LMFDB - Embedded newform 2450.4.a.cn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1555279308.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 59x^{2} + 268 $ x^4 - 59*x^2 + 268
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.35124$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -7.35124 q^{3} +4.00000 q^{4} +14.7025 q^{6} -8.00000 q^{8} +27.0408 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -7.35124 q^{3} +4.00000 q^{4} +14.7025 q^{6} -8.00000 q^{8} +27.0408 q^{9} +57.0816 q^{11} -29.4050 q^{12} -66.4610 q^{13} +16.0000 q^{16} -58.5101 q^{17} -54.0816 q^{18} -15.0023 q^{19} -114.163 q^{22} -143.204 q^{23} +58.8099 q^{24} +132.922 q^{26} -0.299801 q^{27} -17.0408 q^{29} -139.973 q^{31} -32.0000 q^{32} -419.620 q^{33} +117.020 q^{34} +108.163 q^{36} +227.367 q^{37} +30.0046 q^{38} +488.571 q^{39} +37.3558 q^{41} +147.530 q^{43} +228.326 q^{44} +286.408 q^{46} +161.727 q^{47} -117.620 q^{48} +430.122 q^{51} -265.844 q^{52} +542.571 q^{53} +0.599602 q^{54} +110.285 q^{57} +34.0816 q^{58} -903.903 q^{59} +505.137 q^{61} +279.947 q^{62} +64.0000 q^{64} +839.241 q^{66} -729.163 q^{67} -234.041 q^{68} +1052.73 q^{69} -36.7145 q^{71} -216.326 q^{72} -1141.24 q^{73} -454.734 q^{74} -60.0092 q^{76} -977.142 q^{78} -243.938 q^{79} -727.897 q^{81} -74.7116 q^{82} +818.986 q^{83} -295.060 q^{86} +125.271 q^{87} -456.653 q^{88} -300.202 q^{89} -572.816 q^{92} +1028.98 q^{93} -323.455 q^{94} +235.240 q^{96} +1729.04 q^{97} +1543.53 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 10 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 + 10 * q^9	$ 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 10 q^{9} + 32 q^{11} + 64 q^{16} - 20 q^{18} - 64 q^{22} - 82 q^{23} + 30 q^{29} - 128 q^{32} + 40 q^{36} + 26 q^{37} + 580 q^{39} - 686 q^{43} + 128 q^{44} + 164 q^{46} + 1426 q^{51} + 796 q^{53} - 246 q^{57} - 60 q^{58} + 256 q^{64} - 2524 q^{67} - 834 q^{71} - 80 q^{72} - 52 q^{74} - 1160 q^{78} + 1282 q^{79} - 752 q^{81} + 1372 q^{86} - 256 q^{88} - 328 q^{92} + 1760 q^{93} + 4898 q^{99}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 + 10 * q^9 + 32 * q^11 + 64 * q^16 - 20 * q^18 - 64 * q^22 - 82 * q^23 + 30 * q^29 - 128 * q^32 + 40 * q^36 + 26 * q^37 + 580 * q^39 - 686 * q^43 + 128 * q^44 + 164 * q^46 + 1426 * q^51 + 796 * q^53 - 246 * q^57 - 60 * q^58 + 256 * q^64 - 2524 * q^67 - 834 * q^71 - 80 * q^72 - 52 * q^74 - 1160 * q^78 + 1282 * q^79 - 752 * q^81 + 1372 * q^86 - 256 * q^88 - 328 * q^92 + 1760 * q^93 + 4898 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−7.35124	−1.41475	−0.707374	−	0.706840i	$-0.750120\pi$
$3$	−7.35124	−1.41475	−0.707374	+	0.706840i	$0.750120\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	14.7025	1.00038
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	27.0408	1.00151
$10$	0	0
$11$	57.0816	1.56461	0.782306	−	0.622894i	$-0.214043\pi$
$11$	57.0816	1.56461	0.782306	+	0.622894i	$0.214043\pi$
$12$	−29.4050	−0.707374
$13$	−66.4610	−1.41792	−0.708960	−	0.705249i	$-0.750835\pi$
$13$	−66.4610	−1.41792	−0.708960	+	0.705249i	$0.750835\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−58.5101	−0.834753	−0.417376	−	0.908734i	$-0.637050\pi$
$17$	−58.5101	−0.834753	−0.417376	+	0.908734i	$0.637050\pi$
$18$	−54.0816	−0.708175
$19$	−15.0023	−0.181145	−0.0905727	−	0.995890i	$-0.528870\pi$
$19$	−15.0023	−0.181145	−0.0905727	+	0.995890i	$0.528870\pi$
$20$	0	0
$21$	0	0
$22$	−114.163	−1.10635
$23$	−143.204	−1.29826	−0.649132	−	0.760676i	$-0.724868\pi$
$23$	−143.204	−1.29826	−0.649132	+	0.760676i	$0.724868\pi$
$24$	58.8099	0.500189
$25$	0	0
$26$	132.922	1.00262
$27$	−0.299801	−0.00213692
$28$	0	0
$29$	−17.0408	−0.109117	−0.0545585	−	0.998511i	$-0.517375\pi$
$29$	−17.0408	−0.109117	−0.0545585	+	0.998511i	$0.517375\pi$
$30$	0	0
$31$	−139.973	−0.810967	−0.405483	−	0.914102i	$-0.632897\pi$
$31$	−139.973	−0.810967	−0.405483	+	0.914102i	$0.632897\pi$
$32$	−32.0000	−0.176777
$33$	−419.620	−2.21353
$34$	117.020	0.590259
$35$	0	0
$36$	108.163	0.500755
$37$	227.367	1.01024	0.505120	−	0.863049i	$-0.331448\pi$
$37$	227.367	1.01024	0.505120	+	0.863049i	$0.331448\pi$
$38$	30.0046	0.128089
$39$	488.571	2.00600
$40$	0	0
$41$	37.3558	0.142293	0.0711463	−	0.997466i	$-0.477334\pi$
$41$	37.3558	0.142293	0.0711463	+	0.997466i	$0.477334\pi$
$42$	0	0
$43$	147.530	0.523213	0.261606	−	0.965175i	$-0.415748\pi$
$43$	147.530	0.523213	0.261606	+	0.965175i	$0.415748\pi$
$44$	228.326	0.782306
$45$	0	0
$46$	286.408	0.918012
$47$	161.727	0.501923	0.250961	−	0.967997i	$-0.419253\pi$
$47$	161.727	0.501923	0.250961	+	0.967997i	$0.419253\pi$
$48$	−117.620	−0.353687
$49$	0	0
$50$	0	0
$51$	430.122	1.18096
$52$	−265.844	−0.708960
$53$	542.571	1.40619	0.703093	−	0.711098i	$-0.251802\pi$
$53$	542.571	1.40619	0.703093	+	0.711098i	$0.251802\pi$
$54$	0.599602	0.00151103
$55$	0	0
$56$	0	0
$57$	110.285	0.256275
$58$	34.0816	0.0771574
$59$	−903.903	−1.99455	−0.997273	−	0.0738061i	$-0.976485\pi$
$59$	−903.903	−1.99455	−0.997273	+	0.0738061i	$0.976485\pi$
$60$	0	0
$61$	505.137	1.06027	0.530133	−	0.847915i	$-0.322142\pi$
$61$	505.137	1.06027	0.530133	+	0.847915i	$0.322142\pi$
$62$	279.947	0.573440
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	839.241	1.56520
$67$	−729.163	−1.32957	−0.664787	−	0.747033i	$-0.731478\pi$
$67$	−729.163	−1.32957	−0.664787	+	0.747033i	$0.731478\pi$
$68$	−234.041	−0.417376
$69$	1052.73	1.83672
$70$	0	0
$71$	−36.7145	−0.0613692	−0.0306846	−	0.999529i	$-0.509769\pi$
$71$	−36.7145	−0.0613692	−0.0306846	+	0.999529i	$0.509769\pi$
$72$	−216.326	−0.354087
$73$	−1141.24	−1.82976	−0.914878	−	0.403730i	$-0.867714\pi$
$73$	−1141.24	−1.82976	−0.914878	+	0.403730i	$0.867714\pi$
$74$	−454.734	−0.714348
$75$	0	0
$76$	−60.0092	−0.0905727
$77$	0	0
$78$	−977.142	−1.41846
$79$	−243.938	−0.347407	−0.173704	−	0.984798i	$-0.555573\pi$
$79$	−243.938	−0.347407	−0.173704	+	0.984798i	$0.555573\pi$
$80$	0	0
$81$	−727.897	−0.998487
$82$	−74.7116	−0.100616
$83$	818.986	1.08308	0.541539	−	0.840676i	$-0.317842\pi$
$83$	818.986	1.08308	0.541539	+	0.840676i	$0.317842\pi$
$84$	0	0
$85$	0	0
$86$	−295.060	−0.369967
$87$	125.271	0.154373
$88$	−456.653	−0.553174
$89$	−300.202	−0.357543	−0.178771	−	0.983891i	$-0.557212\pi$
$89$	−300.202	−0.357543	−0.178771	+	0.983891i	$0.557212\pi$
$90$	0	0
$91$	0	0
$92$	−572.816	−0.649132
$93$	1028.98	1.14731
$94$	−323.455	−0.354913
$95$	0	0
$96$	235.240	0.250094
$97$	1729.04	1.80987	0.904936	−	0.425548i	$-0.139919\pi$
$97$	1729.04	1.80987	0.904936	+	0.425548i	$0.139919\pi$
$98$	0	0
$99$	1543.53	1.56698
$100$	0	0
$101$	579.849	0.571259	0.285629	−	0.958340i	$-0.407797\pi$
$101$	579.849	0.571259	0.285629	+	0.958340i	$0.407797\pi$
$102$	−860.245	−0.835068
$103$	−1967.73	−1.88239	−0.941197	−	0.337857i	$-0.890298\pi$
$103$	−1967.73	−1.88239	−0.941197	+	0.337857i	$0.890298\pi$
$104$	531.688	0.501311
$105$	0	0
$106$	−1085.14	−0.994324
$107$	−363.509	−0.328427	−0.164214	−	0.986425i	$-0.552509\pi$
$107$	−363.509	−0.328427	−0.164214	+	0.986425i	$0.552509\pi$
$108$	−1.19920	−0.00106846
$109$	−818.998	−0.719686	−0.359843	−	0.933013i	$-0.617170\pi$
$109$	−818.998	−0.719686	−0.359843	+	0.933013i	$0.617170\pi$
$110$	0	0
$111$	−1671.43	−1.42924
$112$	0	0
$113$	−1369.41	−1.14003	−0.570014	−	0.821635i	$-0.693062\pi$
$113$	−1369.41	−1.14003	−0.570014	+	0.821635i	$0.693062\pi$
$114$	−220.571	−0.181214
$115$	0	0
$116$	−68.1631	−0.0545585
$117$	−1797.16	−1.42006
$118$	1807.81	1.41036
$119$	0	0
$120$	0	0
$121$	1927.31	1.44801
$122$	−1010.27	−0.749721
$123$	−274.612	−0.201308
$124$	−559.894	−0.405483
$125$	0	0
$126$	0	0
$127$	1547.32	1.08112	0.540562	−	0.841304i	$-0.318211\pi$
$127$	1547.32	1.08112	0.540562	+	0.841304i	$0.318211\pi$
$128$	−128.000	−0.0883883
$129$	−1084.53	−0.740214
$130$	0	0
$131$	2094.35	1.39683	0.698413	−	0.715695i	$-0.253890\pi$
$131$	2094.35	1.39683	0.698413	+	0.715695i	$0.253890\pi$
$132$	−1678.48	−1.10677
$133$	0	0
$134$	1458.33	0.940151
$135$	0	0
$136$	468.081	0.295130
$137$	−1975.39	−1.23189	−0.615945	−	0.787790i	$-0.711225\pi$
$137$	−1975.39	−1.23189	−0.615945	+	0.787790i	$0.711225\pi$
$138$	−2105.45	−1.29875
$139$	−1941.17	−1.18452	−0.592259	−	0.805748i	$-0.701764\pi$
$139$	−1941.17	−1.18452	−0.592259	+	0.805748i	$0.701764\pi$
$140$	0	0
$141$	−1188.90	−0.710094
$142$	73.4290	0.0433946
$143$	−3793.70	−2.21850
$144$	432.653	0.250378
$145$	0	0
$146$	2282.48	1.29383
$147$	0	0
$148$	909.468	0.505120
$149$	1981.29	1.08935	0.544675	−	0.838647i	$-0.316653\pi$
$149$	1981.29	1.08935	0.544675	+	0.838647i	$0.316653\pi$
$150$	0	0
$151$	−2518.59	−1.35735	−0.678675	−	0.734438i	$-0.737446\pi$
$151$	−2518.59	−1.35735	−0.678675	+	0.734438i	$0.737446\pi$
$152$	120.018	0.0640445
$153$	−1582.16	−0.836014
$154$	0	0
$155$	0	0
$156$	1954.28	1.00300
$157$	−703.465	−0.357596	−0.178798	−	0.983886i	$-0.557221\pi$
$157$	−703.465	−0.357596	−0.178798	+	0.983886i	$0.557221\pi$
$158$	487.876	0.245654
$159$	−3988.57	−1.98940
$160$	0	0
$161$	0	0
$162$	1455.79	0.706037
$163$	−591.183	−0.284080	−0.142040	−	0.989861i	$-0.545366\pi$
$163$	−591.183	−0.284080	−0.142040	+	0.989861i	$0.545366\pi$
$164$	149.423	0.0711463
$165$	0	0
$166$	−1637.97	−0.765851
$167$	−1496.20	−0.693290	−0.346645	−	0.937996i	$-0.612679\pi$
$167$	−1496.20	−0.693290	−0.346645	+	0.937996i	$0.612679\pi$
$168$	0	0
$169$	2220.06	1.01050
$170$	0	0
$171$	−405.674	−0.181419
$172$	590.121	0.261606
$173$	−255.938	−0.112478	−0.0562388	−	0.998417i	$-0.517911\pi$
$173$	−255.938	−0.112478	−0.0562388	+	0.998417i	$0.517911\pi$
$174$	−250.542	−0.109158
$175$	0	0
$176$	913.305	0.391153
$177$	6644.81	2.82178
$178$	600.404	0.252821
$179$	3214.20	1.34213	0.671063	−	0.741400i	$-0.265838\pi$
$179$	3214.20	1.34213	0.671063	+	0.741400i	$0.265838\pi$
$180$	0	0
$181$	−3072.68	−1.26182	−0.630912	−	0.775854i	$-0.717319\pi$
$181$	−3072.68	−1.26182	−0.630912	+	0.775854i	$0.717319\pi$
$182$	0	0
$183$	−3713.39	−1.50001
$184$	1145.63	0.459006
$185$	0	0
$186$	−2057.96	−0.811273
$187$	−3339.85	−1.30606
$188$	646.909	0.250961
$189$	0	0
$190$	0	0
$191$	−2183.83	−0.827312	−0.413656	−	0.910433i	$-0.635748\pi$
$191$	−2183.83	−0.827312	−0.413656	+	0.910433i	$0.635748\pi$
$192$	−470.480	−0.176843
$193$	−2292.92	−0.855169	−0.427585	−	0.903975i	$-0.640635\pi$
$193$	−2292.92	−0.855169	−0.427585	+	0.903975i	$0.640635\pi$
$194$	−3458.08	−1.27977
$195$	0	0
$196$	0	0
$197$	−5362.46	−1.93939	−0.969695	−	0.244320i	$-0.921435\pi$
$197$	−5362.46	−1.93939	−0.969695	+	0.244320i	$0.921435\pi$
$198$	−3087.06	−1.10802
$199$	1785.14	0.635906	0.317953	−	0.948106i	$-0.397005\pi$
$199$	1785.14	0.635906	0.317953	+	0.948106i	$0.397005\pi$
$200$	0	0
$201$	5360.26	1.88101
$202$	−1159.70	−0.403941
$203$	0	0
$204$	1720.49	0.590482
$205$	0	0
$206$	3935.47	1.33105
$207$	−3872.35	−1.30023
$208$	−1063.38	−0.354480
$209$	−856.354	−0.283422
$210$	0	0
$211$	2014.53	0.657280	0.328640	−	0.944455i	$-0.393410\pi$
$211$	2014.53	0.657280	0.328640	+	0.944455i	$0.393410\pi$
$212$	2170.28	0.703093
$213$	269.897	0.0868219
$214$	727.018	0.232233
$215$	0	0
$216$	2.39841	0.000755514 0
$217$	0	0
$218$	1638.00	0.508895
$219$	8389.54	2.58864
$220$	0	0
$221$	3888.64	1.18361
$222$	3342.86	1.01062
$223$	2617.79	0.786098	0.393049	−	0.919517i	$-0.371420\pi$
$223$	2617.79	0.786098	0.393049	+	0.919517i	$0.371420\pi$
$224$	0	0
$225$	0	0
$226$	2738.82	0.806121
$227$	5061.85	1.48003	0.740015	−	0.672590i	$-0.234818\pi$
$227$	5061.85	1.48003	0.740015	+	0.672590i	$0.234818\pi$
$228$	441.142	0.128137
$229$	2646.14	0.763588	0.381794	−	0.924248i	$-0.375306\pi$
$229$	2646.14	0.763588	0.381794	+	0.924248i	$0.375306\pi$
$230$	0	0
$231$	0	0
$232$	136.326	0.0385787
$233$	−3016.88	−0.848250	−0.424125	−	0.905604i	$-0.639418\pi$
$233$	−3016.88	−0.848250	−0.424125	+	0.905604i	$0.639418\pi$
$234$	3594.31	1.00414
$235$	0	0
$236$	−3615.61	−0.997273
$237$	1793.25	0.491493
$238$	0	0
$239$	6473.96	1.75216	0.876078	−	0.482169i	$-0.160151\pi$
$239$	6473.96	1.75216	0.876078	+	0.482169i	$0.160151\pi$
$240$	0	0
$241$	−5149.77	−1.37646	−0.688228	−	0.725495i	$-0.741611\pi$
$241$	−5149.77	−1.37646	−0.688228	+	0.725495i	$0.741611\pi$
$242$	−3854.61	−1.02390
$243$	5359.04	1.41474
$244$	2020.55	0.530133
$245$	0	0
$246$	549.223	0.142346
$247$	997.067	0.256850
$248$	1119.79	0.286720
$249$	−6020.57	−1.53228
$250$	0	0
$251$	−3664.50	−0.921520	−0.460760	−	0.887525i	$-0.652423\pi$
$251$	−3664.50	−0.921520	−0.460760	+	0.887525i	$0.652423\pi$
$252$	0	0
$253$	−8174.30	−2.03128
$254$	−3094.65	−0.764471
$255$	0	0
$256$	256.000	0.0625000
$257$	2420.95	0.587605	0.293802	−	0.955866i	$-0.405079\pi$
$257$	2420.95	0.587605	0.293802	+	0.955866i	$0.405079\pi$
$258$	2169.06	0.523410
$259$	0	0
$260$	0	0
$261$	−460.796	−0.109282
$262$	−4188.70	−0.987705
$263$	6020.51	1.41156	0.705780	−	0.708431i	$-0.250597\pi$
$263$	6020.51	1.41156	0.705780	+	0.708431i	$0.250597\pi$
$264$	3356.96	0.782602
$265$	0	0
$266$	0	0
$267$	2206.86	0.505833
$268$	−2916.65	−0.664787
$269$	−8330.75	−1.88823	−0.944116	−	0.329613i	$-0.893082\pi$
$269$	−8330.75	−1.88823	−0.944116	+	0.329613i	$0.893082\pi$
$270$	0	0
$271$	−8300.14	−1.86051	−0.930254	−	0.366916i	$-0.880414\pi$
$271$	−8300.14	−1.86051	−0.930254	+	0.366916i	$0.880414\pi$
$272$	−936.162	−0.208688
$273$	0	0
$274$	3950.78	0.871077
$275$	0	0
$276$	4210.91	0.918358
$277$	463.305	0.100496	0.0502478	−	0.998737i	$-0.483999\pi$
$277$	463.305	0.100496	0.0502478	+	0.998737i	$0.483999\pi$
$278$	3882.34	0.837581
$279$	−3784.99	−0.812192
$280$	0	0
$281$	8071.85	1.71362	0.856809	−	0.515635i	$-0.172444\pi$
$281$	8071.85	1.71362	0.856809	+	0.515635i	$0.172444\pi$
$282$	2377.79	0.502112
$283$	−1856.10	−0.389871	−0.194936	−	0.980816i	$-0.562450\pi$
$283$	−1856.10	−0.389871	−0.194936	+	0.980816i	$0.562450\pi$
$284$	−146.858	−0.0306846
$285$	0	0
$286$	7587.40	1.56871
$287$	0	0
$288$	−865.305	−0.177044
$289$	−1489.56	−0.303188
$290$	0	0
$291$	−12710.6	−2.56051
$292$	−4564.97	−0.914878
$293$	−4751.16	−0.947323	−0.473661	−	0.880707i	$-0.657068\pi$
$293$	−4751.16	−0.947323	−0.473661	+	0.880707i	$0.657068\pi$
$294$	0	0
$295$	0	0
$296$	−1818.94	−0.357174
$297$	−17.1131	−0.00334345
$298$	−3962.57	−0.770287
$299$	9517.47	1.84084
$300$	0	0
$301$	0	0
$302$	5037.18	0.959792
$303$	−4262.61	−0.808187
$304$	−240.037	−0.0452863
$305$	0	0
$306$	3164.32	0.591151
$307$	5113.17	0.950566	0.475283	−	0.879833i	$-0.342346\pi$
$307$	5113.17	0.950566	0.475283	+	0.879833i	$0.342346\pi$
$308$	0	0
$309$	14465.3	2.66311
$310$	0	0
$311$	−7805.82	−1.42324	−0.711620	−	0.702565i	$-0.752038\pi$
$311$	−7805.82	−1.42324	−0.711620	+	0.702565i	$0.752038\pi$
$312$	−3908.57	−0.709228
$313$	4005.98	0.723424	0.361712	−	0.932290i	$-0.382192\pi$
$313$	4005.98	0.723424	0.361712	+	0.932290i	$0.382192\pi$
$314$	1406.93	0.252859
$315$	0	0
$316$	−975.752	−0.173704
$317$	4920.96	0.871888	0.435944	−	0.899974i	$-0.356415\pi$
$317$	4920.96	0.871888	0.435944	+	0.899974i	$0.356415\pi$
$318$	7977.14	1.40672
$319$	−972.715	−0.170726
$320$	0	0
$321$	2672.24	0.464642
$322$	0	0
$323$	877.786	0.151212
$324$	−2911.59	−0.499244
$325$	0	0
$326$	1182.37	0.200875
$327$	6020.66	1.01817
$328$	−298.847	−0.0503081
$329$	0	0
$330$	0	0
$331$	507.816	0.0843265	0.0421633	−	0.999111i	$-0.486575\pi$
$331$	507.816	0.0843265	0.0421633	+	0.999111i	$0.486575\pi$
$332$	3275.94	0.541539
$333$	6148.18	1.01177
$334$	2992.40	0.490230
$335$	0	0
$336$	0	0
$337$	8019.71	1.29632	0.648162	−	0.761502i	$-0.275538\pi$
$337$	8019.71	1.29632	0.648162	+	0.761502i	$0.275538\pi$
$338$	−4440.13	−0.714530
$339$	10066.9	1.61285
$340$	0	0
$341$	−7989.90	−1.26885
$342$	811.347	0.128283
$343$	0	0
$344$	−1180.24	−0.184984
$345$	0	0
$346$	511.877	0.0795337
$347$	−4103.65	−0.634857	−0.317429	−	0.948282i	$-0.602819\pi$
$347$	−4103.65	−0.634857	−0.317429	+	0.948282i	$0.602819\pi$
$348$	501.084	0.0771865
$349$	5784.67	0.887239	0.443620	−	0.896215i	$-0.353694\pi$
$349$	5784.67	0.887239	0.443620	+	0.896215i	$0.353694\pi$
$350$	0	0
$351$	19.9251	0.00302998
$352$	−1826.61	−0.276587
$353$	10667.2	1.60839	0.804193	−	0.594369i	$-0.202598\pi$
$353$	10667.2	1.60839	0.804193	+	0.594369i	$0.202598\pi$
$354$	−13289.6	−1.99530
$355$	0	0
$356$	−1200.81	−0.178771
$357$	0	0
$358$	−6428.40	−0.949027
$359$	7806.55	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$359$	7806.55	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$360$	0	0
$361$	−6633.93	−0.967186
$362$	6145.35	0.892245
$363$	−14168.1	−2.04857
$364$	0	0
$365$	0	0
$366$	7426.77	1.06067
$367$	9321.23	1.32579	0.662894	−	0.748713i	$-0.269328\pi$
$367$	9321.23	1.32579	0.662894	+	0.748713i	$0.269328\pi$
$368$	−2291.26	−0.324566
$369$	1010.13	0.142508
$370$	0	0
$371$	0	0
$372$	4115.92	0.573657
$373$	11486.7	1.59452	0.797261	−	0.603635i	$-0.206281\pi$
$373$	11486.7	1.59452	0.797261	+	0.603635i	$0.206281\pi$
$374$	6679.70	0.923527
$375$	0	0
$376$	−1293.82	−0.177456
$377$	1132.55	0.154719
$378$	0	0
$379$	4019.09	0.544714	0.272357	−	0.962196i	$-0.412197\pi$
$379$	4019.09	0.544714	0.272357	+	0.962196i	$0.412197\pi$
$380$	0	0
$381$	−11374.8	−1.52952
$382$	4367.67	0.584998
$383$	666.889	0.0889724	0.0444862	−	0.999010i	$-0.485835\pi$
$383$	666.889	0.0889724	0.0444862	+	0.999010i	$0.485835\pi$
$384$	940.959	0.125047
$385$	0	0
$386$	4585.83	0.604696
$387$	3989.33	0.524003
$388$	6916.17	0.904936
$389$	−2626.02	−0.342274	−0.171137	−	0.985247i	$-0.554744\pi$
$389$	−2626.02	−0.342274	−0.171137	+	0.985247i	$0.554744\pi$
$390$	0	0
$391$	8378.88	1.08373
$392$	0	0
$393$	−15396.1	−1.97615
$394$	10724.9	1.37136
$395$	0	0
$396$	6174.12	0.783488
$397$	3182.96	0.402388	0.201194	−	0.979551i	$-0.435518\pi$
$397$	3182.96	0.402388	0.201194	+	0.979551i	$0.435518\pi$
$398$	−3570.28	−0.449653
$399$	0	0
$400$	0	0
$401$	−163.574	−0.0203703	−0.0101852	−	0.999948i	$-0.503242\pi$
$401$	−163.574	−0.0203703	−0.0101852	+	0.999948i	$0.503242\pi$
$402$	−10720.5	−1.33008
$403$	9302.77	1.14989
$404$	2319.40	0.285629
$405$	0	0
$406$	0	0
$407$	12978.5	1.58064
$408$	−3440.98	−0.417534
$409$	1103.27	0.133382	0.0666912	−	0.997774i	$-0.478756\pi$
$409$	1103.27	0.133382	0.0666912	+	0.997774i	$0.478756\pi$
$410$	0	0
$411$	14521.6	1.74281
$412$	−7870.94	−0.941197
$413$	0	0
$414$	7744.69	0.919398
$415$	0	0
$416$	2126.75	0.250655
$417$	14270.0	1.67579
$418$	1712.71	0.200410
$419$	−4114.41	−0.479718	−0.239859	−	0.970808i	$-0.577101\pi$
$419$	−4114.41	−0.479718	−0.239859	+	0.970808i	$0.577101\pi$
$420$	0	0
$421$	−2271.69	−0.262982	−0.131491	−	0.991317i	$-0.541976\pi$
$421$	−2271.69	−0.262982	−0.131491	+	0.991317i	$0.541976\pi$
$422$	−4029.06	−0.464767
$423$	4373.23	0.502681
$424$	−4340.57	−0.497162
$425$	0	0
$426$	−539.795	−0.0613924
$427$	0	0
$428$	−1454.04	−0.164214
$429$	27888.4	3.13861
$430$	0	0
$431$	−2297.31	−0.256745	−0.128373	−	0.991726i	$-0.540975\pi$
$431$	−2297.31	−0.256745	−0.128373	+	0.991726i	$0.540975\pi$
$432$	−4.79682	−0.000534229 0
$433$	−8528.94	−0.946593	−0.473297	−	0.880903i	$-0.656936\pi$
$433$	−8528.94	−0.946593	−0.473297	+	0.880903i	$0.656936\pi$
$434$	0	0
$435$	0	0
$436$	−3275.99	−0.359843
$437$	2148.39	0.235175
$438$	−16779.1	−1.83045
$439$	10343.8	1.12456	0.562280	−	0.826947i	$-0.309924\pi$
$439$	10343.8	1.12456	0.562280	+	0.826947i	$0.309924\pi$
$440$	0	0
$441$	0	0
$442$	−7777.29	−0.836941
$443$	−12604.9	−1.35186	−0.675932	−	0.736964i	$-0.736259\pi$
$443$	−12604.9	−1.35186	−0.675932	+	0.736964i	$0.736259\pi$
$444$	−6685.72	−0.714618
$445$	0	0
$446$	−5235.57	−0.555855
$447$	−14564.9	−1.54116
$448$	0	0
$449$	−3950.22	−0.415194	−0.207597	−	0.978214i	$-0.566564\pi$
$449$	−3950.22	−0.415194	−0.207597	+	0.978214i	$0.566564\pi$
$450$	0	0
$451$	2132.33	0.222633
$452$	−5477.63	−0.570014
$453$	18514.8	1.92031
$454$	−10123.7	−1.04654
$455$	0	0
$456$	−882.284	−0.0906069
$457$	12864.5	1.31679	0.658396	−	0.752672i	$-0.271235\pi$
$457$	12864.5	1.31679	0.658396	+	0.752672i	$0.271235\pi$
$458$	−5292.27	−0.539938
$459$	17.5414	0.00178380
$460$	0	0
$461$	4775.19	0.482436	0.241218	−	0.970471i	$-0.422453\pi$
$461$	4775.19	0.482436	0.241218	+	0.970471i	$0.422453\pi$
$462$	0	0
$463$	5949.13	0.597148	0.298574	−	0.954387i	$-0.403489\pi$
$463$	5949.13	0.597148	0.298574	+	0.954387i	$0.403489\pi$
$464$	−272.653	−0.0272793
$465$	0	0
$466$	6033.76	0.599803
$467$	5960.50	0.590619	0.295310	−	0.955402i	$-0.404577\pi$
$467$	5960.50	0.590619	0.295310	+	0.955402i	$0.404577\pi$
$468$	−7188.63	−0.710031
$469$	0	0
$470$	0	0
$471$	5171.34	0.505908
$472$	7231.23	0.705178
$473$	8421.25	0.818625
$474$	−3586.50	−0.347538
$475$	0	0
$476$	0	0
$477$	14671.5	1.40831
$478$	−12947.9	−1.23896
$479$	1603.96	0.153000	0.0764999	−	0.997070i	$-0.475625\pi$
$479$	1603.96	0.153000	0.0764999	+	0.997070i	$0.475625\pi$
$480$	0	0
$481$	−15111.0	−1.43244
$482$	10299.5	0.973301
$483$	0	0
$484$	7709.22	0.724006
$485$	0	0
$486$	−10718.1	−1.00038
$487$	4384.68	0.407985	0.203992	−	0.978972i	$-0.434608\pi$
$487$	4384.68	0.407985	0.203992	+	0.978972i	$0.434608\pi$
$488$	−4041.10	−0.374860
$489$	4345.93	0.401901
$490$	0	0
$491$	11788.3	1.08350	0.541752	−	0.840538i	$-0.317761\pi$
$491$	11788.3	1.08350	0.541752	+	0.840538i	$0.317761\pi$
$492$	−1098.45	−0.100654
$493$	997.059	0.0910857
$494$	−1994.13	−0.181620
$495$	0	0
$496$	−2239.57	−0.202742
$497$	0	0
$498$	12041.1	1.08349
$499$	11419.3	1.02444	0.512221	−	0.858854i	$-0.328823\pi$
$499$	11419.3	1.02444	0.512221	+	0.858854i	$0.328823\pi$
$500$	0	0
$501$	10998.9	0.980830
$502$	7329.01	0.651613
$503$	20425.4	1.81058	0.905290	−	0.424795i	$-0.139654\pi$
$503$	20425.4	1.81058	0.905290	+	0.424795i	$0.139654\pi$
$504$	0	0
$505$	0	0
$506$	16348.6	1.43633
$507$	−16320.2	−1.42960
$508$	6189.30	0.540562
$509$	−16158.8	−1.40712	−0.703562	−	0.710634i	$-0.748408\pi$
$509$	−16158.8	−1.40712	−0.703562	+	0.710634i	$0.748408\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	4.49770	0.000387093 0
$514$	−4841.89	−0.415499
$515$	0	0
$516$	−4338.12	−0.370107
$517$	9231.65	0.785314
$518$	0	0
$519$	1881.46	0.159127
$520$	0	0
$521$	6484.05	0.545242	0.272621	−	0.962121i	$-0.412109\pi$
$521$	6484.05	0.545242	0.272621	+	0.962121i	$0.412109\pi$
$522$	921.592	0.0772739
$523$	−5953.94	−0.497797	−0.248898	−	0.968530i	$-0.580069\pi$
$523$	−5953.94	−0.497797	−0.248898	+	0.968530i	$0.580069\pi$
$524$	8377.40	0.698413
$525$	0	0
$526$	−12041.0	−0.998124
$527$	8189.87	0.676957
$528$	−6713.93	−0.553383
$529$	8340.36	0.685490
$530$	0	0
$531$	−24442.2	−1.99756
$532$	0	0
$533$	−2482.70	−0.201760
$534$	−4413.71	−0.357678
$535$	0	0
$536$	5833.31	0.470075
$537$	−23628.4	−1.89877
$538$	16661.5	1.33518
$539$	0	0
$540$	0	0
$541$	449.856	0.0357502	0.0178751	−	0.999840i	$-0.494310\pi$
$541$	449.856	0.0357502	0.0178751	+	0.999840i	$0.494310\pi$
$542$	16600.3	1.31558
$543$	22588.0	1.78516
$544$	1872.32	0.147565
$545$	0	0
$546$	0	0
$547$	20803.1	1.62610	0.813049	−	0.582196i	$-0.197806\pi$
$547$	20803.1	1.62610	0.813049	+	0.582196i	$0.197806\pi$
$548$	−7901.55	−0.615945
$549$	13659.3	1.06187
$550$	0	0
$551$	255.651	0.0197660
$552$	−8421.81	−0.649377
$553$	0	0
$554$	−926.610	−0.0710612
$555$	0	0
$556$	−7764.69	−0.592259
$557$	−7910.37	−0.601747	−0.300873	−	0.953664i	$-0.597278\pi$
$557$	−7910.37	−0.601747	−0.300873	+	0.953664i	$0.597278\pi$
$558$	7569.98	0.574306
$559$	−9805.00	−0.741874
$560$	0	0
$561$	24552.1	1.84775
$562$	−16143.7	−1.21171
$563$	174.056	0.0130295	0.00651473	−	0.999979i	$-0.497926\pi$
$563$	174.056	0.0130295	0.00651473	+	0.999979i	$0.497926\pi$
$564$	−4755.59	−0.355047
$565$	0	0
$566$	3712.20	0.275681
$567$	0	0
$568$	293.716	0.0216973
$569$	6613.74	0.487280	0.243640	−	0.969866i	$-0.421658\pi$
$569$	6613.74	0.487280	0.243640	+	0.969866i	$0.421658\pi$
$570$	0	0
$571$	17994.0	1.31878	0.659392	−	0.751799i	$-0.270814\pi$
$571$	17994.0	1.31878	0.659392	+	0.751799i	$0.270814\pi$
$572$	−15174.8	−1.10925
$573$	16053.9	1.17044
$574$	0	0
$575$	0	0
$576$	1730.61	0.125189
$577$	1050.75	0.0758114	0.0379057	−	0.999281i	$-0.487931\pi$
$577$	1050.75	0.0758114	0.0379057	+	0.999281i	$0.487931\pi$
$578$	2979.13	0.214386
$579$	16855.8	1.20985
$580$	0	0
$581$	0	0
$582$	25421.2	1.81055
$583$	30970.8	2.20014
$584$	9129.93	0.646917
$585$	0	0
$586$	9502.32	0.669858
$587$	8013.30	0.563448	0.281724	−	0.959495i	$-0.409094\pi$
$587$	8013.30	0.563448	0.281724	+	0.959495i	$0.409094\pi$
$588$	0	0
$589$	2099.92	0.146903
$590$	0	0
$591$	39420.8	2.74375
$592$	3637.87	0.252560
$593$	−5508.67	−0.381474	−0.190737	−	0.981641i	$-0.561088\pi$
$593$	−5508.67	−0.381474	−0.190737	+	0.981641i	$0.561088\pi$
$594$	34.2262	0.00236417
$595$	0	0
$596$	7925.14	0.544675
$597$	−13123.0	−0.899646
$598$	−19034.9	−1.30167
$599$	−14559.5	−0.993131	−0.496565	−	0.867999i	$-0.665406\pi$
$599$	−14559.5	−0.993131	−0.496565	+	0.867999i	$0.665406\pi$
$600$	0	0
$601$	5744.20	0.389868	0.194934	−	0.980816i	$-0.437551\pi$
$601$	5744.20	0.389868	0.194934	+	0.980816i	$0.437551\pi$
$602$	0	0
$603$	−19717.1	−1.33158
$604$	−10074.4	−0.678675
$605$	0	0
$606$	8525.22	0.571474
$607$	9280.07	0.620538	0.310269	−	0.950649i	$-0.399581\pi$
$607$	9280.07	0.620538	0.310269	+	0.950649i	$0.399581\pi$
$608$	480.073	0.0320223
$609$	0	0
$610$	0	0
$611$	−10748.6	−0.711686
$612$	−6328.64	−0.418007
$613$	2618.56	0.172533	0.0862665	−	0.996272i	$-0.472506\pi$
$613$	2618.56	0.172533	0.0862665	+	0.996272i	$0.472506\pi$
$614$	−10226.3	−0.672152
$615$	0	0
$616$	0	0
$617$	−22096.6	−1.44178	−0.720888	−	0.693051i	$-0.756266\pi$
$617$	−22096.6	−1.44178	−0.720888	+	0.693051i	$0.756266\pi$
$618$	−28930.6	−1.88311
$619$	−14903.4	−0.967716	−0.483858	−	0.875146i	$-0.660765\pi$
$619$	−14903.4	−0.967716	−0.483858	+	0.875146i	$0.660765\pi$
$620$	0	0
$621$	42.9327	0.00277428
$622$	15611.6	1.00638
$623$	0	0
$624$	7817.14	0.501500
$625$	0	0
$626$	−8011.97	−0.511538
$627$	6295.27	0.400971
$628$	−2813.86	−0.178798
$629$	−13303.3	−0.843301
$630$	0	0
$631$	22683.0	1.43105	0.715527	−	0.698585i	$-0.246187\pi$
$631$	22683.0	1.43105	0.715527	+	0.698585i	$0.246187\pi$
$632$	1951.50	0.122827
$633$	−14809.3	−0.929885
$634$	−9841.92	−0.616518
$635$	0	0
$636$	−15954.3	−0.994699
$637$	0	0
$638$	1945.43	0.120721
$639$	−992.789	−0.0614619
$640$	0	0
$641$	−14738.3	−0.908158	−0.454079	−	0.890961i	$-0.650032\pi$
$641$	−14738.3	−0.908158	−0.454079	+	0.890961i	$0.650032\pi$
$642$	−5344.49	−0.328551
$643$	3959.81	0.242861	0.121431	−	0.992600i	$-0.461252\pi$
$643$	3959.81	0.242861	0.121431	+	0.992600i	$0.461252\pi$
$644$	0	0
$645$	0	0
$646$	−1755.57	−0.106923
$647$	−790.575	−0.0480382	−0.0240191	−	0.999711i	$-0.507646\pi$
$647$	−790.575	−0.0480382	−0.0240191	+	0.999711i	$0.507646\pi$
$648$	5823.18	0.353019
$649$	−51596.2	−3.12069
$650$	0	0
$651$	0	0
$652$	−2364.73	−0.142040
$653$	8990.15	0.538762	0.269381	−	0.963034i	$-0.413181\pi$
$653$	8990.15	0.538762	0.269381	+	0.963034i	$0.413181\pi$
$654$	−12041.3	−0.719958
$655$	0	0
$656$	597.693	0.0355732
$657$	−30860.1	−1.83252
$658$	0	0
$659$	−11856.9	−0.700880	−0.350440	−	0.936585i	$-0.613968\pi$
$659$	−11856.9	−0.700880	−0.350440	+	0.936585i	$0.613968\pi$
$660$	0	0
$661$	18580.9	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$661$	18580.9	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$662$	−1015.63	−0.0596278
$663$	−28586.4	−1.67451
$664$	−6551.89	−0.382926
$665$	0	0
$666$	−12296.4	−0.715427
$667$	2440.31	0.141663
$668$	−5984.80	−0.346645
$669$	−19244.0	−1.11213
$670$	0	0
$671$	28834.0	1.65890
$672$	0	0
$673$	−11231.1	−0.643282	−0.321641	−	0.946862i	$-0.604234\pi$
$673$	−11231.1	−0.643282	−0.321641	+	0.946862i	$0.604234\pi$
$674$	−16039.4	−0.916640
$675$	0	0
$676$	8880.25	0.505249
$677$	−18809.8	−1.06783	−0.533915	−	0.845538i	$-0.679280\pi$
$677$	−18809.8	−1.06783	−0.533915	+	0.845538i	$0.679280\pi$
$678$	−20133.7	−1.14046
$679$	0	0
$680$	0	0
$681$	−37210.9	−2.09387
$682$	15979.8	0.897212
$683$	4109.83	0.230246	0.115123	−	0.993351i	$-0.463274\pi$
$683$	4109.83	0.230246	0.115123	+	0.993351i	$0.463274\pi$
$684$	−1622.69	−0.0907095
$685$	0	0
$686$	0	0
$687$	−19452.4	−1.08028
$688$	2360.48	0.130803
$689$	−36059.8	−1.99386
$690$	0	0
$691$	27895.8	1.53576	0.767878	−	0.640596i	$-0.221313\pi$
$691$	27895.8	1.53576	0.767878	+	0.640596i	$0.221313\pi$
$692$	−1023.75	−0.0562388
$693$	0	0
$694$	8207.30	0.448912
$695$	0	0
$696$	−1002.17	−0.0545791
$697$	−2185.69	−0.118779
$698$	−11569.3	−0.627373
$699$	22177.8	1.20006
$700$	0	0
$701$	−14679.4	−0.790920	−0.395460	−	0.918483i	$-0.629415\pi$
$701$	−14679.4	−0.790920	−0.395460	+	0.918483i	$0.629415\pi$
$702$	−39.8502	−0.00214252
$703$	−3411.03	−0.183000
$704$	3653.22	0.195577
$705$	0	0
$706$	−21334.5	−1.13730
$707$	0	0
$708$	26579.2	1.41089
$709$	11212.5	0.593928	0.296964	−	0.954889i	$-0.404026\pi$
$709$	11212.5	0.593928	0.296964	+	0.954889i	$0.404026\pi$
$710$	0	0
$711$	−6596.27	−0.347932
$712$	2401.61	0.126411
$713$	20044.7	1.05285
$714$	0	0
$715$	0	0
$716$	12856.8	0.671063
$717$	−47591.6	−2.47886
$718$	−15613.1	−0.811527
$719$	6754.56	0.350351	0.175176	−	0.984537i	$-0.443951\pi$
$719$	6754.56	0.350351	0.175176	+	0.984537i	$0.443951\pi$
$720$	0	0
$721$	0	0
$722$	13267.9	0.683904
$723$	37857.2	1.94734
$724$	−12290.7	−0.630912
$725$	0	0
$726$	28336.2	1.44856
$727$	−9988.70	−0.509574	−0.254787	−	0.966997i	$-0.582005\pi$
$727$	−9988.70	−0.509574	−0.254787	+	0.966997i	$0.582005\pi$
$728$	0	0
$729$	−19742.4	−1.00302
$730$	0	0
$731$	−8632.01	−0.436753
$732$	−14853.5	−0.750004
$733$	−27901.1	−1.40594	−0.702968	−	0.711222i	$-0.748142\pi$
$733$	−27901.1	−1.40594	−0.702968	+	0.711222i	$0.748142\pi$
$734$	−18642.5	−0.937474
$735$	0	0
$736$	4582.53	0.229503
$737$	−41621.8	−2.08027
$738$	−2020.26	−0.100768
$739$	21482.4	1.06934	0.534670	−	0.845061i	$-0.320436\pi$
$739$	21482.4	1.06934	0.534670	+	0.845061i	$0.320436\pi$
$740$	0	0
$741$	−7329.68	−0.363377
$742$	0	0
$743$	−30550.5	−1.50846	−0.754232	−	0.656608i	$-0.771991\pi$
$743$	−30550.5	−1.50846	−0.754232	+	0.656608i	$0.771991\pi$
$744$	−8231.83	−0.405636
$745$	0	0
$746$	−22973.3	−1.12750
$747$	22146.0	1.08471
$748$	−13359.4	−0.653032
$749$	0	0
$750$	0	0
$751$	11923.3	0.579345	0.289672	−	0.957126i	$-0.406454\pi$
$751$	11923.3	0.579345	0.289672	+	0.957126i	$0.406454\pi$
$752$	2587.64	0.125481
$753$	26938.7	1.30372
$754$	−2265.09	−0.109403
$755$	0	0
$756$	0	0
$757$	23776.6	1.14158	0.570790	−	0.821096i	$-0.306637\pi$
$757$	23776.6	1.14158	0.570790	+	0.821096i	$0.306637\pi$
$758$	−8038.18	−0.385171
$759$	60091.3	2.87375
$760$	0	0
$761$	−426.647	−0.0203232	−0.0101616	−	0.999948i	$-0.503235\pi$
$761$	−426.647	−0.0203232	−0.0101616	+	0.999948i	$0.503235\pi$
$762$	22749.5	1.08153
$763$	0	0
$764$	−8735.33	−0.413656
$765$	0	0
$766$	−1333.78	−0.0629130
$767$	60074.3	2.82811
$768$	−1881.92	−0.0884217
$769$	26741.9	1.25402	0.627009	−	0.779012i	$-0.284279\pi$
$769$	26741.9	1.25402	0.627009	+	0.779012i	$0.284279\pi$
$770$	0	0
$771$	−17797.0	−0.831312
$772$	−9171.66	−0.427585
$773$	13714.9	0.638151	0.319076	−	0.947729i	$-0.396628\pi$
$773$	13714.9	0.638151	0.319076	+	0.947729i	$0.396628\pi$
$774$	−7978.66	−0.370526
$775$	0	0
$776$	−13832.3	−0.639886
$777$	0	0
$778$	5252.05	0.242024
$779$	−560.423	−0.0257756
$780$	0	0
$781$	−2095.72	−0.0960190
$782$	−16757.8	−0.766313
$783$	5.10885	0.000233174 0
$784$	0	0
$785$	0	0
$786$	30792.1	1.39735
$787$	36263.9	1.64253	0.821263	−	0.570550i	$-0.193270\pi$
$787$	36263.9	1.64253	0.821263	+	0.570550i	$0.193270\pi$
$788$	−21449.9	−0.969695
$789$	−44258.2	−1.99700
$790$	0	0
$791$	0	0
$792$	−12348.2	−0.554010
$793$	−33571.9	−1.50337
$794$	−6365.91	−0.284531
$795$	0	0
$796$	7140.56	0.317953
$797$	15338.3	0.681696	0.340848	−	0.940118i	$-0.389286\pi$
$797$	15338.3	0.681696	0.340848	+	0.940118i	$0.389286\pi$
$798$	0	0
$799$	−9462.69	−0.418981
$800$	0	0
$801$	−8117.69	−0.358083
$802$	327.149	0.0144040
$803$	−65143.9	−2.86286
$804$	21441.0	0.940506
$805$	0	0
$806$	−18605.5	−0.813092
$807$	61241.3	2.67137
$808$	−4638.79	−0.201970
$809$	14755.6	0.641258	0.320629	−	0.947205i	$-0.396106\pi$
$809$	14755.6	0.641258	0.320629	+	0.947205i	$0.396106\pi$
$810$	0	0
$811$	23001.7	0.995928	0.497964	−	0.867198i	$-0.334081\pi$
$811$	23001.7	0.995928	0.497964	+	0.867198i	$0.334081\pi$
$812$	0	0
$813$	61016.4	2.63215
$814$	−25956.9	−1.11768
$815$	0	0
$816$	6881.96	0.295241
$817$	−2213.29	−0.0947775
$818$	−2206.55	−0.0943156
$819$	0	0
$820$	0	0
$821$	−15806.2	−0.671911	−0.335956	−	0.941878i	$-0.609059\pi$
$821$	−15806.2	−0.671911	−0.335956	+	0.941878i	$0.609059\pi$
$822$	−29043.1	−1.23235
$823$	−32882.0	−1.39270	−0.696352	−	0.717700i	$-0.745195\pi$
$823$	−32882.0	−1.39270	−0.696352	+	0.717700i	$0.745195\pi$
$824$	15741.9	0.665527
$825$	0	0
$826$	0	0
$827$	−12293.1	−0.516894	−0.258447	−	0.966025i	$-0.583211\pi$
$827$	−12293.1	−0.516894	−0.258447	+	0.966025i	$0.583211\pi$
$828$	−15489.4	−0.650113
$829$	−42328.4	−1.77337	−0.886686	−	0.462372i	$-0.846999\pi$
$829$	−42328.4	−1.77337	−0.886686	+	0.462372i	$0.846999\pi$
$830$	0	0
$831$	−3405.87	−0.142176
$832$	−4253.50	−0.177240
$833$	0	0
$834$	−28540.1	−1.18497
$835$	0	0
$836$	−3425.42	−0.141711
$837$	41.9642	0.00173297
$838$	8228.81	0.339212
$839$	27477.2	1.13065	0.565327	−	0.824867i	$-0.308750\pi$
$839$	27477.2	1.13065	0.565327	+	0.824867i	$0.308750\pi$
$840$	0	0
$841$	−24098.6	−0.988093
$842$	4543.37	0.185956
$843$	−59338.2	−2.42434
$844$	8058.12	0.328640
$845$	0	0
$846$	−8746.47	−0.355449
$847$	0	0
$848$	8681.14	0.351546
$849$	13644.6	0.551570
$850$	0	0
$851$	−32559.8	−1.31156
$852$	1079.59	0.0434110
$853$	−3925.61	−0.157574	−0.0787869	−	0.996891i	$-0.525105\pi$
$853$	−3925.61	−0.157574	−0.0787869	+	0.996891i	$0.525105\pi$
$854$	0	0
$855$	0	0
$856$	2908.07	0.116117
$857$	−22620.7	−0.901643	−0.450821	−	0.892614i	$-0.648869\pi$
$857$	−22620.7	−0.901643	−0.450821	+	0.892614i	$0.648869\pi$
$858$	−55776.8	−2.21933
$859$	41185.0	1.63587	0.817936	−	0.575310i	$-0.195118\pi$
$859$	41185.0	1.63587	0.817936	+	0.575310i	$0.195118\pi$
$860$	0	0
$861$	0	0
$862$	4594.61	0.181546
$863$	6304.19	0.248664	0.124332	−	0.992241i	$-0.460321\pi$
$863$	6304.19	0.248664	0.124332	+	0.992241i	$0.460321\pi$
$864$	9.59364	0.000377757 0
$865$	0	0
$866$	17057.9	0.669342
$867$	10950.1	0.428934
$868$	0	0
$869$	−13924.4	−0.543558
$870$	0	0
$871$	48460.9	1.88523
$872$	6551.99	0.254448
$873$	46754.6	1.81261
$874$	−4296.77	−0.166293
$875$	0	0
$876$	33558.2	1.29432
$877$	2346.71	0.0903568	0.0451784	−	0.998979i	$-0.485614\pi$
$877$	2346.71	0.0903568	0.0451784	+	0.998979i	$0.485614\pi$
$878$	−20687.6	−0.795184
$879$	34926.9	1.34022
$880$	0	0
$881$	16882.2	0.645604	0.322802	−	0.946466i	$-0.395375\pi$
$881$	16882.2	0.645604	0.322802	+	0.946466i	$0.395375\pi$
$882$	0	0
$883$	19073.0	0.726907	0.363453	−	0.931612i	$-0.381598\pi$
$883$	19073.0	0.726907	0.363453	+	0.931612i	$0.381598\pi$
$884$	15554.6	0.591806
$885$	0	0
$886$	25209.8	0.955913
$887$	13067.0	0.494642	0.247321	−	0.968934i	$-0.420450\pi$
$887$	13067.0	0.494642	0.247321	+	0.968934i	$0.420450\pi$
$888$	13371.4	0.505311
$889$	0	0
$890$	0	0
$891$	−41549.5	−1.56225
$892$	10471.1	0.393049
$893$	−2426.28	−0.0909209
$894$	29129.8	1.08976
$895$	0	0
$896$	0	0
$897$	−69965.3	−2.60432
$898$	7900.43	0.293587
$899$	2385.26	0.0884903
$900$	0	0
$901$	−31745.9	−1.17382
$902$	−4264.66	−0.157425
$903$	0	0
$904$	10955.3	0.403060
$905$	0	0
$906$	−37029.5	−1.35786
$907$	−21404.3	−0.783593	−0.391796	−	0.920052i	$-0.628146\pi$
$907$	−21404.3	−0.783593	−0.391796	+	0.920052i	$0.628146\pi$
$908$	20247.4	0.740015
$909$	15679.6	0.572121
$910$	0	0
$911$	−18604.9	−0.676627	−0.338314	−	0.941033i	$-0.609856\pi$
$911$	−18604.9	−0.676627	−0.338314	+	0.941033i	$0.609856\pi$
$912$	1764.57	0.0640687
$913$	46749.0	1.69460
$914$	−25728.9	−0.931112
$915$	0	0
$916$	10584.5	0.381794
$917$	0	0
$918$	−35.0828	−0.00126134
$919$	−34739.8	−1.24697	−0.623483	−	0.781837i	$-0.714283\pi$
$919$	−34739.8	−1.24697	−0.623483	+	0.781837i	$0.714283\pi$
$920$	0	0
$921$	−37588.1	−1.34481
$922$	−9550.38	−0.341133
$923$	2440.08	0.0870166
$924$	0	0
$925$	0	0
$926$	−11898.3	−0.422247
$927$	−53209.1	−1.88524
$928$	545.305	0.0192893
$929$	31907.2	1.12685	0.563423	−	0.826168i	$-0.309484\pi$
$929$	31907.2	1.12685	0.563423	+	0.826168i	$0.309484\pi$
$930$	0	0
$931$	0	0
$932$	−12067.5	−0.424125
$933$	57382.5	2.01352
$934$	−11921.0	−0.417631
$935$	0	0
$936$	14377.3	0.502068
$937$	25960.6	0.905119	0.452560	−	0.891734i	$-0.350511\pi$
$937$	25960.6	0.905119	0.452560	+	0.891734i	$0.350511\pi$
$938$	0	0
$939$	−29449.0	−1.02346
$940$	0	0
$941$	26798.9	0.928393	0.464197	−	0.885732i	$-0.346343\pi$
$941$	26798.9	0.928393	0.464197	+	0.885732i	$0.346343\pi$
$942$	−10342.7	−0.357731
$943$	−5349.50	−0.184733
$944$	−14462.5	−0.498636
$945$	0	0
$946$	−16842.5	−0.578855
$947$	−10459.5	−0.358910	−0.179455	−	0.983766i	$-0.557433\pi$
$947$	−10459.5	−0.358910	−0.179455	+	0.983766i	$0.557433\pi$
$948$	7172.99	0.245747
$949$	75848.0	2.59445
$950$	0	0
$951$	−36175.2	−1.23350
$952$	0	0
$953$	−37686.8	−1.28100	−0.640501	−	0.767958i	$-0.721273\pi$
$953$	−37686.8	−1.28100	−0.640501	+	0.767958i	$0.721273\pi$
$954$	−29343.1	−0.995825
$955$	0	0
$956$	25895.8	0.876078
$957$	7150.66	0.241534
$958$	−3207.93	−0.108187
$959$	0	0
$960$	0	0
$961$	−10198.4	−0.342333
$962$	30222.1	1.01289
$963$	−9829.57	−0.328924
$964$	−20599.1	−0.688228
$965$	0	0
$966$	0	0
$967$	−38757.5	−1.28889	−0.644445	−	0.764650i	$-0.722912\pi$
$967$	−38757.5	−1.28889	−0.644445	+	0.764650i	$0.722912\pi$
$968$	−15418.4	−0.511950
$969$	−6452.82	−0.213926
$970$	0	0
$971$	−27300.1	−0.902269	−0.451134	−	0.892456i	$-0.648980\pi$
$971$	−27300.1	−0.902269	−0.451134	+	0.892456i	$0.648980\pi$
$972$	21436.2	0.707372
$973$	0	0
$974$	−8769.35	−0.288489
$975$	0	0
$976$	8082.20	0.265066
$977$	26532.3	0.868828	0.434414	−	0.900713i	$-0.356956\pi$
$977$	26532.3	0.868828	0.434414	+	0.900713i	$0.356956\pi$
$978$	−8691.86	−0.284187
$979$	−17136.0	−0.559416
$980$	0	0
$981$	−22146.4	−0.720773
$982$	−23576.7	−0.766153
$983$	18255.4	0.592326	0.296163	−	0.955137i	$-0.404293\pi$
$983$	18255.4	0.592326	0.296163	+	0.955137i	$0.404293\pi$
$984$	2196.89	0.0711732
$985$	0	0
$986$	−1994.12	−0.0644073
$987$	0	0
$988$	3988.27	0.128425
$989$	−21126.9	−0.679268
$990$	0	0
$991$	45772.7	1.46722	0.733612	−	0.679569i	$-0.237833\pi$
$991$	45772.7	1.46722	0.733612	+	0.679569i	$0.237833\pi$
$992$	4479.15	0.143360
$993$	−3733.08	−0.119301
$994$	0	0
$995$	0	0
$996$	−24082.3	−0.766140
$997$	26370.6	0.837678	0.418839	−	0.908061i	$-0.362437\pi$
$997$	26370.6	0.837678	0.418839	+	0.908061i	$0.362437\pi$
$998$	−22838.5	−0.724390
$999$	−68.1649	−0.00215880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cn.1.1	✓	4
5.4	even	2		2450.4.a.ct.1.4	yes	4
7.6	odd	2	inner	2450.4.a.cn.1.4	yes	4
35.34	odd	2		2450.4.a.ct.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.4.a.cn.1.1	✓	4	1.1	even	1	trivial
2450.4.a.cn.1.4	yes	4	7.6	odd	2	inner
2450.4.a.ct.1.1	yes	4	35.34	odd	2
2450.4.a.ct.1.4	yes	4	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -7.35124 q^{3} +4.00000 q^{4} +14.7025 q^{6} -8.00000 q^{8} +27.0408 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -7.35124 q^{3} +4.00000 q^{4} +14.7025 q^{6} -8.00000 q^{8} +27.0408 q^{9} +57.0816 q^{11} -29.4050 q^{12} -66.4610 q^{13} +16.0000 q^{16} -58.5101 q^{17} -54.0816 q^{18} -15.0023 q^{19} -114.163 q^{22} -143.204 q^{23} +58.8099 q^{24} +132.922 q^{26} -0.299801 q^{27} -17.0408 q^{29} -139.973 q^{31} -32.0000 q^{32} -419.620 q^{33} +117.020 q^{34} +108.163 q^{36} +227.367 q^{37} +30.0046 q^{38} +488.571 q^{39} +37.3558 q^{41} +147.530 q^{43} +228.326 q^{44} +286.408 q^{46} +161.727 q^{47} -117.620 q^{48} +430.122 q^{51} -265.844 q^{52} +542.571 q^{53} +0.599602 q^{54} +110.285 q^{57} +34.0816 q^{58} -903.903 q^{59} +505.137 q^{61} +279.947 q^{62} +64.0000 q^{64} +839.241 q^{66} -729.163 q^{67} -234.041 q^{68} +1052.73 q^{69} -36.7145 q^{71} -216.326 q^{72} -1141.24 q^{73} -454.734 q^{74} -60.0092 q^{76} -977.142 q^{78} -243.938 q^{79} -727.897 q^{81} -74.7116 q^{82} +818.986 q^{83} -295.060 q^{86} +125.271 q^{87} -456.653 q^{88} -300.202 q^{89} -572.816 q^{92} +1028.98 q^{93} -323.455 q^{94} +235.240 q^{96} +1729.04 q^{97} +1543.53 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 10 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 + 10 * q^9	\( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 10 q^{9} + 32 q^{11} + 64 q^{16} - 20 q^{18} - 64 q^{22} - 82 q^{23} + 30 q^{29} - 128 q^{32} + 40 q^{36} + 26 q^{37} + 580 q^{39} - 686 q^{43} + 128 q^{44} + 164 q^{46} + 1426 q^{51} + 796 q^{53} - 246 q^{57} - 60 q^{58} + 256 q^{64} - 2524 q^{67} - 834 q^{71} - 80 q^{72} - 52 q^{74} - 1160 q^{78} + 1282 q^{79} - 752 q^{81} + 1372 q^{86} - 256 q^{88} - 328 q^{92} + 1760 q^{93} + 4898 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 32 * q^8 + 10 * q^9 + 32 * q^11 + 64 * q^16 - 20 * q^18 - 64 * q^22 - 82 * q^23 + 30 * q^29 - 128 * q^32 + 40 * q^36 + 26 * q^37 + 580 * q^39 - 686 * q^43 + 128 * q^44 + 164 * q^46 + 1426 * q^51 + 796 * q^53 - 246 * q^57 - 60 * q^58 + 256 * q^64 - 2524 * q^67 - 834 * q^71 - 80 * q^72 - 52 * q^74 - 1160 * q^78 + 1282 * q^79 - 752 * q^81 + 1372 * q^86 - 256 * q^88 - 328 * q^92 + 1760 * q^93 + 4898 * q^99

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