611\pi$
$978$	0	0
$979$	−6134.92	−0.200279
$980$	0	0
$981$	0	0
$982$	−3046.73	−0.0990072
$983$	−22704.7	−0.736692	−0.368346	−	0.929689i	$-0.620076\pi$
$983$	−22704.7	−0.736692	−0.368346	+	0.929689i	$0.620076\pi$
$984$	0	0
$985$	0	0
$986$	−12543.6	−0.405140
$987$	0	0
$988$	−61622.9	−1.98430
$989$	−74295.7	−2.38874
$990$	0	0
$991$	8792.88	0.281852	0.140926	−	0.990020i	$-0.454992\pi$
$991$	8792.88	0.281852	0.140926	+	0.990020i	$0.454992\pi$
$992$	−47698.2	−1.52663
$993$	0	0
$994$	−22241.6	−0.709717
$995$	0	0
$996$	0	0
$997$	−14403.0	−0.457519	−0.228760	−	0.973483i	$-0.573467\pi$
$997$	−14403.0	−0.457519	−0.228760	+	0.973483i	$0.573467\pi$
$998$	11643.4	0.369303
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bt.1.4		7
3.2	odd	2		2475.4.a.bp.1.4		7
5.4	even	2		495.4.a.o.1.4	✓	7
15.14	odd	2		495.4.a.p.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.4	✓	7	5.4	even	2
495.4.a.p.1.4	yes	7	15.14	odd	2
2475.4.a.bp.1.4		7	3.2	odd	2
2475.4.a.bt.1.4		7	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)

\(f(q)\)	\(=\)	\(q+1.43406 q^{2} -5.94347 q^{4} +23.1010 q^{7} -19.9958 q^{8} +O(q^{10})\)	\(q+1.43406 q^{2} -5.94347 q^{4} +23.1010 q^{7} -19.9958 q^{8} -11.0000 q^{11} +85.3360 q^{13} +33.1283 q^{14} +18.8725 q^{16} +109.083 q^{17} +121.498 q^{19} -15.7747 q^{22} +171.169 q^{23} +122.377 q^{26} -137.300 q^{28} -80.1854 q^{29} -255.029 q^{31} +187.031 q^{32} +156.432 q^{34} -103.684 q^{37} +174.236 q^{38} +194.671 q^{41} -434.049 q^{43} +65.3781 q^{44} +245.467 q^{46} -459.463 q^{47} +190.656 q^{49} -507.191 q^{52} +656.774 q^{53} -461.923 q^{56} -114.991 q^{58} +158.957 q^{59} +107.192 q^{61} -365.727 q^{62} +117.234 q^{64} -685.810 q^{67} -648.332 q^{68} -671.377 q^{71} +260.221 q^{73} -148.689 q^{74} -722.121 q^{76} -254.111 q^{77} +361.102 q^{79} +279.170 q^{82} +706.521 q^{83} -622.453 q^{86} +219.954 q^{88} +557.720 q^{89} +1971.35 q^{91} -1017.34 q^{92} -658.898 q^{94} -482.897 q^{97} +273.413 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8}+O(q^{10}) \) 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8	\( 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8} - 77 q^{11} - 38 q^{13} - 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} - 55 q^{22} + 334 q^{23} + 372 q^{26} - 812 q^{28} + 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} - 466 q^{37} - 494 q^{38} + 258 q^{41} - 308 q^{43} - 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} - 512 q^{52} + 110 q^{53} - 20 q^{56} - 1362 q^{58} + 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} - 2268 q^{67} - 1186 q^{68} + 166 q^{71} - 200 q^{73} + 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1006 q^{82} + 370 q^{83} - 106 q^{86} - 495 q^{88} + 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} - 3698 q^{97} + 697 q^{98}+O(q^{100}) \) 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8 - 77 * q^11 - 38 * q^13 - 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 - 55 * q^22 + 334 * q^23 + 372 * q^26 - 812 * q^28 + 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 - 466 * q^37 - 494 * q^38 + 258 * q^41 - 308 * q^43 - 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 - 512 * q^52 + 110 * q^53 - 20 * q^56 - 1362 * q^58 + 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 - 2268 * q^67 - 1186 * q^68 + 166 * q^71 - 200 * q^73 + 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1006 * q^82 + 370 * q^83 - 106 * q^86 - 495 * q^88 + 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 - 3698 * q^97 + 697 * q^98

Introduction

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