LMFDB - Embedded newform 2450.4.a.cv.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2450,4,Mod(1,2450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [6,-12,-7,24,0,14,0,-48,31,0,31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$144.554679514$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 92x^{4} + 26x^{3} + 2116x^{2} + 80x - 7800 $ x^6 - x^5 - 92x^4 + 26x^3 + 2116x^2 + 80x - 7800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.06759$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} -3.06759 q^{3} +4.00000 q^{4} +6.13518 q^{6} -8.00000 q^{8} -17.5899 q^{9} -49.5016 q^{11} -12.2704 q^{12} +44.6154 q^{13} +16.0000 q^{16} -39.2912 q^{17} +35.1798 q^{18} -56.3157 q^{19} +99.0033 q^{22} -91.6603 q^{23} +24.5407 q^{24} -89.2307 q^{26} +136.783 q^{27} +281.845 q^{29} +70.5654 q^{31} -32.0000 q^{32} +151.851 q^{33} +78.5824 q^{34} -70.3596 q^{36} +197.089 q^{37} +112.631 q^{38} -136.862 q^{39} +399.021 q^{41} -203.567 q^{43} -198.007 q^{44} +183.321 q^{46} +68.4988 q^{47} -49.0814 q^{48} +120.529 q^{51} +178.461 q^{52} -617.486 q^{53} -273.567 q^{54} +172.753 q^{57} -563.690 q^{58} +480.513 q^{59} -23.4974 q^{61} -141.131 q^{62} +64.0000 q^{64} -303.701 q^{66} +252.124 q^{67} -157.165 q^{68} +281.176 q^{69} +835.640 q^{71} +140.719 q^{72} -257.786 q^{73} -394.179 q^{74} -225.263 q^{76} +273.723 q^{78} -773.700 q^{79} +55.3315 q^{81} -798.043 q^{82} -1341.21 q^{83} +407.134 q^{86} -864.584 q^{87} +396.013 q^{88} +1301.12 q^{89} -366.641 q^{92} -216.466 q^{93} -136.998 q^{94} +98.1629 q^{96} -323.729 q^{97} +870.729 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} - 7 q^{3} + 24 q^{4} + 14 q^{6} - 48 q^{8} + 31 q^{9} + 31 q^{11} - 28 q^{12} - 59 q^{13} + 96 q^{16} - 68 q^{17} - 62 q^{18} + 93 q^{19} - 62 q^{22} + 94 q^{23} + 56 q^{24} + 118 q^{26}+ \cdots + 4273 q^{99}+O(q^{100}) $ 6 * q - 12 * q^2 - 7 * q^3 + 24 * q^4 + 14 * q^6 - 48 * q^8 + 31 * q^9 + 31 * q^11 - 28 * q^12 - 59 * q^13 + 96 * q^16 - 68 * q^17 - 62 * q^18 + 93 * q^19 - 62 * q^22 + 94 * q^23 + 56 * q^24 + 118 * q^26 - 385 * q^27 + 169 * q^29 + 326 * q^31 - 192 * q^32 - 400 * q^33 + 136 * q^34 + 124 * q^36 + 253 * q^37 - 186 * q^38 - 434 * q^39 + 198 * q^41 + 99 * q^43 + 124 * q^44 - 188 * q^46 - 901 * q^47 - 112 * q^48 + 724 * q^51 - 236 * q^52 - 233 * q^53 + 770 * q^54 + 518 * q^57 - 338 * q^58 + 668 * q^59 + 157 * q^61 - 652 * q^62 + 384 * q^64 + 800 * q^66 - 1193 * q^67 - 272 * q^68 - 45 * q^69 + 1108 * q^71 - 248 * q^72 - 1458 * q^73 - 506 * q^74 + 372 * q^76 + 868 * q^78 - 886 * q^79 - 614 * q^81 - 396 * q^82 - 1799 * q^83 - 198 * q^86 - 789 * q^87 - 248 * q^88 + 3047 * q^89 + 376 * q^92 - 1902 * q^93 + 1802 * q^94 + 224 * q^96 - 3404 * q^97 + 4273 * q^99

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
$3$	−3.06759	−0.590358	−0.295179	−	0.955442i	$-0.595379\pi$
$3$	−3.06759	−0.590358	−0.295179	+	0.955442i	$0.595379\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	6.13518	0.417446
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−17.5899	−0.651478
$10$	0	0
$11$	−49.5016	−1.35685	−0.678423	−	0.734672i	$-0.737336\pi$
$11$	−49.5016	−1.35685	−0.678423	+	0.734672i	$0.737336\pi$
$12$	−12.2704	−0.295179
$13$	44.6154	0.951852	0.475926	−	0.879485i	$-0.342113\pi$
$13$	44.6154	0.951852	0.475926	+	0.879485i	$0.342113\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−39.2912	−0.560560	−0.280280	−	0.959918i	$-0.590427\pi$
$17$	−39.2912	−0.560560	−0.280280	+	0.959918i	$0.590427\pi$
$18$	35.1798	0.460664
$19$	−56.3157	−0.679984	−0.339992	−	0.940428i	$-0.610424\pi$
$19$	−56.3157	−0.679984	−0.339992	+	0.940428i	$0.610424\pi$
$20$	0	0
$21$	0	0
$22$	99.0033	0.959435
$23$	−91.6603	−0.830978	−0.415489	−	0.909598i	$-0.636390\pi$
$23$	−91.6603	−0.830978	−0.415489	+	0.909598i	$0.636390\pi$
$24$	24.5407	0.208723
$25$	0	0
$26$	−89.2307	−0.673061
$27$	136.783	0.974963
$28$	0	0
$29$	281.845	1.80473	0.902367	−	0.430969i	$-0.141828\pi$
$29$	281.845	1.80473	0.902367	+	0.430969i	$0.141828\pi$
$30$	0	0
$31$	70.5654	0.408836	0.204418	−	0.978884i	$-0.434470\pi$
$31$	70.5654	0.408836	0.204418	+	0.978884i	$0.434470\pi$
$32$	−32.0000	−0.176777
$33$	151.851	0.801025
$34$	78.5824	0.396376
$35$	0	0
$36$	−70.3596	−0.325739
$37$	197.089	0.875710	0.437855	−	0.899046i	$-0.355738\pi$
$37$	197.089	0.875710	0.437855	+	0.899046i	$0.355738\pi$
$38$	112.631	0.480822
$39$	−136.862	−0.561933
$40$	0	0
$41$	399.021	1.51992	0.759959	−	0.649971i	$-0.225219\pi$
$41$	399.021	1.51992	0.759959	+	0.649971i	$0.225219\pi$
$42$	0	0
$43$	−203.567	−0.721946	−0.360973	−	0.932576i	$-0.617555\pi$
$43$	−203.567	−0.721946	−0.360973	+	0.932576i	$0.617555\pi$
$44$	−198.007	−0.678423
$45$	0	0
$46$	183.321	0.587590
$47$	68.4988	0.212587	0.106293	−	0.994335i	$-0.466102\pi$
$47$	68.4988	0.212587	0.106293	+	0.994335i	$0.466102\pi$
$48$	−49.0814	−0.147589
$49$	0	0
$50$	0	0
$51$	120.529	0.330931
$52$	178.461	0.475926
$53$	−617.486	−1.60034	−0.800172	−	0.599771i	$-0.795258\pi$
$53$	−617.486	−1.60034	−0.800172	+	0.599771i	$0.795258\pi$
$54$	−273.567	−0.689403
$55$	0	0
$56$	0	0
$57$	172.753	0.401434
$58$	−563.690	−1.27614
$59$	480.513	1.06029	0.530147	−	0.847905i	$-0.322137\pi$
$59$	480.513	1.06029	0.530147	+	0.847905i	$0.322137\pi$
$60$	0	0
$61$	−23.4974	−0.0493202	−0.0246601	−	0.999696i	$-0.507850\pi$
$61$	−23.4974	−0.0493202	−0.0246601	+	0.999696i	$0.507850\pi$
$62$	−141.131	−0.289091
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−303.701	−0.566410
$67$	252.124	0.459729	0.229865	−	0.973223i	$-0.426172\pi$
$67$	252.124	0.459729	0.229865	+	0.973223i	$0.426172\pi$
$68$	−157.165	−0.280280
$69$	281.176	0.490574
$70$	0	0
$71$	835.640	1.39679	0.698396	−	0.715711i	$-0.253897\pi$
$71$	835.640	1.39679	0.698396	+	0.715711i	$0.253897\pi$
$72$	140.719	0.230332
$73$	−257.786	−0.413309	−0.206654	−	0.978414i	$-0.566258\pi$
$73$	−257.786	−0.413309	−0.206654	+	0.978414i	$0.566258\pi$
$74$	−394.179	−0.619221
$75$	0	0
$76$	−225.263	−0.339992
$77$	0	0
$78$	273.723	0.397347
$79$	−773.700	−1.10187	−0.550937	−	0.834547i	$-0.685730\pi$
$79$	−773.700	−1.10187	−0.550937	+	0.834547i	$0.685730\pi$
$80$	0	0
$81$	55.3315	0.0759006
$82$	−798.043	−1.07474
$83$	−1341.21	−1.77369	−0.886846	−	0.462065i	$-0.847109\pi$
$83$	−1341.21	−1.77369	−0.886846	+	0.462065i	$0.847109\pi$
$84$	0	0
$85$	0	0
$86$	407.134	0.510493
$87$	−864.584	−1.06544
$88$	396.013	0.479717
$89$	1301.12	1.54964	0.774821	−	0.632181i	$-0.217840\pi$
$89$	1301.12	1.54964	0.774821	+	0.632181i	$0.217840\pi$
$90$	0	0
$91$	0	0
$92$	−366.641	−0.415489
$93$	−216.466	−0.241360
$94$	−136.998	−0.150322
$95$	0	0
$96$	98.1629	0.104362
$97$	−323.729	−0.338862	−0.169431	−	0.985542i	$-0.554193\pi$
$97$	−323.729	−0.338862	−0.169431	+	0.985542i	$0.554193\pi$
$98$	0	0
$99$	870.729	0.883955
$100$	0	0
$101$	1181.02	1.16352	0.581761	−	0.813360i	$-0.302364\pi$
$101$	1181.02	1.16352	0.581761	+	0.813360i	$0.302364\pi$
$102$	−241.059	−0.234003
$103$	−732.042	−0.700294	−0.350147	−	0.936695i	$-0.613868\pi$
$103$	−732.042	−0.700294	−0.350147	+	0.936695i	$0.613868\pi$
$104$	−356.923	−0.336530
$105$	0	0
$106$	1234.97	1.13161
$107$	1338.04	1.20891	0.604455	−	0.796639i	$-0.293391\pi$
$107$	1338.04	1.20891	0.604455	+	0.796639i	$0.293391\pi$
$108$	547.134	0.487481
$109$	−51.7571	−0.0454810	−0.0227405	−	0.999741i	$-0.507239\pi$
$109$	−51.7571	−0.0454810	−0.0227405	+	0.999741i	$0.507239\pi$
$110$	0	0
$111$	−604.589	−0.516982
$112$	0	0
$113$	471.803	0.392774	0.196387	−	0.980526i	$-0.437079\pi$
$113$	471.803	0.392774	0.196387	+	0.980526i	$0.437079\pi$
$114$	−345.507	−0.283857
$115$	0	0
$116$	1127.38	0.902367
$117$	−784.780	−0.620110
$118$	−961.025	−0.749742
$119$	0	0
$120$	0	0
$121$	1119.41	0.841031
$122$	46.9948	0.0348746
$123$	−1224.03	−0.897296
$124$	282.262	0.204418
$125$	0	0
$126$	0	0
$127$	202.294	0.141344	0.0706721	−	0.997500i	$-0.477486\pi$
$127$	202.294	0.141344	0.0706721	+	0.997500i	$0.477486\pi$
$128$	−128.000	−0.0883883
$129$	624.460	0.426206
$130$	0	0
$131$	−1168.96	−0.779635	−0.389817	−	0.920892i	$-0.627462\pi$
$131$	−1168.96	−0.779635	−0.389817	+	0.920892i	$0.627462\pi$
$132$	607.403	0.400512
$133$	0	0
$134$	−504.248	−0.325078
$135$	0	0
$136$	314.330	0.198188
$137$	146.718	0.0914958	0.0457479	−	0.998953i	$-0.485433\pi$
$137$	146.718	0.0914958	0.0457479	+	0.998953i	$0.485433\pi$
$138$	−562.352	−0.346888
$139$	−2505.64	−1.52896	−0.764479	−	0.644648i	$-0.777004\pi$
$139$	−2505.64	−1.52896	−0.764479	+	0.644648i	$0.777004\pi$
$140$	0	0
$141$	−210.126	−0.125502
$142$	−1671.28	−0.987681
$143$	−2208.53	−1.29152
$144$	−281.438	−0.162869
$145$	0	0
$146$	515.571	0.292253
$147$	0	0
$148$	788.357	0.437855
$149$	2108.18	1.15912	0.579559	−	0.814930i	$-0.303225\pi$
$149$	2108.18	1.15912	0.579559	+	0.814930i	$0.303225\pi$
$150$	0	0
$151$	−2215.51	−1.19401	−0.597004	−	0.802238i	$-0.703642\pi$
$151$	−2215.51	−1.19401	−0.597004	+	0.802238i	$0.703642\pi$
$152$	450.525	0.240411
$153$	691.128	0.365192
$154$	0	0
$155$	0	0
$156$	−547.446	−0.280967
$157$	−2262.03	−1.14987	−0.574935	−	0.818199i	$-0.694972\pi$
$157$	−2262.03	−1.14987	−0.574935	+	0.818199i	$0.694972\pi$
$158$	1547.40	0.779143
$159$	1894.19	0.944775
$160$	0	0
$161$	0	0
$162$	−110.663	−0.0536698
$163$	87.7158	0.0421499	0.0210750	−	0.999778i	$-0.493291\pi$
$163$	87.7158	0.0421499	0.0210750	+	0.999778i	$0.493291\pi$
$164$	1596.09	0.759959
$165$	0	0
$166$	2682.41	1.25419
$167$	−1851.38	−0.857868	−0.428934	−	0.903336i	$-0.641111\pi$
$167$	−1851.38	−0.857868	−0.428934	+	0.903336i	$0.641111\pi$
$168$	0	0
$169$	−206.469	−0.0939779
$170$	0	0
$171$	990.587	0.442995
$172$	−814.268	−0.360973
$173$	1842.08	0.809541	0.404771	−	0.914418i	$-0.367351\pi$
$173$	1842.08	0.809541	0.404771	+	0.914418i	$0.367351\pi$
$174$	1729.17	0.753379
$175$	0	0
$176$	−792.026	−0.339211
$177$	−1474.02	−0.625953
$178$	−2602.23	−1.09576
$179$	748.532	0.312558	0.156279	−	0.987713i	$-0.450050\pi$
$179$	748.532	0.312558	0.156279	+	0.987713i	$0.450050\pi$
$180$	0	0
$181$	−1393.90	−0.572418	−0.286209	−	0.958167i	$-0.592395\pi$
$181$	−1393.90	−0.572418	−0.286209	+	0.958167i	$0.592395\pi$
$182$	0	0
$183$	72.0803	0.0291166
$184$	733.282	0.293795
$185$	0	0
$186$	432.931	0.170667
$187$	1944.98	0.760593
$188$	273.995	0.106293
$189$	0	0
$190$	0	0
$191$	−958.174	−0.362990	−0.181495	−	0.983392i	$-0.558094\pi$
$191$	−958.174	−0.362990	−0.181495	+	0.983392i	$0.558094\pi$
$192$	−196.326	−0.0737947
$193$	4577.04	1.70706	0.853530	−	0.521044i	$-0.174457\pi$
$193$	4577.04	1.70706	0.853530	+	0.521044i	$0.174457\pi$
$194$	647.457	0.239612
$195$	0	0
$196$	0	0
$197$	3429.27	1.24023	0.620115	−	0.784511i	$-0.287086\pi$
$197$	3429.27	1.24023	0.620115	+	0.784511i	$0.287086\pi$
$198$	−1741.46	−0.625050
$199$	−146.016	−0.0520140	−0.0260070	−	0.999662i	$-0.508279\pi$
$199$	−146.016	−0.0520140	−0.0260070	+	0.999662i	$0.508279\pi$
$200$	0	0
$201$	−773.413	−0.271405
$202$	−2362.04	−0.822734
$203$	0	0
$204$	482.117	0.165465
$205$	0	0
$206$	1464.08	0.495182
$207$	1612.29	0.541363
$208$	713.846	0.237963
$209$	2787.72	0.922634
$210$	0	0
$211$	−705.410	−0.230154	−0.115077	−	0.993357i	$-0.536711\pi$
$211$	−705.410	−0.230154	−0.115077	+	0.993357i	$0.536711\pi$
$212$	−2469.94	−0.800172
$213$	−2563.40	−0.824607
$214$	−2676.08	−0.854829
$215$	0	0
$216$	−1094.27	−0.344701
$217$	0	0
$218$	103.514	0.0321600
$219$	790.781	0.244000
$220$	0	0
$221$	−1752.99	−0.533570
$222$	1209.18	0.365562
$223$	2232.07	0.670271	0.335135	−	0.942170i	$-0.391218\pi$
$223$	2232.07	0.670271	0.335135	+	0.942170i	$0.391218\pi$
$224$	0	0
$225$	0	0
$226$	−943.606	−0.277733
$227$	5464.96	1.59789	0.798947	−	0.601401i	$-0.205391\pi$
$227$	5464.96	1.59789	0.798947	+	0.601401i	$0.205391\pi$
$228$	691.014	0.200717
$229$	1943.89	0.560942	0.280471	−	0.959862i	$-0.409509\pi$
$229$	1943.89	0.560942	0.280471	+	0.959862i	$0.409509\pi$
$230$	0	0
$231$	0	0
$232$	−2254.76	−0.638070
$233$	−3307.25	−0.929893	−0.464946	−	0.885339i	$-0.653926\pi$
$233$	−3307.25	−0.929893	−0.464946	+	0.885339i	$0.653926\pi$
$234$	1569.56	0.438484
$235$	0	0
$236$	1922.05	0.530147
$237$	2373.39	0.650500
$238$	0	0
$239$	6107.99	1.65311	0.826554	−	0.562857i	$-0.190298\pi$
$239$	6107.99	1.65311	0.826554	+	0.562857i	$0.190298\pi$
$240$	0	0
$241$	−3962.60	−1.05914	−0.529571	−	0.848265i	$-0.677647\pi$
$241$	−3962.60	−1.05914	−0.529571	+	0.848265i	$0.677647\pi$
$242$	−2238.82	−0.594699
$243$	−3862.89	−1.01977
$244$	−93.9895	−0.0246601
$245$	0	0
$246$	2448.07	0.634484
$247$	−2512.54	−0.647244
$248$	−564.523	−0.144545
$249$	4114.27	1.04711
$250$	0	0
$251$	−3.92299	−0.000986522 0	−0.000493261	−	1.00000i	$-0.500157\pi$
$251$	−3.92299	−0.000986522 0	−0.000493261		1.00000i	$0.500157\pi$
$252$	0	0
$253$	4537.33	1.12751
$254$	−404.589	−0.0999455
$255$	0	0
$256$	256.000	0.0625000
$257$	219.595	0.0532994	0.0266497	−	0.999645i	$-0.491516\pi$
$257$	219.595	0.0532994	0.0266497	+	0.999645i	$0.491516\pi$
$258$	−1248.92	−0.301373
$259$	0	0
$260$	0	0
$261$	−4957.62	−1.17574
$262$	2337.91	0.551285
$263$	5467.72	1.28195	0.640977	−	0.767560i	$-0.278529\pi$
$263$	5467.72	1.28195	0.640977	+	0.767560i	$0.278529\pi$
$264$	−1214.81	−0.283205
$265$	0	0
$266$	0	0
$267$	−3991.29	−0.914843
$268$	1008.50	0.229865
$269$	−291.731	−0.0661233	−0.0330616	−	0.999453i	$-0.510526\pi$
$269$	−291.731	−0.0661233	−0.0330616	+	0.999453i	$0.510526\pi$
$270$	0	0
$271$	2661.02	0.596478	0.298239	−	0.954491i	$-0.403601\pi$
$271$	2661.02	0.596478	0.298239	+	0.954491i	$0.403601\pi$
$272$	−628.659	−0.140140
$273$	0	0
$274$	−293.435	−0.0646973
$275$	0	0
$276$	1124.70	0.245287
$277$	4502.66	0.976673	0.488336	−	0.872655i	$-0.337604\pi$
$277$	4502.66	0.976673	0.488336	+	0.872655i	$0.337604\pi$
$278$	5011.27	1.08114
$279$	−1241.24	−0.266348
$280$	0	0
$281$	−462.388	−0.0981628	−0.0490814	−	0.998795i	$-0.515629\pi$
$281$	−462.388	−0.0981628	−0.0490814	+	0.998795i	$0.515629\pi$
$282$	420.253	0.0887435
$283$	−7145.71	−1.50095	−0.750474	−	0.660899i	$-0.770175\pi$
$283$	−7145.71	−1.50095	−0.750474	+	0.660899i	$0.770175\pi$
$284$	3342.56	0.698396
$285$	0	0
$286$	4417.07	0.913240
$287$	0	0
$288$	562.877	0.115166
$289$	−3369.20	−0.685773
$290$	0	0
$291$	993.066	0.200050
$292$	−1031.14	−0.206654
$293$	−8656.21	−1.72594	−0.862971	−	0.505254i	$-0.831399\pi$
$293$	−8656.21	−1.72594	−0.862971	+	0.505254i	$0.831399\pi$
$294$	0	0
$295$	0	0
$296$	−1576.71	−0.309610
$297$	−6771.01	−1.32287
$298$	−4216.36	−0.819621
$299$	−4089.46	−0.790968
$300$	0	0
$301$	0	0
$302$	4431.01	0.844291
$303$	−3622.88	−0.686894
$304$	−901.051	−0.169996
$305$	0	0
$306$	−1382.26	−0.258230
$307$	−4296.26	−0.798699	−0.399349	−	0.916799i	$-0.630764\pi$
$307$	−4296.26	−0.798699	−0.399349	+	0.916799i	$0.630764\pi$
$308$	0	0
$309$	2245.60	0.413424
$310$	0	0
$311$	3935.13	0.717494	0.358747	−	0.933435i	$-0.383204\pi$
$311$	3935.13	0.717494	0.358747	+	0.933435i	$0.383204\pi$
$312$	1094.89	0.198673
$313$	−10498.9	−1.89596	−0.947979	−	0.318334i	$-0.896877\pi$
$313$	−10498.9	−1.89596	−0.947979	+	0.318334i	$0.896877\pi$
$314$	4524.05	0.813080
$315$	0	0
$316$	−3094.80	−0.550937
$317$	−5747.17	−1.01828	−0.509138	−	0.860685i	$-0.670036\pi$
$317$	−5747.17	−1.01828	−0.509138	+	0.860685i	$0.670036\pi$
$318$	−3788.39	−0.668057
$319$	−13951.8	−2.44874
$320$	0	0
$321$	−4104.56	−0.713690
$322$	0	0
$323$	2212.71	0.381172
$324$	221.326	0.0379503
$325$	0	0
$326$	−175.432	−0.0298045
$327$	158.770	0.0268501
$328$	−3192.17	−0.537372
$329$	0	0
$330$	0	0
$331$	9408.09	1.56228	0.781141	−	0.624355i	$-0.214638\pi$
$331$	9408.09	1.56228	0.781141	+	0.624355i	$0.214638\pi$
$332$	−5364.82	−0.886846
$333$	−3466.78	−0.570506
$334$	3702.76	0.606605
$335$	0	0
$336$	0	0
$337$	4501.95	0.727706	0.363853	−	0.931456i	$-0.381461\pi$
$337$	4501.95	0.727706	0.363853	+	0.931456i	$0.381461\pi$
$338$	412.939	0.0664524
$339$	−1447.30	−0.231877
$340$	0	0
$341$	−3493.10	−0.554728
$342$	−1981.17	−0.313244
$343$	0	0
$344$	1628.54	0.255246
$345$	0	0
$346$	−3684.16	−0.572432
$347$	−4993.36	−0.772500	−0.386250	−	0.922394i	$-0.626230\pi$
$347$	−4993.36	−0.772500	−0.386250	+	0.922394i	$0.626230\pi$
$348$	−3458.34	−0.532719
$349$	6675.06	1.02381	0.511903	−	0.859044i	$-0.328941\pi$
$349$	6675.06	1.02381	0.511903	+	0.859044i	$0.328941\pi$
$350$	0	0
$351$	6102.65	0.928020
$352$	1584.05	0.239859
$353$	−2839.32	−0.428107	−0.214054	−	0.976822i	$-0.568667\pi$
$353$	−2839.32	−0.428107	−0.214054	+	0.976822i	$0.568667\pi$
$354$	2948.03	0.442616
$355$	0	0
$356$	5204.47	0.774821
$357$	0	0
$358$	−1497.06	−0.221012
$359$	−8484.41	−1.24733	−0.623663	−	0.781693i	$-0.714356\pi$
$359$	−8484.41	−1.24733	−0.623663	+	0.781693i	$0.714356\pi$
$360$	0	0
$361$	−3687.54	−0.537621
$362$	2787.80	0.404761
$363$	−3433.90	−0.496509
$364$	0	0
$365$	0	0
$366$	−144.161	−0.0205885
$367$	−11319.7	−1.61004	−0.805020	−	0.593247i	$-0.797846\pi$
$367$	−11319.7	−1.61004	−0.805020	+	0.593247i	$0.797846\pi$
$368$	−1466.56	−0.207744
$369$	−7018.74	−0.990193
$370$	0	0
$371$	0	0
$372$	−865.863	−0.120680
$373$	−11306.2	−1.56947	−0.784735	−	0.619832i	$-0.787201\pi$
$373$	−11306.2	−1.56947	−0.784735	+	0.619832i	$0.787201\pi$
$374$	−3889.96	−0.537821
$375$	0	0
$376$	−547.991	−0.0751608
$377$	12574.6	1.71784
$378$	0	0
$379$	−10626.4	−1.44022	−0.720109	−	0.693861i	$-0.755908\pi$
$379$	−10626.4	−1.44022	−0.720109	+	0.693861i	$0.755908\pi$
$380$	0	0
$381$	−620.556	−0.0834437
$382$	1916.35	0.256672
$383$	7582.88	1.01166	0.505832	−	0.862632i	$-0.331186\pi$
$383$	7582.88	1.01166	0.505832	+	0.862632i	$0.331186\pi$
$384$	392.651	0.0521808
$385$	0	0
$386$	−9154.08	−1.20707
$387$	3580.72	0.470332
$388$	−1294.91	−0.169431
$389$	3212.83	0.418758	0.209379	−	0.977835i	$-0.432856\pi$
$389$	3212.83	0.418758	0.209379	+	0.977835i	$0.432856\pi$
$390$	0	0
$391$	3601.44	0.465813
$392$	0	0
$393$	3585.88	0.460264
$394$	−6858.54	−0.876976
$395$	0	0
$396$	3482.91	0.441977
$397$	−13180.8	−1.66631	−0.833155	−	0.553039i	$-0.813468\pi$
$397$	−13180.8	−1.66631	−0.833155	+	0.553039i	$0.813468\pi$
$398$	292.031	0.0367794
$399$	0	0
$400$	0	0
$401$	−7194.23	−0.895917	−0.447959	−	0.894054i	$-0.647849\pi$
$401$	−7194.23	−0.895917	−0.447959	+	0.894054i	$0.647849\pi$
$402$	1546.83	0.191912
$403$	3148.30	0.389152
$404$	4724.07	0.581761
$405$	0	0
$406$	0	0
$407$	−9756.24	−1.18820
$408$	−964.234	−0.117002
$409$	−1089.15	−0.131675	−0.0658376	−	0.997830i	$-0.520972\pi$
$409$	−1089.15	−0.131675	−0.0658376	+	0.997830i	$0.520972\pi$
$410$	0	0
$411$	−450.069	−0.0540153
$412$	−2928.17	−0.350147
$413$	0	0
$414$	−3224.59	−0.382802
$415$	0	0
$416$	−1427.69	−0.168265
$417$	7686.26	0.902633
$418$	−5575.44	−0.652401
$419$	−12514.5	−1.45913	−0.729563	−	0.683914i	$-0.760276\pi$
$419$	−12514.5	−1.45913	−0.729563	+	0.683914i	$0.760276\pi$
$420$	0	0
$421$	−827.022	−0.0957401	−0.0478701	−	0.998854i	$-0.515243\pi$
$421$	−827.022	−0.0957401	−0.0478701	+	0.998854i	$0.515243\pi$
$422$	1410.82	0.162743
$423$	−1204.89	−0.138496
$424$	4939.89	0.565807
$425$	0	0
$426$	5126.80	0.583085
$427$	0	0
$428$	5352.17	0.604455
$429$	6774.87	0.762457
$430$	0	0
$431$	−1680.44	−0.187805	−0.0939025	−	0.995581i	$-0.529934\pi$
$431$	−1680.44	−0.187805	−0.0939025	+	0.995581i	$0.529934\pi$
$432$	2188.54	0.243741
$433$	−7421.56	−0.823689	−0.411845	−	0.911254i	$-0.635115\pi$
$433$	−7421.56	−0.823689	−0.411845	+	0.911254i	$0.635115\pi$
$434$	0	0
$435$	0	0
$436$	−207.028	−0.0227405
$437$	5161.91	0.565052
$438$	−1581.56	−0.172534
$439$	4798.23	0.521656	0.260828	−	0.965385i	$-0.416004\pi$
$439$	4798.23	0.521656	0.260828	+	0.965385i	$0.416004\pi$
$440$	0	0
$441$	0	0
$442$	3505.98	0.377291
$443$	−15935.6	−1.70908	−0.854541	−	0.519384i	$-0.826161\pi$
$443$	−15935.6	−1.70908	−0.854541	+	0.519384i	$0.826161\pi$
$444$	−2418.36	−0.258491
$445$	0	0
$446$	−4464.14	−0.473953
$447$	−6467.03	−0.684295
$448$	0	0
$449$	−9352.06	−0.982964	−0.491482	−	0.870888i	$-0.663545\pi$
$449$	−9352.06	−0.982964	−0.491482	+	0.870888i	$0.663545\pi$
$450$	0	0
$451$	−19752.2	−2.06230
$452$	1887.21	0.196387
$453$	6796.26	0.704892
$454$	−10929.9	−1.12988
$455$	0	0
$456$	−1382.03	−0.141928
$457$	8577.22	0.877955	0.438977	−	0.898498i	$-0.355341\pi$
$457$	8577.22	0.877955	0.438977	+	0.898498i	$0.355341\pi$
$458$	−3887.78	−0.396646
$459$	−5374.39	−0.546525
$460$	0	0
$461$	−8463.98	−0.855113	−0.427556	−	0.903989i	$-0.640625\pi$
$461$	−8463.98	−0.855113	−0.427556	+	0.903989i	$0.640625\pi$
$462$	0	0
$463$	−4297.74	−0.431388	−0.215694	−	0.976461i	$-0.569201\pi$
$463$	−4297.74	−0.431388	−0.215694	+	0.976461i	$0.569201\pi$
$464$	4509.52	0.451183
$465$	0	0
$466$	6614.50	0.657533
$467$	−1030.13	−0.102074	−0.0510371	−	0.998697i	$-0.516253\pi$
$467$	−1030.13	−0.102074	−0.0510371	+	0.998697i	$0.516253\pi$
$468$	−3139.12	−0.310055
$469$	0	0
$470$	0	0
$471$	6938.97	0.678834
$472$	−3844.10	−0.374871
$473$	10076.9	0.979569
$474$	−4746.79	−0.459973
$475$	0	0
$476$	0	0
$477$	10861.5	1.04259
$478$	−12216.0	−1.16892
$479$	5137.48	0.490058	0.245029	−	0.969516i	$-0.421203\pi$
$479$	5137.48	0.490058	0.245029	+	0.969516i	$0.421203\pi$
$480$	0	0
$481$	8793.21	0.833546
$482$	7925.20	0.748927
$483$	0	0
$484$	4477.65	0.420515
$485$	0	0
$486$	7725.78	0.721087
$487$	526.537	0.0489932	0.0244966	−	0.999700i	$-0.492202\pi$
$487$	526.537	0.0489932	0.0244966	+	0.999700i	$0.492202\pi$
$488$	187.979	0.0174373
$489$	−269.076	−0.0248835
$490$	0	0
$491$	−1546.87	−0.142178	−0.0710889	−	0.997470i	$-0.522647\pi$
$491$	−1546.87	−0.142178	−0.0710889	+	0.997470i	$0.522647\pi$
$492$	−4896.13	−0.448648
$493$	−11074.0	−1.01166
$494$	5025.09	0.457671
$495$	0	0
$496$	1129.05	0.102209
$497$	0	0
$498$	−8228.54	−0.740421
$499$	9492.84	0.851619	0.425809	−	0.904813i	$-0.359989\pi$
$499$	9492.84	0.851619	0.425809	+	0.904813i	$0.359989\pi$
$500$	0	0
$501$	5679.27	0.506449
$502$	7.84599	0.000697577 0
$503$	−9819.97	−0.870479	−0.435239	−	0.900315i	$-0.643336\pi$
$503$	−9819.97	−0.870479	−0.435239	+	0.900315i	$0.643336\pi$
$504$	0	0
$505$	0	0
$506$	−9074.67	−0.797269
$507$	633.363	0.0554806
$508$	809.177	0.0706721
$509$	−2638.05	−0.229724	−0.114862	−	0.993381i	$-0.536643\pi$
$509$	−2638.05	−0.229724	−0.114862	+	0.993381i	$0.536643\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−7703.06	−0.662960
$514$	−439.189	−0.0376884
$515$	0	0
$516$	2497.84	0.213103
$517$	−3390.80	−0.288448
$518$	0	0
$519$	−5650.74	−0.477919
$520$	0	0
$521$	4426.35	0.372211	0.186106	−	0.982530i	$-0.440413\pi$
$521$	4426.35	0.372211	0.186106	+	0.982530i	$0.440413\pi$
$522$	9915.24	0.831376
$523$	905.764	0.0757291	0.0378645	−	0.999283i	$-0.487944\pi$
$523$	905.764	0.0757291	0.0378645	+	0.999283i	$0.487944\pi$
$524$	−4675.82	−0.389817
$525$	0	0
$526$	−10935.4	−0.906478
$527$	−2772.60	−0.229177
$528$	2429.61	0.200256
$529$	−3765.39	−0.309476
$530$	0	0
$531$	−8452.16	−0.690758
$532$	0	0
$533$	17802.5	1.44674
$534$	7982.58	0.646892
$535$	0	0
$536$	−2016.99	−0.162539
$537$	−2296.19	−0.184521
$538$	583.462	0.0467562
$539$	0	0
$540$	0	0
$541$	19620.6	1.55926	0.779628	−	0.626243i	$-0.215408\pi$
$541$	19620.6	1.55926	0.779628	+	0.626243i	$0.215408\pi$
$542$	−5322.04	−0.421774
$543$	4275.91	0.337932
$544$	1257.32	0.0990939
$545$	0	0
$546$	0	0
$547$	6717.26	0.525062	0.262531	−	0.964923i	$-0.415443\pi$
$547$	6717.26	0.525062	0.262531	+	0.964923i	$0.415443\pi$
$548$	586.870	0.0457479
$549$	413.316	0.0321310
$550$	0	0
$551$	−15872.3	−1.22719
$552$	−2249.41	−0.173444
$553$	0	0
$554$	−9005.31	−0.690612
$555$	0	0
$556$	−10022.5	−0.764479
$557$	−13104.0	−0.996827	−0.498413	−	0.866940i	$-0.666084\pi$
$557$	−13104.0	−0.996827	−0.498413	+	0.866940i	$0.666084\pi$
$558$	2482.48	0.188336
$559$	−9082.21	−0.687186
$560$	0	0
$561$	−5966.40	−0.449022
$562$	924.776	0.0694116
$563$	−16648.7	−1.24629	−0.623143	−	0.782108i	$-0.714145\pi$
$563$	−16648.7	−1.24629	−0.623143	+	0.782108i	$0.714145\pi$
$564$	−840.505	−0.0627512
$565$	0	0
$566$	14291.4	1.06133
$567$	0	0
$568$	−6685.12	−0.493841
$569$	14697.7	1.08288	0.541441	−	0.840739i	$-0.317879\pi$
$569$	14697.7	1.08288	0.541441	+	0.840739i	$0.317879\pi$
$570$	0	0
$571$	2997.78	0.219708	0.109854	−	0.993948i	$-0.464962\pi$
$571$	2997.78	0.219708	0.109854	+	0.993948i	$0.464962\pi$
$572$	−8834.13	−0.645758
$573$	2939.28	0.214294
$574$	0	0
$575$	0	0
$576$	−1125.75	−0.0814347
$577$	4873.27	0.351607	0.175803	−	0.984425i	$-0.443748\pi$
$577$	4873.27	0.351607	0.175803	+	0.984425i	$0.443748\pi$
$578$	6738.40	0.484915
$579$	−14040.5	−1.00778
$580$	0	0
$581$	0	0
$582$	−1986.13	−0.141457
$583$	30566.6	2.17142
$584$	2062.29	0.146127
$585$	0	0
$586$	17312.4	1.22043
$587$	−23971.4	−1.68553	−0.842764	−	0.538283i	$-0.819073\pi$
$587$	−23971.4	−1.68553	−0.842764	+	0.538283i	$0.819073\pi$
$588$	0	0
$589$	−3973.94	−0.278002
$590$	0	0
$591$	−10519.6	−0.732180
$592$	3153.43	0.218928
$593$	−1561.67	−0.108145	−0.0540726	−	0.998537i	$-0.517220\pi$
$593$	−1561.67	−0.108145	−0.0540726	+	0.998537i	$0.517220\pi$
$594$	13542.0	0.935413
$595$	0	0
$596$	8432.71	0.579559
$597$	447.916	0.0307069
$598$	8178.91	0.559299
$599$	17401.4	1.18698	0.593491	−	0.804841i	$-0.297749\pi$
$599$	17401.4	1.18698	0.593491	+	0.804841i	$0.297749\pi$
$600$	0	0
$601$	−26823.7	−1.82057	−0.910283	−	0.413987i	$-0.864136\pi$
$601$	−26823.7	−1.82057	−0.910283	+	0.413987i	$0.864136\pi$
$602$	0	0
$603$	−4434.84	−0.299503
$604$	−8862.02	−0.597004
$605$	0	0
$606$	7245.76	0.485707
$607$	22721.4	1.51933	0.759665	−	0.650315i	$-0.225363\pi$
$607$	22721.4	1.51933	0.759665	+	0.650315i	$0.225363\pi$
$608$	1802.10	0.120205
$609$	0	0
$610$	0	0
$611$	3056.10	0.202351
$612$	2764.51	0.182596
$613$	−22931.6	−1.51092	−0.755462	−	0.655192i	$-0.772588\pi$
$613$	−22931.6	−1.51092	−0.755462	+	0.655192i	$0.772588\pi$
$614$	8592.52	0.564765
$615$	0	0
$616$	0	0
$617$	4299.04	0.280507	0.140253	−	0.990116i	$-0.455208\pi$
$617$	4299.04	0.280507	0.140253	+	0.990116i	$0.455208\pi$
$618$	−4491.21	−0.292335
$619$	−5321.89	−0.345565	−0.172783	−	0.984960i	$-0.555276\pi$
$619$	−5321.89	−0.345565	−0.172783	+	0.984960i	$0.555276\pi$
$620$	0	0
$621$	−12537.6	−0.810173
$622$	−7870.25	−0.507345
$623$	0	0
$624$	−2189.79	−0.140483
$625$	0	0
$626$	20997.9	1.34064
$627$	−8551.58	−0.544684
$628$	−9048.11	−0.574935
$629$	−7743.87	−0.490888
$630$	0	0
$631$	19486.3	1.22938	0.614688	−	0.788771i	$-0.289282\pi$
$631$	19486.3	1.22938	0.614688	+	0.788771i	$0.289282\pi$
$632$	6189.60	0.389571
$633$	2163.91	0.135873
$634$	11494.3	0.720030
$635$	0	0
$636$	7576.77	0.472388
$637$	0	0
$638$	27903.6	1.73152
$639$	−14698.8	−0.909979
$640$	0	0
$641$	−28574.7	−1.76074	−0.880368	−	0.474290i	$-0.842705\pi$
$641$	−28574.7	−1.76074	−0.880368	+	0.474290i	$0.842705\pi$
$642$	8209.13	0.504655
$643$	−6347.10	−0.389277	−0.194639	−	0.980875i	$-0.562353\pi$
$643$	−6347.10	−0.389277	−0.194639	+	0.980875i	$0.562353\pi$
$644$	0	0
$645$	0	0
$646$	−4425.42	−0.269529
$647$	−2649.69	−0.161005	−0.0805024	−	0.996754i	$-0.525652\pi$
$647$	−2649.69	−0.161005	−0.0805024	+	0.996754i	$0.525652\pi$
$648$	−442.652	−0.0268349
$649$	−23786.2	−1.43866
$650$	0	0
$651$	0	0
$652$	350.863	0.0210750
$653$	7061.62	0.423189	0.211595	−	0.977358i	$-0.432134\pi$
$653$	7061.62	0.423189	0.211595	+	0.977358i	$0.432134\pi$
$654$	−317.539	−0.0189859
$655$	0	0
$656$	6384.34	0.379980
$657$	4534.42	0.269261
$658$	0	0
$659$	11227.0	0.663646	0.331823	−	0.943342i	$-0.392336\pi$
$659$	11227.0	0.663646	0.331823	+	0.943342i	$0.392336\pi$
$660$	0	0
$661$	28750.0	1.69175	0.845873	−	0.533385i	$-0.179080\pi$
$661$	28750.0	1.69175	0.845873	+	0.533385i	$0.179080\pi$
$662$	−18816.2	−1.10470
$663$	5377.46	0.314997
$664$	10729.6	0.627095
$665$	0	0
$666$	6933.56	0.403408
$667$	−25834.0	−1.49969
$668$	−7405.52	−0.428934
$669$	−6847.07	−0.395700
$670$	0	0
$671$	1163.16	0.0669199
$672$	0	0
$673$	−4440.57	−0.254341	−0.127170	−	0.991881i	$-0.540589\pi$
$673$	−4440.57	−0.254341	−0.127170	+	0.991881i	$0.540589\pi$
$674$	−9003.90	−0.514566
$675$	0	0
$676$	−825.878	−0.0469889
$677$	−16309.8	−0.925905	−0.462953	−	0.886383i	$-0.653210\pi$
$677$	−16309.8	−0.925905	−0.462953	+	0.886383i	$0.653210\pi$
$678$	2894.59	0.163962
$679$	0	0
$680$	0	0
$681$	−16764.3	−0.943330
$682$	6986.21	0.392252
$683$	4866.44	0.272634	0.136317	−	0.990665i	$-0.456473\pi$
$683$	4866.44	0.272634	0.136317	+	0.990665i	$0.456473\pi$
$684$	3962.35	0.221497
$685$	0	0
$686$	0	0
$687$	−5963.05	−0.331157
$688$	−3257.07	−0.180486
$689$	−27549.4	−1.52329
$690$	0	0
$691$	−15027.1	−0.827291	−0.413646	−	0.910438i	$-0.635745\pi$
$691$	−15027.1	−0.827291	−0.413646	+	0.910438i	$0.635745\pi$
$692$	7368.31	0.404771
$693$	0	0
$694$	9986.72	0.546240
$695$	0	0
$696$	6916.67	0.376689
$697$	−15678.0	−0.852005
$698$	−13350.1	−0.723939
$699$	10145.3	0.548970
$700$	0	0
$701$	−10821.8	−0.583073	−0.291536	−	0.956560i	$-0.594166\pi$
$701$	−10821.8	−0.583073	−0.291536	+	0.956560i	$0.594166\pi$
$702$	−12205.3	−0.656209
$703$	−11099.2	−0.595469
$704$	−3168.10	−0.169606
$705$	0	0
$706$	5678.64	0.302718
$707$	0	0
$708$	−5896.06	−0.312977
$709$	20502.7	1.08603	0.543016	−	0.839722i	$-0.317282\pi$
$709$	20502.7	1.08603	0.543016	+	0.839722i	$0.317282\pi$
$710$	0	0
$711$	13609.3	0.717846
$712$	−10408.9	−0.547881
$713$	−6468.05	−0.339734
$714$	0	0
$715$	0	0
$716$	2994.13	0.156279
$717$	−18736.8	−0.975926
$718$	16968.8	0.881993
$719$	273.658	0.0141943	0.00709717	−	0.999975i	$-0.497741\pi$
$719$	273.658	0.0141943	0.00709717	+	0.999975i	$0.497741\pi$
$720$	0	0
$721$	0	0
$722$	7375.09	0.380156
$723$	12155.6	0.625273
$724$	−5575.60	−0.286209
$725$	0	0
$726$	6867.79	0.351085
$727$	1244.59	0.0634931	0.0317465	−	0.999496i	$-0.489893\pi$
$727$	1244.59	0.0634931	0.0317465	+	0.999496i	$0.489893\pi$
$728$	0	0
$729$	10355.8	0.526129
$730$	0	0
$731$	7998.39	0.404694
$732$	288.321	0.0145583
$733$	−15653.2	−0.788765	−0.394383	−	0.918946i	$-0.629042\pi$
$733$	−15653.2	−0.788765	−0.394383	+	0.918946i	$0.629042\pi$
$734$	22639.5	1.13847
$735$	0	0
$736$	2933.13	0.146898
$737$	−12480.6	−0.623782
$738$	14037.5	0.700172
$739$	−5366.29	−0.267121	−0.133560	−	0.991041i	$-0.542641\pi$
$739$	−5366.29	−0.267121	−0.133560	+	0.991041i	$0.542641\pi$
$740$	0	0
$741$	7707.46	0.382106
$742$	0	0
$743$	7927.48	0.391428	0.195714	−	0.980661i	$-0.437298\pi$
$743$	7927.48	0.391428	0.195714	+	0.980661i	$0.437298\pi$
$744$	1731.73	0.0853335
$745$	0	0
$746$	22612.4	1.10978
$747$	23591.7	1.15552
$748$	7779.91	0.380297
$749$	0	0
$750$	0	0
$751$	7386.52	0.358906	0.179453	−	0.983767i	$-0.442567\pi$
$751$	7386.52	0.358906	0.179453	+	0.983767i	$0.442567\pi$
$752$	1095.98	0.0531467
$753$	12.0341	0.000582401 0
$754$	−25149.2	−1.21470
$755$	0	0
$756$	0	0
$757$	−13406.0	−0.643659	−0.321829	−	0.946798i	$-0.604298\pi$
$757$	−13406.0	−0.643659	−0.321829	+	0.946798i	$0.604298\pi$
$758$	21252.8	1.01839
$759$	−13918.7	−0.665634
$760$	0	0
$761$	−25885.0	−1.23302	−0.616510	−	0.787347i	$-0.711454\pi$
$761$	−25885.0	−1.23302	−0.616510	+	0.787347i	$0.711454\pi$
$762$	1241.11	0.0590036
$763$	0	0
$764$	−3832.69	−0.181495
$765$	0	0
$766$	−15165.8	−0.715354
$767$	21438.2	1.00924
$768$	−785.303	−0.0368974
$769$	11600.6	0.543991	0.271996	−	0.962299i	$-0.412316\pi$
$769$	11600.6	0.543991	0.271996	+	0.962299i	$0.412316\pi$
$770$	0	0
$771$	−673.626	−0.0314657
$772$	18308.2	0.853530
$773$	28247.3	1.31434	0.657170	−	0.753742i	$-0.271753\pi$
$773$	28247.3	1.31434	0.657170	+	0.753742i	$0.271753\pi$
$774$	−7161.44	−0.332575
$775$	0	0
$776$	2589.83	0.119806
$777$	0	0
$778$	−6425.65	−0.296106
$779$	−22471.2	−1.03352
$780$	0	0
$781$	−41365.6	−1.89523
$782$	−7202.89	−0.329379
$783$	38551.7	1.75955
$784$	0	0
$785$	0	0
$786$	−7171.75	−0.325456
$787$	10505.3	0.475822	0.237911	−	0.971287i	$-0.423537\pi$
$787$	10505.3	0.475822	0.237911	+	0.971287i	$0.423537\pi$
$788$	13717.1	0.620115
$789$	−16772.7	−0.756812
$790$	0	0
$791$	0	0
$792$	−6965.83	−0.312525
$793$	−1048.34	−0.0469455
$794$	26361.6	1.17826
$795$	0	0
$796$	−584.063	−0.0260070
$797$	−2165.99	−0.0962651	−0.0481325	−	0.998841i	$-0.515327\pi$
$797$	−2165.99	−0.0962651	−0.0481325	+	0.998841i	$0.515327\pi$
$798$	0	0
$799$	−2691.40	−0.119168
$800$	0	0
$801$	−22886.5	−1.00956
$802$	14388.5	0.633509
$803$	12760.8	0.560796
$804$	−3093.65	−0.135702
$805$	0	0
$806$	−6296.60	−0.275172
$807$	894.911	0.0390364
$808$	−9448.14	−0.411367
$809$	5348.67	0.232446	0.116223	−	0.993223i	$-0.462921\pi$
$809$	5348.67	0.232446	0.116223	+	0.993223i	$0.462921\pi$
$810$	0	0
$811$	−33841.2	−1.46526	−0.732629	−	0.680628i	$-0.761707\pi$
$811$	−33841.2	−1.46526	−0.732629	+	0.680628i	$0.761707\pi$
$812$	0	0
$813$	−8162.92	−0.352135
$814$	19512.5	0.840187
$815$	0	0
$816$	1928.47	0.0827327
$817$	11464.0	0.490912
$818$	2178.31	0.0931084
$819$	0	0
$820$	0	0
$821$	14581.5	0.619850	0.309925	−	0.950761i	$-0.399696\pi$
$821$	14581.5	0.619850	0.309925	+	0.950761i	$0.399696\pi$
$822$	900.138	0.0381946
$823$	−3078.27	−0.130379	−0.0651894	−	0.997873i	$-0.520765\pi$
$823$	−3078.27	−0.130379	−0.0651894	+	0.997873i	$0.520765\pi$
$824$	5856.34	0.247591
$825$	0	0
$826$	0	0
$827$	−40275.0	−1.69347	−0.846734	−	0.532017i	$-0.821434\pi$
$827$	−40275.0	−1.69347	−0.846734	+	0.532017i	$0.821434\pi$
$828$	6449.18	0.270682
$829$	29186.9	1.22280	0.611402	−	0.791320i	$-0.290606\pi$
$829$	29186.9	1.22280	0.611402	+	0.791320i	$0.290606\pi$
$830$	0	0
$831$	−13812.3	−0.576587
$832$	2855.38	0.118981
$833$	0	0
$834$	−15372.5	−0.638258
$835$	0	0
$836$	11150.9	0.461317
$837$	9652.18	0.398600
$838$	25029.0	1.03176
$839$	9396.07	0.386637	0.193318	−	0.981136i	$-0.438075\pi$
$839$	9396.07	0.386637	0.193318	+	0.981136i	$0.438075\pi$
$840$	0	0
$841$	55047.5	2.25706
$842$	1654.04	0.0676985
$843$	1418.42	0.0579512
$844$	−2821.64	−0.115077
$845$	0	0
$846$	2409.77	0.0979312
$847$	0	0
$848$	−9879.77	−0.400086
$849$	21920.1	0.886097
$850$	0	0
$851$	−18065.3	−0.727696
$852$	−10253.6	−0.412304
$853$	20167.0	0.809502	0.404751	−	0.914427i	$-0.367358\pi$
$853$	20167.0	0.809502	0.404751	+	0.914427i	$0.367358\pi$
$854$	0	0
$855$	0	0
$856$	−10704.3	−0.427414
$857$	42479.4	1.69320	0.846599	−	0.532232i	$-0.178647\pi$
$857$	42479.4	1.69320	0.846599	+	0.532232i	$0.178647\pi$
$858$	−13549.7	−0.539138
$859$	21043.7	0.835857	0.417929	−	0.908480i	$-0.362756\pi$
$859$	21043.7	0.835857	0.417929	+	0.908480i	$0.362756\pi$
$860$	0	0
$861$	0	0
$862$	3360.88	0.132798
$863$	−8451.15	−0.333349	−0.166675	−	0.986012i	$-0.553303\pi$
$863$	−8451.15	−0.333349	−0.166675	+	0.986012i	$0.553303\pi$
$864$	−4377.07	−0.172351
$865$	0	0
$866$	14843.1	0.582436
$867$	10335.3	0.404851
$868$	0	0
$869$	38299.4	1.49507
$870$	0	0
$871$	11248.6	0.437594
$872$	414.057	0.0160800
$873$	5694.35	0.220761
$874$	−10323.8	−0.399552
$875$	0	0
$876$	3163.12	0.122000
$877$	34430.9	1.32571	0.662856	−	0.748747i	$-0.269344\pi$
$877$	34430.9	1.32571	0.662856	+	0.748747i	$0.269344\pi$
$878$	−9596.46	−0.368867
$879$	26553.7	1.01892
$880$	0	0
$881$	−11602.4	−0.443695	−0.221848	−	0.975081i	$-0.571209\pi$
$881$	−11602.4	−0.443695	−0.221848	+	0.975081i	$0.571209\pi$
$882$	0	0
$883$	−37934.8	−1.44576	−0.722880	−	0.690974i	$-0.757182\pi$
$883$	−37934.8	−1.44576	−0.722880	+	0.690974i	$0.757182\pi$
$884$	−7011.96	−0.266785
$885$	0	0
$886$	31871.2	1.20850
$887$	−25436.3	−0.962871	−0.481436	−	0.876481i	$-0.659885\pi$
$887$	−25436.3	−0.962871	−0.481436	+	0.876481i	$0.659885\pi$
$888$	4836.71	0.182781
$889$	0	0
$890$	0	0
$891$	−2739.00	−0.102985
$892$	8928.28	0.335135
$893$	−3857.56	−0.144556
$894$	12934.1	0.483870
$895$	0	0
$896$	0	0
$897$	12544.8	0.466954
$898$	18704.1	0.695061
$899$	19888.5	0.737840
$900$	0	0
$901$	24261.8	0.897088
$902$	39504.4	1.45826
$903$	0	0
$904$	−3774.42	−0.138867
$905$	0	0
$906$	−13592.5	−0.498434
$907$	−6031.85	−0.220821	−0.110410	−	0.993886i	$-0.535217\pi$
$907$	−6031.85	−0.220821	−0.110410	+	0.993886i	$0.535217\pi$
$908$	21859.8	0.798947
$909$	−20774.0	−0.758008
$910$	0	0
$911$	−21631.5	−0.786699	−0.393349	−	0.919389i	$-0.628684\pi$
$911$	−21631.5	−0.786699	−0.393349	+	0.919389i	$0.628684\pi$
$912$	2764.05	0.100359
$913$	66391.9	2.40663
$914$	−17154.4	−0.620808
$915$	0	0
$916$	7775.56	0.280471
$917$	0	0
$918$	10748.8	0.386451
$919$	−28244.7	−1.01383	−0.506913	−	0.861997i	$-0.669214\pi$
$919$	−28244.7	−1.01383	−0.506913	+	0.861997i	$0.669214\pi$
$920$	0	0
$921$	13179.2	0.471518
$922$	16928.0	0.604656
$923$	37282.4	1.32954
$924$	0	0
$925$	0	0
$926$	8595.47	0.305038
$927$	12876.5	0.456226
$928$	−9019.03	−0.319035
$929$	44944.2	1.58727	0.793633	−	0.608396i	$-0.208187\pi$
$929$	44944.2	1.58727	0.793633	+	0.608396i	$0.208187\pi$
$930$	0	0
$931$	0	0
$932$	−13229.0	−0.464946
$933$	−12071.4	−0.423578
$934$	2060.25	0.0721773
$935$	0	0
$936$	6278.24	0.219242
$937$	−3777.71	−0.131710	−0.0658551	−	0.997829i	$-0.520978\pi$
$937$	−3777.71	−0.131710	−0.0658551	+	0.997829i	$0.520978\pi$
$938$	0	0
$939$	32206.4	1.11929
$940$	0	0
$941$	−352.846	−0.0122236	−0.00611182	−	0.999981i	$-0.501945\pi$
$941$	−352.846	−0.0122236	−0.00611182	+	0.999981i	$0.501945\pi$
$942$	−13877.9	−0.480008
$943$	−36574.4	−1.26302
$944$	7688.20	0.265074
$945$	0	0
$946$	−20153.8	−0.692660
$947$	−13534.8	−0.464438	−0.232219	−	0.972664i	$-0.574599\pi$
$947$	−13534.8	−0.464438	−0.232219	+	0.972664i	$0.574599\pi$
$948$	9493.58	0.325250
$949$	−11501.2	−0.393409
$950$	0	0
$951$	17630.0	0.601147
$952$	0	0
$953$	−51886.1	−1.76365	−0.881824	−	0.471579i	$-0.843684\pi$
$953$	−51886.1	−1.76365	−0.881824	+	0.471579i	$0.843684\pi$
$954$	−21723.0	−0.737221
$955$	0	0
$956$	24432.0	0.826554
$957$	42798.3	1.44564
$958$	−10275.0	−0.346523
$959$	0	0
$960$	0	0
$961$	−24811.5	−0.832853
$962$	−17586.4	−0.589406
$963$	−23536.0	−0.787578
$964$	−15850.4	−0.529571
$965$	0	0
$966$	0	0
$967$	45517.8	1.51371	0.756853	−	0.653585i	$-0.226736\pi$
$967$	45517.8	1.51371	0.756853	+	0.653585i	$0.226736\pi$
$968$	−8955.30	−0.297349
$969$	−6787.69	−0.225028
$970$	0	0
$971$	23398.3	0.773313	0.386656	−	0.922224i	$-0.373630\pi$
$971$	23398.3	0.773313	0.386656	+	0.922224i	$0.373630\pi$
$972$	−15451.6	−0.509886
$973$	0	0
$974$	−1053.07	−0.0346434
$975$	0	0
$976$	−375.958	−0.0123300
$977$	34323.9	1.12397	0.561985	−	0.827148i	$-0.310038\pi$
$977$	34323.9	1.12397	0.561985	+	0.827148i	$0.310038\pi$
$978$	538.152	0.0175953
$979$	−64407.4	−2.10262
$980$	0	0
$981$	910.402	0.0296299
$982$	3093.74	0.100535
$983$	37157.0	1.20562	0.602809	−	0.797885i	$-0.294048\pi$
$983$	37157.0	1.20562	0.602809	+	0.797885i	$0.294048\pi$
$984$	9792.27	0.317242
$985$	0	0
$986$	22148.0	0.715352
$987$	0	0
$988$	−10050.2	−0.323622
$989$	18659.0	0.599921
$990$	0	0
$991$	−10542.5	−0.337935	−0.168968	−	0.985622i	$-0.554043\pi$
$991$	−10542.5	−0.337935	−0.168968	+	0.985622i	$0.554043\pi$
$992$	−2258.09	−0.0722727
$993$	−28860.2	−0.922306
$994$	0	0
$995$	0	0
$996$	16457.1	0.523556
$997$	−36255.9	−1.15169	−0.575846	−	0.817558i	$-0.695327\pi$
$997$	−36255.9	−1.15169	−0.575846	+	0.817558i	$0.695327\pi$
$998$	−18985.7	−0.602185
$999$	26958.6	0.853785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cv.1.3		6
5.2	odd	4		490.4.c.f.99.4		12
5.3	odd	4		490.4.c.f.99.9		12
5.4	even	2		2450.4.a.cy.1.4		6
7.2	even	3		350.4.e.o.151.4		12
7.4	even	3		350.4.e.o.51.4		12
7.6	odd	2		2450.4.a.cw.1.4		6
35.2	odd	12		70.4.i.a.39.3	yes	24
35.4	even	6		350.4.e.n.51.3		12
35.9	even	6		350.4.e.n.151.3		12
35.13	even	4		490.4.c.e.99.10		12
35.18	odd	12		70.4.i.a.9.3	✓	24
35.23	odd	12		70.4.i.a.39.10	yes	24
35.27	even	4		490.4.c.e.99.3		12
35.32	odd	12		70.4.i.a.9.10	yes	24
35.34	odd	2		2450.4.a.cx.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.i.a.9.3	✓	24	35.18	odd	12
70.4.i.a.9.10	yes	24	35.32	odd	12
70.4.i.a.39.3	yes	24	35.2	odd	12
70.4.i.a.39.10	yes	24	35.23	odd	12
350.4.e.n.51.3		12	35.4	even	6
350.4.e.n.151.3		12	35.9	even	6
350.4.e.o.51.4		12	7.4	even	3
350.4.e.o.151.4		12	7.2	even	3
490.4.c.e.99.3		12	35.27	even	4
490.4.c.e.99.10		12	35.13	even	4
490.4.c.f.99.4		12	5.2	odd	4
490.4.c.f.99.9		12	5.3	odd	4
2450.4.a.cv.1.3		6	1.1	even	1	trivial
2450.4.a.cw.1.4		6	7.6	odd	2
2450.4.a.cx.1.3		6	35.34	odd	2
2450.4.a.cy.1.4		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.06759 q^{3} +4.00000 q^{4} +6.13518 q^{6} -8.00000 q^{8} -17.5899 q^{9} -49.5016 q^{11} -12.2704 q^{12} +44.6154 q^{13} +16.0000 q^{16} -39.2912 q^{17} +35.1798 q^{18} -56.3157 q^{19} +99.0033 q^{22} -91.6603 q^{23} +24.5407 q^{24} -89.2307 q^{26} +136.783 q^{27} +281.845 q^{29} +70.5654 q^{31} -32.0000 q^{32} +151.851 q^{33} +78.5824 q^{34} -70.3596 q^{36} +197.089 q^{37} +112.631 q^{38} -136.862 q^{39} +399.021 q^{41} -203.567 q^{43} -198.007 q^{44} +183.321 q^{46} +68.4988 q^{47} -49.0814 q^{48} +120.529 q^{51} +178.461 q^{52} -617.486 q^{53} -273.567 q^{54} +172.753 q^{57} -563.690 q^{58} +480.513 q^{59} -23.4974 q^{61} -141.131 q^{62} +64.0000 q^{64} -303.701 q^{66} +252.124 q^{67} -157.165 q^{68} +281.176 q^{69} +835.640 q^{71} +140.719 q^{72} -257.786 q^{73} -394.179 q^{74} -225.263 q^{76} +273.723 q^{78} -773.700 q^{79} +55.3315 q^{81} -798.043 q^{82} -1341.21 q^{83} +407.134 q^{86} -864.584 q^{87} +396.013 q^{88} +1301.12 q^{89} -366.641 q^{92} -216.466 q^{93} -136.998 q^{94} +98.1629 q^{96} -323.729 q^{97} +870.729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} - 7 q^{3} + 24 q^{4} + 14 q^{6} - 48 q^{8} + 31 q^{9} + 31 q^{11} - 28 q^{12} - 59 q^{13} + 96 q^{16} - 68 q^{17} - 62 q^{18} + 93 q^{19} - 62 q^{22} + 94 q^{23} + 56 q^{24} + 118 q^{26}+ \cdots + 4273 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 - 7 * q^3 + 24 * q^4 + 14 * q^6 - 48 * q^8 + 31 * q^9 + 31 * q^11 - 28 * q^12 - 59 * q^13 + 96 * q^16 - 68 * q^17 - 62 * q^18 + 93 * q^19 - 62 * q^22 + 94 * q^23 + 56 * q^24 + 118 * q^26 - 385 * q^27 + 169 * q^29 + 326 * q^31 - 192 * q^32 - 400 * q^33 + 136 * q^34 + 124 * q^36 + 253 * q^37 - 186 * q^38 - 434 * q^39 + 198 * q^41 + 99 * q^43 + 124 * q^44 - 188 * q^46 - 901 * q^47 - 112 * q^48 + 724 * q^51 - 236 * q^52 - 233 * q^53 + 770 * q^54 + 518 * q^57 - 338 * q^58 + 668 * q^59 + 157 * q^61 - 652 * q^62 + 384 * q^64 + 800 * q^66 - 1193 * q^67 - 272 * q^68 - 45 * q^69 + 1108 * q^71 - 248 * q^72 - 1458 * q^73 - 506 * q^74 + 372 * q^76 + 868 * q^78 - 886 * q^79 - 614 * q^81 - 396 * q^82 - 1799 * q^83 - 198 * q^86 - 789 * q^87 - 248 * q^88 + 3047 * q^89 + 376 * q^92 - 1902 * q^93 + 1802 * q^94 + 224 * q^96 - 3404 * q^97 + 4273 * q^99

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