LMFDB - Embedded newform 2175.4.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,4,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2175.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:	$128.329154262$
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} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 $ x^7 - 2x^6 - 37x^5 + 55x^4 + 336x^3 - 227x^2 - 824x - 166
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.40909$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.40909 q^{2} +3.00000 q^{3} -6.01448 q^{4} -4.22726 q^{6} -22.4156 q^{7} +19.7476 q^{8} +9.00000 q^{9} +11.9568 q^{11} -18.0434 q^{12} +24.3982 q^{13} +31.5855 q^{14} +20.2898 q^{16} +57.0575 q^{17} -12.6818 q^{18} -101.555 q^{19} -67.2467 q^{21} -16.8481 q^{22} -133.400 q^{23} +59.2428 q^{24} -34.3792 q^{26} +27.0000 q^{27} +134.818 q^{28} +29.0000 q^{29} +292.953 q^{31} -186.571 q^{32} +35.8703 q^{33} -80.3989 q^{34} -54.1303 q^{36} -393.581 q^{37} +143.100 q^{38} +73.1947 q^{39} +237.918 q^{41} +94.7564 q^{42} +82.3986 q^{43} -71.9137 q^{44} +187.972 q^{46} -490.781 q^{47} +60.8693 q^{48} +159.458 q^{49} +171.173 q^{51} -146.743 q^{52} +416.624 q^{53} -38.0453 q^{54} -442.654 q^{56} -304.666 q^{57} -40.8635 q^{58} +320.424 q^{59} +612.103 q^{61} -412.796 q^{62} -201.740 q^{63} +100.576 q^{64} -50.5443 q^{66} -569.634 q^{67} -343.171 q^{68} -400.199 q^{69} -689.224 q^{71} +177.728 q^{72} -125.224 q^{73} +554.589 q^{74} +610.801 q^{76} -268.018 q^{77} -103.138 q^{78} -356.958 q^{79} +81.0000 q^{81} -335.247 q^{82} -947.383 q^{83} +404.454 q^{84} -116.107 q^{86} +87.0000 q^{87} +236.117 q^{88} -1212.05 q^{89} -546.901 q^{91} +802.330 q^{92} +878.858 q^{93} +691.553 q^{94} -559.712 q^{96} +597.340 q^{97} -224.690 q^{98} +107.611 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) $ 7 * q + 2 * q^2 + 21 * q^3 + 22 * q^4 + 6 * q^6 + 50 * q^7 + 33 * q^8 + 63 * q^9 + 76 * q^11 + 66 * q^12 - 30 * q^13 + 89 * q^14 + 138 * q^16 + 140 * q^17 + 18 * q^18 + 90 * q^19 + 150 * q^21 - 61 * q^22 - 34 * q^23 + 99 * q^24 - 241 * q^26 + 189 * q^27 + 57 * q^28 + 203 * q^29 + 524 * q^31 + 6 * q^32 + 228 * q^33 + 255 * q^34 + 198 * q^36 + 28 * q^37 - 222 * q^38 - 90 * q^39 + 1532 * q^41 + 267 * q^42 + 464 * q^43 + 1475 * q^44 + 72 * q^46 + 360 * q^47 + 414 * q^48 + 569 * q^49 + 420 * q^51 + 205 * q^52 - 282 * q^53 + 54 * q^54 + 1102 * q^56 + 270 * q^57 + 58 * q^58 + 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 450 * q^63 + 701 * q^64 - 183 * q^66 - 1546 * q^67 - 1801 * q^68 - 102 * q^69 + 1802 * q^71 + 297 * q^72 + 220 * q^73 + 1594 * q^74 + 1960 * q^76 - 3222 * q^77 - 723 * q^78 + 1298 * q^79 + 567 * q^81 - 856 * q^82 - 1652 * q^83 + 171 * q^84 + 7628 * q^86 + 609 * q^87 - 550 * q^88 + 2846 * q^89 - 816 * q^91 - 472 * q^92 + 1572 * q^93 + 745 * q^94 + 18 * q^96 - 1110 * q^97 + 761 * q^98 + 684 * 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 $

$2$	−1.40909	−0.498187	−0.249094	−	0.968479i	$-0.580133\pi$
$2$	−1.40909	−0.498187	−0.249094	+	0.968479i	$0.580133\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−6.01448	−0.751810
$5$	0	0
$5$	0	0
$6$	−4.22726	−0.287628
$7$	−22.4156	−1.21033	−0.605164	−	0.796101i	$-0.706892\pi$
$7$	−22.4156	−1.21033	−0.605164	+	0.796101i	$0.706892\pi$
$8$	19.7476	0.872729
$9$	9.00000	0.333333
$10$	0	0
$11$	11.9568	0.327736	0.163868	−	0.986482i	$-0.447603\pi$
$11$	11.9568	0.327736	0.163868	+	0.986482i	$0.447603\pi$
$12$	−18.0434	−0.434058
$13$	24.3982	0.520527	0.260264	−	0.965538i	$-0.416190\pi$
$13$	24.3982	0.520527	0.260264	+	0.965538i	$0.416190\pi$
$14$	31.5855	0.602969
$15$	0	0
$16$	20.2898	0.317027
$17$	57.0575	0.814028	0.407014	−	0.913422i	$-0.366570\pi$
$17$	57.0575	0.814028	0.407014	+	0.913422i	$0.366570\pi$
$18$	−12.6818	−0.166062
$19$	−101.555	−1.22623	−0.613115	−	0.789994i	$-0.710084\pi$
$19$	−101.555	−1.22623	−0.613115	+	0.789994i	$0.710084\pi$
$20$	0	0
$21$	−67.2467	−0.698783
$22$	−16.8481	−0.163274
$23$	−133.400	−1.20938	−0.604691	−	0.796460i	$-0.706703\pi$
$23$	−133.400	−1.20938	−0.604691	+	0.796460i	$0.706703\pi$
$24$	59.2428	0.503870
$25$	0	0
$26$	−34.3792	−0.259320
$27$	27.0000	0.192450
$28$	134.818	0.909936
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	292.953	1.69729	0.848643	−	0.528966i	$-0.177420\pi$
$31$	292.953	1.69729	0.848643	+	0.528966i	$0.177420\pi$
$32$	−186.571	−1.03067
$33$	35.8703	0.189219
$34$	−80.3989	−0.405538
$35$	0	0
$36$	−54.1303	−0.250603
$37$	−393.581	−1.74876	−0.874382	−	0.485238i	$-0.838733\pi$
$37$	−393.581	−1.74876	−0.874382	+	0.485238i	$0.838733\pi$
$38$	143.100	0.610892
$39$	73.1947	0.300527
$40$	0	0
$41$	237.918	0.906257	0.453128	−	0.891445i	$-0.350308\pi$
$41$	237.918	0.906257	0.453128	+	0.891445i	$0.350308\pi$
$42$	94.7564	0.348125
$43$	82.3986	0.292225	0.146112	−	0.989268i	$-0.453324\pi$
$43$	82.3986	0.292225	0.146112	+	0.989268i	$0.453324\pi$
$44$	−71.9137	−0.246395
$45$	0	0
$46$	187.972	0.602498
$47$	−490.781	−1.52314	−0.761572	−	0.648080i	$-0.775572\pi$
$47$	−490.781	−1.52314	−0.761572	+	0.648080i	$0.775572\pi$
$48$	60.8693	0.183036
$49$	159.458	0.464893
$50$	0	0
$51$	171.173	0.469979
$52$	−146.743	−0.391338
$53$	416.624	1.07977	0.539885	−	0.841739i	$-0.318468\pi$
$53$	416.624	1.07977	0.539885	+	0.841739i	$0.318468\pi$
$54$	−38.0453	−0.0958761
$55$	0	0
$56$	−442.654	−1.05629
$57$	−304.666	−0.707964
$58$	−40.8635	−0.0925110
$59$	320.424	0.707046	0.353523	−	0.935426i	$-0.384984\pi$
$59$	320.424	0.707046	0.353523	+	0.935426i	$0.384984\pi$
$60$	0	0
$61$	612.103	1.28478	0.642392	−	0.766376i	$-0.277942\pi$
$61$	612.103	1.28478	0.642392	+	0.766376i	$0.277942\pi$
$62$	−412.796	−0.845566
$63$	−201.740	−0.403442
$64$	100.576	0.196438
$65$	0	0
$66$	−50.5443	−0.0942663
$67$	−569.634	−1.03868	−0.519342	−	0.854567i	$-0.673823\pi$
$67$	−569.634	−1.03868	−0.519342	+	0.854567i	$0.673823\pi$
$68$	−343.171	−0.611994
$69$	−400.199	−0.698237
$70$	0	0
$71$	−689.224	−1.15205	−0.576027	−	0.817431i	$-0.695398\pi$
$71$	−689.224	−1.15205	−0.576027	+	0.817431i	$0.695398\pi$
$72$	177.728	0.290910
$73$	−125.224	−0.200772	−0.100386	−	0.994949i	$-0.532008\pi$
$73$	−125.224	−0.200772	−0.100386	+	0.994949i	$0.532008\pi$
$74$	554.589	0.871212
$75$	0	0
$76$	610.801	0.921891
$77$	−268.018	−0.396668
$78$	−103.138	−0.149718
$79$	−356.958	−0.508366	−0.254183	−	0.967156i	$-0.581807\pi$
$79$	−356.958	−0.508366	−0.254183	+	0.967156i	$0.581807\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−335.247	−0.451485
$83$	−947.383	−1.25288	−0.626439	−	0.779471i	$-0.715488\pi$
$83$	−947.383	−1.25288	−0.626439	+	0.779471i	$0.715488\pi$
$84$	404.454	0.525352
$85$	0	0
$86$	−116.107	−0.145583
$87$	87.0000	0.107211
$88$	236.117	0.286025
$89$	−1212.05	−1.44356	−0.721779	−	0.692123i	$-0.756675\pi$
$89$	−1212.05	−1.44356	−0.721779	+	0.692123i	$0.756675\pi$
$90$	0	0
$91$	−546.901	−0.630009
$92$	802.330	0.909224
$93$	878.858	0.979929
$94$	691.553	0.758811
$95$	0	0
$96$	−559.712	−0.595056
$97$	597.340	0.625265	0.312633	−	0.949874i	$-0.398789\pi$
$97$	597.340	0.625265	0.312633	+	0.949874i	$0.398789\pi$
$98$	−224.690	−0.231603
$99$	107.611	0.109245
$100$	0	0
$101$	−318.168	−0.313455	−0.156727	−	0.987642i	$-0.550094\pi$
$101$	−318.168	−0.313455	−0.156727	+	0.987642i	$0.550094\pi$
$102$	−241.197	−0.234138
$103$	629.601	0.602295	0.301148	−	0.953577i	$-0.402630\pi$
$103$	629.601	0.602295	0.301148	+	0.953577i	$0.402630\pi$
$104$	481.807	0.454279
$105$	0	0
$106$	−587.059	−0.537927
$107$	−716.131	−0.647019	−0.323509	−	0.946225i	$-0.604863\pi$
$107$	−716.131	−0.647019	−0.323509	+	0.946225i	$0.604863\pi$
$108$	−162.391	−0.144686
$109$	1969.57	1.73074	0.865371	−	0.501132i	$-0.167083\pi$
$109$	1969.57	1.73074	0.865371	+	0.501132i	$0.167083\pi$
$110$	0	0
$111$	−1180.74	−1.00965
$112$	−454.807	−0.383707
$113$	−1106.67	−0.921295	−0.460647	−	0.887583i	$-0.652383\pi$
$113$	−1106.67	−0.921295	−0.460647	+	0.887583i	$0.652383\pi$
$114$	429.300	0.352698
$115$	0	0
$116$	−174.420	−0.139608
$117$	219.584	0.173509
$118$	−451.505	−0.352241
$119$	−1278.98	−0.985241
$120$	0	0
$121$	−1188.04	−0.892589
$122$	−862.506	−0.640063
$123$	713.753	0.523228
$124$	−1761.96	−1.27604
$125$	0	0
$126$	284.269	0.200990
$127$	2496.52	1.74433	0.872167	−	0.489207i	$-0.162714\pi$
$127$	2496.52	1.74433	0.872167	+	0.489207i	$0.162714\pi$
$128$	1350.85	0.932805
$129$	247.196	0.168716
$130$	0	0
$131$	2155.49	1.43760	0.718800	−	0.695216i	$-0.244691\pi$
$131$	2155.49	1.43760	0.718800	+	0.695216i	$0.244691\pi$
$132$	−215.741	−0.142256
$133$	2276.42	1.48414
$134$	802.663	0.517459
$135$	0	0
$136$	1126.75	0.710426
$137$	−1128.30	−0.703628	−0.351814	−	0.936070i	$-0.614435\pi$
$137$	−1128.30	−0.703628	−0.351814	+	0.936070i	$0.614435\pi$
$138$	563.915	0.347852
$139$	1331.04	0.812212	0.406106	−	0.913826i	$-0.366886\pi$
$139$	1331.04	0.812212	0.406106	+	0.913826i	$0.366886\pi$
$140$	0	0
$141$	−1472.34	−0.879388
$142$	971.176	0.573938
$143$	291.724	0.170596
$144$	182.608	0.105676
$145$	0	0
$146$	176.451	0.100022
$147$	478.374	0.268406
$148$	2367.18	1.31474
$149$	2689.78	1.47890	0.739448	−	0.673214i	$-0.235087\pi$
$149$	2689.78	1.47890	0.739448	+	0.673214i	$0.235087\pi$
$150$	0	0
$151$	−1784.39	−0.961667	−0.480834	−	0.876812i	$-0.659666\pi$
$151$	−1784.39	−0.961667	−0.480834	+	0.876812i	$0.659666\pi$
$152$	−2005.47	−1.07017
$153$	513.518	0.271343
$154$	377.660	0.197615
$155$	0	0
$156$	−440.228	−0.225939
$157$	984.642	0.500529	0.250264	−	0.968178i	$-0.419482\pi$
$157$	984.642	0.500529	0.250264	+	0.968178i	$0.419482\pi$
$158$	502.984	0.253261
$159$	1249.87	0.623405
$160$	0	0
$161$	2990.23	1.46375
$162$	−114.136	−0.0553541
$163$	646.272	0.310552	0.155276	−	0.987871i	$-0.450373\pi$
$163$	646.272	0.310552	0.155276	+	0.987871i	$0.450373\pi$
$164$	−1430.95	−0.681333
$165$	0	0
$166$	1334.94	0.624167
$167$	160.346	0.0742993	0.0371496	−	0.999310i	$-0.488172\pi$
$167$	160.346	0.0742993	0.0371496	+	0.999310i	$0.488172\pi$
$168$	−1327.96	−0.609848
$169$	−1601.73	−0.729051
$170$	0	0
$171$	−913.997	−0.408743
$172$	−495.585	−0.219698
$173$	−2583.11	−1.13520	−0.567602	−	0.823303i	$-0.692129\pi$
$173$	−2583.11	−1.13520	−0.567602	+	0.823303i	$0.692129\pi$
$174$	−122.590	−0.0534113
$175$	0	0
$176$	242.600	0.103901
$177$	961.273	0.408213
$178$	1707.88	0.719162
$179$	3663.85	1.52988	0.764941	−	0.644100i	$-0.222768\pi$
$179$	3663.85	1.52988	0.764941	+	0.644100i	$0.222768\pi$
$180$	0	0
$181$	3394.16	1.39384	0.696922	−	0.717147i	$-0.254553\pi$
$181$	3394.16	1.39384	0.696922	+	0.717147i	$0.254553\pi$
$182$	770.630	0.313862
$183$	1836.31	0.741770
$184$	−2634.32	−1.05546
$185$	0	0
$186$	−1238.39	−0.488188
$187$	682.223	0.266787
$188$	2951.79	1.14511
$189$	−605.221	−0.232928
$190$	0	0
$191$	−634.977	−0.240551	−0.120276	−	0.992741i	$-0.538378\pi$
$191$	−634.977	−0.240551	−0.120276	+	0.992741i	$0.538378\pi$
$192$	301.729	0.113413
$193$	2438.75	0.909559	0.454780	−	0.890604i	$-0.349718\pi$
$193$	2438.75	0.909559	0.454780	+	0.890604i	$0.349718\pi$
$194$	−841.704	−0.311499
$195$	0	0
$196$	−959.057	−0.349511
$197$	4985.61	1.80310	0.901549	−	0.432677i	$-0.142431\pi$
$197$	4985.61	1.80310	0.901549	+	0.432677i	$0.142431\pi$
$198$	−151.633	−0.0544247
$199$	1111.42	0.395913	0.197957	−	0.980211i	$-0.436569\pi$
$199$	1111.42	0.395913	0.197957	+	0.980211i	$0.436569\pi$
$200$	0	0
$201$	−1708.90	−0.599684
$202$	448.326	0.156159
$203$	−650.052	−0.224752
$204$	−1029.51	−0.353335
$205$	0	0
$206$	−887.162	−0.300056
$207$	−1200.60	−0.403127
$208$	495.035	0.165021
$209$	−1214.27	−0.401880
$210$	0	0
$211$	−4654.76	−1.51870	−0.759352	−	0.650680i	$-0.774484\pi$
$211$	−4654.76	−1.51870	−0.759352	+	0.650680i	$0.774484\pi$
$212$	−2505.78	−0.811781
$213$	−2067.67	−0.665139
$214$	1009.09	0.322336
$215$	0	0
$216$	533.185	0.167957
$217$	−6566.71	−2.05427
$218$	−2775.30	−0.862233
$219$	−375.672	−0.115916
$220$	0	0
$221$	1392.10	0.423724
$222$	1663.77	0.502994
$223$	751.947	0.225803	0.112901	−	0.993606i	$-0.463986\pi$
$223$	751.947	0.225803	0.112901	+	0.993606i	$0.463986\pi$
$224$	4182.09	1.24745
$225$	0	0
$226$	1559.39	0.458977
$227$	1076.59	0.314784	0.157392	−	0.987536i	$-0.449691\pi$
$227$	1076.59	0.314784	0.157392	+	0.987536i	$0.449691\pi$
$228$	1832.40	0.532254
$229$	−3048.93	−0.879822	−0.439911	−	0.898041i	$-0.644990\pi$
$229$	−3048.93	−0.879822	−0.439911	+	0.898041i	$0.644990\pi$
$230$	0	0
$231$	−804.053	−0.229017
$232$	572.680	0.162062
$233$	−20.2258	−0.00568686	−0.00284343	−	0.999996i	$-0.500905\pi$
$233$	−20.2258	−0.00568686	−0.00284343	+	0.999996i	$0.500905\pi$
$234$	−309.413	−0.0864400
$235$	0	0
$236$	−1927.19	−0.531564
$237$	−1070.87	−0.293505
$238$	1802.19	0.490834
$239$	307.893	0.0833303	0.0416651	−	0.999132i	$-0.486734\pi$
$239$	307.893	0.0833303	0.0416651	+	0.999132i	$0.486734\pi$
$240$	0	0
$241$	4530.44	1.21092	0.605459	−	0.795877i	$-0.292990\pi$
$241$	4530.44	1.21092	0.605459	+	0.795877i	$0.292990\pi$
$242$	1674.04	0.444676
$243$	243.000	0.0641500
$244$	−3681.48	−0.965913
$245$	0	0
$246$	−1005.74	−0.260665
$247$	−2477.77	−0.638286
$248$	5785.12	1.48127
$249$	−2842.15	−0.723349
$250$	0	0
$251$	−1320.51	−0.332070	−0.166035	−	0.986120i	$-0.553096\pi$
$251$	−1320.51	−0.332070	−0.166035	+	0.986120i	$0.553096\pi$
$252$	1213.36	0.303312
$253$	−1595.03	−0.396358
$254$	−3517.81	−0.869005
$255$	0	0
$256$	−2708.07	−0.661149
$257$	2205.98	0.535430	0.267715	−	0.963498i	$-0.413731\pi$
$257$	2205.98	0.535430	0.267715	+	0.963498i	$0.413731\pi$
$258$	−348.320	−0.0840522
$259$	8822.34	2.11658
$260$	0	0
$261$	261.000	0.0618984
$262$	−3037.26	−0.716194
$263$	6630.48	1.55457	0.777286	−	0.629147i	$-0.216596\pi$
$263$	6630.48	1.55457	0.777286	+	0.629147i	$0.216596\pi$
$264$	708.352	0.165137
$265$	0	0
$266$	−3207.67	−0.739379
$267$	−3636.14	−0.833439
$268$	3426.05	0.780893
$269$	4744.31	1.07534	0.537669	−	0.843156i	$-0.319305\pi$
$269$	4744.31	1.07534	0.537669	+	0.843156i	$0.319305\pi$
$270$	0	0
$271$	−5641.82	−1.26464	−0.632318	−	0.774709i	$-0.717896\pi$
$271$	−5641.82	−1.26464	−0.632318	+	0.774709i	$0.717896\pi$
$272$	1157.68	0.258069
$273$	−1640.70	−0.363736
$274$	1589.87	0.350538
$275$	0	0
$276$	2406.99	0.524941
$277$	5431.05	1.17805	0.589025	−	0.808115i	$-0.299512\pi$
$277$	5431.05	1.17805	0.589025	+	0.808115i	$0.299512\pi$
$278$	−1875.55	−0.404634
$279$	2636.58	0.565762
$280$	0	0
$281$	9392.01	1.99388	0.996940	−	0.0781664i	$-0.0249065\pi$
$281$	9392.01	1.99388	0.996940	+	0.0781664i	$0.0249065\pi$
$282$	2074.66	0.438100
$283$	6584.64	1.38310	0.691548	−	0.722331i	$-0.256929\pi$
$283$	6584.64	1.38310	0.691548	+	0.722331i	$0.256929\pi$
$284$	4145.32	0.866125
$285$	0	0
$286$	−411.064	−0.0849886
$287$	−5333.07	−1.09687
$288$	−1679.14	−0.343556
$289$	−1657.44	−0.337358
$290$	0	0
$291$	1792.02	0.360997
$292$	753.157	0.150942
$293$	1391.13	0.277374	0.138687	−	0.990336i	$-0.455712\pi$
$293$	1391.13	0.277374	0.138687	+	0.990336i	$0.455712\pi$
$294$	−674.071	−0.133716
$295$	0	0
$296$	−7772.27	−1.52620
$297$	322.833	0.0630729
$298$	−3790.13	−0.736766
$299$	−3254.72	−0.629516
$300$	0	0
$301$	−1847.01	−0.353688
$302$	2514.36	0.479090
$303$	−954.505	−0.180973
$304$	−2060.53	−0.388748
$305$	0	0
$306$	−723.590	−0.135179
$307$	5027.84	0.934704	0.467352	−	0.884071i	$-0.345208\pi$
$307$	5027.84	0.934704	0.467352	+	0.884071i	$0.345208\pi$
$308$	1611.99	0.298219
$309$	1888.80	0.347735
$310$	0	0
$311$	939.987	0.171388	0.0856942	−	0.996321i	$-0.472689\pi$
$311$	939.987	0.171388	0.0856942	+	0.996321i	$0.472689\pi$
$312$	1445.42	0.262278
$313$	9028.02	1.63033	0.815166	−	0.579228i	$-0.196646\pi$
$313$	9028.02	1.63033	0.815166	+	0.579228i	$0.196646\pi$
$314$	−1387.45	−0.249357
$315$	0	0
$316$	2146.91	0.382194
$317$	2461.43	0.436112	0.218056	−	0.975936i	$-0.430028\pi$
$317$	2461.43	0.436112	0.218056	+	0.975936i	$0.430028\pi$
$318$	−1761.18	−0.310572
$319$	346.746	0.0608591
$320$	0	0
$321$	−2148.39	−0.373556
$322$	−4213.49	−0.729220
$323$	−5794.49	−0.998185
$324$	−487.173	−0.0835344
$325$	0	0
$326$	−910.652	−0.154713
$327$	5908.72	0.999244
$328$	4698.31	0.790916
$329$	11001.1	1.84350
$330$	0	0
$331$	4350.92	0.722502	0.361251	−	0.932469i	$-0.382350\pi$
$331$	4350.92	0.722502	0.361251	+	0.932469i	$0.382350\pi$
$332$	5698.01	0.941925
$333$	−3542.23	−0.582921
$334$	−225.942	−0.0370149
$335$	0	0
$336$	−1364.42	−0.221533
$337$	−2324.77	−0.375781	−0.187890	−	0.982190i	$-0.560165\pi$
$337$	−2324.77	−0.375781	−0.187890	+	0.982190i	$0.560165\pi$
$338$	2256.97	0.363204
$339$	−3320.00	−0.531910
$340$	0	0
$341$	3502.77	0.556262
$342$	1287.90	0.203631
$343$	4114.20	0.647655
$344$	1627.18	0.255033
$345$	0	0
$346$	3639.82	0.565543
$347$	−3384.11	−0.523541	−0.261770	−	0.965130i	$-0.584306\pi$
$347$	−3384.11	−0.523541	−0.261770	+	0.965130i	$0.584306\pi$
$348$	−523.260	−0.0806025
$349$	3876.35	0.594545	0.297273	−	0.954793i	$-0.403923\pi$
$349$	3876.35	0.594545	0.297273	+	0.954793i	$0.403923\pi$
$350$	0	0
$351$	658.753	0.100176
$352$	−2230.78	−0.337787
$353$	−7741.90	−1.16731	−0.583654	−	0.812003i	$-0.698377\pi$
$353$	−7741.90	−1.16731	−0.583654	+	0.812003i	$0.698377\pi$
$354$	−1354.52	−0.203366
$355$	0	0
$356$	7289.83	1.08528
$357$	−3836.93	−0.568829
$358$	−5162.68	−0.762168
$359$	−6553.34	−0.963432	−0.481716	−	0.876327i	$-0.659986\pi$
$359$	−6553.34	−0.963432	−0.481716	+	0.876327i	$0.659986\pi$
$360$	0	0
$361$	3454.46	0.503639
$362$	−4782.66	−0.694395
$363$	−3564.11	−0.515336
$364$	3289.32	0.473647
$365$	0	0
$366$	−2587.52	−0.369540
$367$	5.48077	0.000779547 0	0.000389774	−	1.00000i	$-0.499876\pi$
$367$	5.48077	0.000779547 0	0.000389774		1.00000i	$0.499876\pi$
$368$	−2706.65	−0.383407
$369$	2141.26	0.302086
$370$	0	0
$371$	−9338.88	−1.30687
$372$	−5285.87	−0.736720
$373$	−12616.7	−1.75139	−0.875696	−	0.482862i	$-0.839597\pi$
$373$	−12616.7	−1.75139	−0.875696	+	0.482862i	$0.839597\pi$
$374$	−961.311	−0.132910
$375$	0	0
$376$	−9691.75	−1.32929
$377$	707.549	0.0966595
$378$	852.808	0.116042
$379$	−7220.33	−0.978585	−0.489292	−	0.872120i	$-0.662745\pi$
$379$	−7220.33	−0.978585	−0.489292	+	0.872120i	$0.662745\pi$
$380$	0	0
$381$	7489.57	1.00709
$382$	894.737	0.119840
$383$	−3257.70	−0.434624	−0.217312	−	0.976102i	$-0.569729\pi$
$383$	−3257.70	−0.434624	−0.217312	+	0.976102i	$0.569729\pi$
$384$	4052.54	0.538555
$385$	0	0
$386$	−3436.41	−0.453131
$387$	741.588	0.0974083
$388$	−3592.69	−0.470080
$389$	12382.6	1.61395	0.806973	−	0.590588i	$-0.201104\pi$
$389$	12382.6	1.61395	0.806973	+	0.590588i	$0.201104\pi$
$390$	0	0
$391$	−7611.46	−0.984470
$392$	3148.92	0.405725
$393$	6466.46	0.829999
$394$	−7025.16	−0.898280
$395$	0	0
$396$	−647.223	−0.0821318
$397$	−9539.00	−1.20592	−0.602958	−	0.797773i	$-0.706011\pi$
$397$	−9539.00	−1.20592	−0.602958	+	0.797773i	$0.706011\pi$
$398$	−1566.09	−0.197239
$399$	6829.26	0.856868
$400$	0	0
$401$	460.813	0.0573863	0.0286932	−	0.999588i	$-0.490865\pi$
$401$	460.813	0.0573863	0.0286932	+	0.999588i	$0.490865\pi$
$402$	2407.99	0.298755
$403$	7147.54	0.883484
$404$	1913.62	0.235658
$405$	0	0
$406$	915.979	0.111969
$407$	−4705.95	−0.573133
$408$	3380.25	0.410165
$409$	12769.1	1.54374	0.771871	−	0.635780i	$-0.219321\pi$
$409$	12769.1	1.54374	0.771871	+	0.635780i	$0.219321\pi$
$410$	0	0
$411$	−3384.89	−0.406240
$412$	−3786.72	−0.452812
$413$	−7182.50	−0.855757
$414$	1691.74	0.200833
$415$	0	0
$416$	−4552.00	−0.536491
$417$	3993.13	0.468931
$418$	1711.01	0.200211
$419$	4192.84	0.488863	0.244432	−	0.969667i	$-0.421399\pi$
$419$	4192.84	0.488863	0.244432	+	0.969667i	$0.421399\pi$
$420$	0	0
$421$	−9960.32	−1.15306	−0.576528	−	0.817078i	$-0.695593\pi$
$421$	−9960.32	−1.15306	−0.576528	+	0.817078i	$0.695593\pi$
$422$	6558.95	0.756599
$423$	−4417.03	−0.507715
$424$	8227.33	0.942346
$425$	0	0
$426$	2913.53	0.331364
$427$	−13720.7	−1.55501
$428$	4307.15	0.486435
$429$	875.172	0.0984935
$430$	0	0
$431$	−898.036	−0.100364	−0.0501820	−	0.998740i	$-0.515980\pi$
$431$	−898.036	−0.100364	−0.0501820	+	0.998740i	$0.515980\pi$
$432$	547.823	0.0610120
$433$	13076.0	1.45126	0.725629	−	0.688086i	$-0.241549\pi$
$433$	13076.0	1.45126	0.725629	+	0.688086i	$0.241549\pi$
$434$	9253.05	1.02341
$435$	0	0
$436$	−11845.9	−1.30119
$437$	13547.4	1.48298
$438$	529.354	0.0577477
$439$	−64.6529	−0.00702897	−0.00351448	−	0.999994i	$-0.501119\pi$
$439$	−64.6529	−0.00702897	−0.00351448	+	0.999994i	$0.501119\pi$
$440$	0	0
$441$	1435.12	0.154964
$442$	−1961.59	−0.211094
$443$	−101.758	−0.0109135	−0.00545676	−	0.999985i	$-0.501737\pi$
$443$	−101.758	−0.0109135	−0.00545676	+	0.999985i	$0.501737\pi$
$444$	7101.55	0.759064
$445$	0	0
$446$	−1059.56	−0.112492
$447$	8069.34	0.853841
$448$	−2254.47	−0.237754
$449$	12597.0	1.32403	0.662016	−	0.749489i	$-0.269701\pi$
$449$	12597.0	1.32403	0.662016	+	0.749489i	$0.269701\pi$
$450$	0	0
$451$	2844.73	0.297013
$452$	6656.01	0.692638
$453$	−5353.18	−0.555219
$454$	−1517.01	−0.156821
$455$	0	0
$456$	−6016.41	−0.617861
$457$	−4595.05	−0.470344	−0.235172	−	0.971954i	$-0.575565\pi$
$457$	−4595.05	−0.470344	−0.235172	+	0.971954i	$0.575565\pi$
$458$	4296.21	0.438316
$459$	1540.55	0.156660
$460$	0	0
$461$	3826.24	0.386564	0.193282	−	0.981143i	$-0.438087\pi$
$461$	3826.24	0.386564	0.193282	+	0.981143i	$0.438087\pi$
$462$	1132.98	0.114093
$463$	−14390.6	−1.44447	−0.722233	−	0.691650i	$-0.756884\pi$
$463$	−14390.6	−1.44447	−0.722233	+	0.691650i	$0.756884\pi$
$464$	588.403	0.0588705
$465$	0	0
$466$	28.4999	0.00283312
$467$	11750.2	1.16431	0.582156	−	0.813077i	$-0.302209\pi$
$467$	11750.2	1.16431	0.582156	+	0.813077i	$0.302209\pi$
$468$	−1320.68	−0.130446
$469$	12768.7	1.25715
$470$	0	0
$471$	2953.93	0.288980
$472$	6327.61	0.617059
$473$	985.221	0.0957727
$474$	1508.95	0.146220
$475$	0	0
$476$	7692.38	0.740714
$477$	3749.62	0.359923
$478$	−433.848	−0.0415141
$479$	−2420.75	−0.230912	−0.115456	−	0.993313i	$-0.536833\pi$
$479$	−2420.75	−0.230912	−0.115456	+	0.993313i	$0.536833\pi$
$480$	0	0
$481$	−9602.68	−0.910280
$482$	−6383.78	−0.603264
$483$	8970.70	0.845095
$484$	7145.41	0.671057
$485$	0	0
$486$	−342.408	−0.0319587
$487$	−5597.36	−0.520822	−0.260411	−	0.965498i	$-0.583858\pi$
$487$	−5597.36	−0.520822	−0.260411	+	0.965498i	$0.583858\pi$
$488$	12087.6	1.12127
$489$	1938.81	0.179297
$490$	0	0
$491$	7787.76	0.715797	0.357899	−	0.933760i	$-0.383493\pi$
$491$	7787.76	0.715797	0.357899	+	0.933760i	$0.383493\pi$
$492$	−4292.85	−0.393368
$493$	1654.67	0.151161
$494$	3491.39	0.317986
$495$	0	0
$496$	5943.94	0.538086
$497$	15449.4	1.39436
$498$	4004.83	0.360363
$499$	758.279	0.0680265	0.0340133	−	0.999421i	$-0.489171\pi$
$499$	758.279	0.0680265	0.0340133	+	0.999421i	$0.489171\pi$
$500$	0	0
$501$	481.039	0.0428967
$502$	1860.71	0.165433
$503$	9841.58	0.872394	0.436197	−	0.899851i	$-0.356325\pi$
$503$	9841.58	0.872394	0.436197	+	0.899851i	$0.356325\pi$
$504$	−3983.88	−0.352096
$505$	0	0
$506$	2247.53	0.197460
$507$	−4805.18	−0.420918
$508$	−15015.3	−1.31141
$509$	10862.8	0.945947	0.472974	−	0.881077i	$-0.343181\pi$
$509$	10862.8	0.945947	0.472974	+	0.881077i	$0.343181\pi$
$510$	0	0
$511$	2806.97	0.243000
$512$	−6990.87	−0.603429
$513$	−2741.99	−0.235988
$514$	−3108.42	−0.266744
$515$	0	0
$516$	−1486.75	−0.126842
$517$	−5868.15	−0.499190
$518$	−12431.4	−1.05445
$519$	−7749.33	−0.655410
$520$	0	0
$521$	16027.8	1.34777	0.673886	−	0.738836i	$-0.264624\pi$
$521$	16027.8	1.34777	0.673886	+	0.738836i	$0.264624\pi$
$522$	−367.771	−0.0308370
$523$	−3028.25	−0.253186	−0.126593	−	0.991955i	$-0.540404\pi$
$523$	−3028.25	−0.253186	−0.126593	+	0.991955i	$0.540404\pi$
$524$	−12964.1	−1.08080
$525$	0	0
$526$	−9342.91	−0.774468
$527$	16715.2	1.38164
$528$	727.799	0.0599875
$529$	5628.49	0.462603
$530$	0	0
$531$	2883.82	0.235682
$532$	−13691.5	−1.11579
$533$	5804.78	0.471731
$534$	5123.63	0.415208
$535$	0	0
$536$	−11248.9	−0.906489
$537$	10991.5	0.883278
$538$	−6685.14	−0.535719
$539$	1906.60	0.152362
$540$	0	0
$541$	−5465.05	−0.434308	−0.217154	−	0.976137i	$-0.569677\pi$
$541$	−5465.05	−0.434308	−0.217154	+	0.976137i	$0.569677\pi$
$542$	7949.81	0.630025
$543$	10182.5	0.804736
$544$	−10645.3	−0.838993
$545$	0	0
$546$	2311.89	0.181208
$547$	−5025.89	−0.392855	−0.196427	−	0.980518i	$-0.562934\pi$
$547$	−5025.89	−0.392855	−0.196427	+	0.980518i	$0.562934\pi$
$548$	6786.12	0.528994
$549$	5508.93	0.428261
$550$	0	0
$551$	−2945.10	−0.227705
$552$	−7902.97	−0.609371
$553$	8001.42	0.615289
$554$	−7652.81	−0.586890
$555$	0	0
$556$	−8005.52	−0.610629
$557$	13090.4	0.995794	0.497897	−	0.867236i	$-0.334106\pi$
$557$	13090.4	0.995794	0.497897	+	0.867236i	$0.334106\pi$
$558$	−3715.16	−0.281855
$559$	2010.38	0.152111
$560$	0	0
$561$	2046.67	0.154029
$562$	−13234.1	−0.993325
$563$	18020.8	1.34900	0.674501	−	0.738274i	$-0.264359\pi$
$563$	18020.8	1.34900	0.674501	+	0.738274i	$0.264359\pi$
$564$	8855.38	0.661132
$565$	0	0
$566$	−9278.32	−0.689040
$567$	−1815.66	−0.134481
$568$	−13610.5	−1.00543
$569$	1614.50	0.118951	0.0594755	−	0.998230i	$-0.481057\pi$
$569$	1614.50	0.118951	0.0594755	+	0.998230i	$0.481057\pi$
$570$	0	0
$571$	20589.2	1.50899	0.754493	−	0.656308i	$-0.227883\pi$
$571$	20589.2	1.50899	0.754493	+	0.656308i	$0.227883\pi$
$572$	−1754.57	−0.128256
$573$	−1904.93	−0.138882
$574$	7514.75	0.546445
$575$	0	0
$576$	905.186	0.0654793
$577$	7402.23	0.534071	0.267035	−	0.963687i	$-0.413956\pi$
$577$	7402.23	0.534071	0.267035	+	0.963687i	$0.413956\pi$
$578$	2335.47	0.168067
$579$	7316.25	0.525134
$580$	0	0
$581$	21236.1	1.51639
$582$	−2525.11	−0.179844
$583$	4981.48	0.353880
$584$	−2472.87	−0.175220
$585$	0	0
$586$	−1960.22	−0.138184
$587$	3667.75	0.257895	0.128947	−	0.991651i	$-0.458840\pi$
$587$	3667.75	0.257895	0.128947	+	0.991651i	$0.458840\pi$
$588$	−2877.17	−0.201790
$589$	−29750.9	−2.08126
$590$	0	0
$591$	14956.8	1.04102
$592$	−7985.66	−0.554406
$593$	5461.87	0.378233	0.189116	−	0.981955i	$-0.439438\pi$
$593$	5461.87	0.378233	0.189116	+	0.981955i	$0.439438\pi$
$594$	−454.899	−0.0314221
$595$	0	0
$596$	−16177.6	−1.11185
$597$	3334.27	0.228581
$598$	4586.18	0.313617
$599$	−20067.3	−1.36883	−0.684415	−	0.729092i	$-0.739942\pi$
$599$	−20067.3	−1.36883	−0.684415	+	0.729092i	$0.739942\pi$
$600$	0	0
$601$	15179.5	1.03026	0.515130	−	0.857112i	$-0.327744\pi$
$601$	15179.5	1.03026	0.515130	+	0.857112i	$0.327744\pi$
$602$	2602.60	0.176203
$603$	−5126.70	−0.346228
$604$	10732.2	0.722991
$605$	0	0
$606$	1344.98	0.0901585
$607$	28792.4	1.92528	0.962642	−	0.270777i	$-0.0872805\pi$
$607$	28792.4	1.92528	0.962642	+	0.270777i	$0.0872805\pi$
$608$	18947.2	1.26384
$609$	−1950.16	−0.129761
$610$	0	0
$611$	−11974.2	−0.792838
$612$	−3088.54	−0.203998
$613$	8519.17	0.561315	0.280657	−	0.959808i	$-0.409448\pi$
$613$	8519.17	0.561315	0.280657	+	0.959808i	$0.409448\pi$
$614$	−7084.66	−0.465657
$615$	0	0
$616$	−5292.71	−0.346184
$617$	2412.39	0.157405	0.0787026	−	0.996898i	$-0.474922\pi$
$617$	2412.39	0.157405	0.0787026	+	0.996898i	$0.474922\pi$
$618$	−2661.49	−0.173237
$619$	3151.33	0.204625	0.102312	−	0.994752i	$-0.467376\pi$
$619$	3151.33	0.204625	0.102312	+	0.994752i	$0.467376\pi$
$620$	0	0
$621$	−3601.79	−0.232746
$622$	−1324.52	−0.0853835
$623$	27168.7	1.74718
$624$	1485.10	0.0952752
$625$	0	0
$626$	−12721.2	−0.812210
$627$	−3642.81	−0.232025
$628$	−5922.11	−0.376302
$629$	−22456.7	−1.42354
$630$	0	0
$631$	−21769.1	−1.37340	−0.686700	−	0.726941i	$-0.740942\pi$
$631$	−21769.1	−1.37340	−0.686700	+	0.726941i	$0.740942\pi$
$632$	−7049.06	−0.443665
$633$	−13964.3	−0.876824
$634$	−3468.36	−0.217265
$635$	0	0
$636$	−7517.33	−0.468682
$637$	3890.50	0.241989
$638$	−488.595	−0.0303192
$639$	−6203.02	−0.384018
$640$	0	0
$641$	12175.8	0.750260	0.375130	−	0.926972i	$-0.377598\pi$
$641$	12175.8	0.750260	0.375130	+	0.926972i	$0.377598\pi$
$642$	3027.27	0.186101
$643$	25531.4	1.56588	0.782938	−	0.622100i	$-0.213720\pi$
$643$	25531.4	1.56588	0.782938	+	0.622100i	$0.213720\pi$
$644$	−17984.7	−1.10046
$645$	0	0
$646$	8164.93	0.497283
$647$	−15379.5	−0.934512	−0.467256	−	0.884122i	$-0.654757\pi$
$647$	−15379.5	−0.934512	−0.467256	+	0.884122i	$0.654757\pi$
$648$	1599.56	0.0969699
$649$	3831.24	0.231725
$650$	0	0
$651$	−19700.1	−1.18603
$652$	−3886.99	−0.233476
$653$	23529.2	1.41006	0.705029	−	0.709179i	$-0.250934\pi$
$653$	23529.2	1.41006	0.705029	+	0.709179i	$0.250934\pi$
$654$	−8325.89	−0.497810
$655$	0	0
$656$	4827.30	0.287308
$657$	−1127.02	−0.0669240
$658$	−15501.6	−0.918409
$659$	−5060.62	−0.299141	−0.149570	−	0.988751i	$-0.547789\pi$
$659$	−5060.62	−0.299141	−0.149570	+	0.988751i	$0.547789\pi$
$660$	0	0
$661$	−17178.7	−1.01085	−0.505426	−	0.862870i	$-0.668665\pi$
$661$	−17178.7	−1.01085	−0.505426	+	0.862870i	$0.668665\pi$
$662$	−6130.82	−0.359941
$663$	4176.31	0.244637
$664$	−18708.5	−1.09342
$665$	0	0
$666$	4991.30	0.290404
$667$	−3868.59	−0.224576
$668$	−964.400	−0.0558589
$669$	2255.84	0.130367
$670$	0	0
$671$	7318.77	0.421070
$672$	12546.3	0.720213
$673$	31040.4	1.77789	0.888944	−	0.458016i	$-0.151440\pi$
$673$	31040.4	1.77789	0.888944	+	0.458016i	$0.151440\pi$
$674$	3275.80	0.187209
$675$	0	0
$676$	9633.54	0.548108
$677$	−11484.3	−0.651960	−0.325980	−	0.945377i	$-0.605694\pi$
$677$	−11484.3	−0.651960	−0.325980	+	0.945377i	$0.605694\pi$
$678$	4678.16	0.264991
$679$	−13389.7	−0.756775
$680$	0	0
$681$	3229.77	0.181740
$682$	−4935.70	−0.277123
$683$	−9670.23	−0.541758	−0.270879	−	0.962613i	$-0.587314\pi$
$683$	−9670.23	−0.541758	−0.270879	+	0.962613i	$0.587314\pi$
$684$	5497.21	0.307297
$685$	0	0
$686$	−5797.26	−0.322653
$687$	−9146.80	−0.507965
$688$	1671.85	0.0926433
$689$	10164.9	0.562049
$690$	0	0
$691$	32367.9	1.78196	0.890979	−	0.454045i	$-0.150019\pi$
$691$	32367.9	1.78196	0.890979	+	0.454045i	$0.150019\pi$
$692$	15536.1	0.853457
$693$	−2412.16	−0.132223
$694$	4768.50	0.260821
$695$	0	0
$696$	1718.04	0.0935664
$697$	13575.0	0.737719
$698$	−5462.11	−0.296195
$699$	−60.6775	−0.00328331
$700$	0	0
$701$	10458.3	0.563489	0.281744	−	0.959490i	$-0.409087\pi$
$701$	10458.3	0.563489	0.281744	+	0.959490i	$0.409087\pi$
$702$	−928.239	−0.0499062
$703$	39970.2	2.14439
$704$	1202.57	0.0643798
$705$	0	0
$706$	10909.0	0.581537
$707$	7131.93	0.379383
$708$	−5781.56	−0.306899
$709$	−12021.8	−0.636794	−0.318397	−	0.947958i	$-0.603144\pi$
$709$	−12021.8	−0.636794	−0.318397	+	0.947958i	$0.603144\pi$
$710$	0	0
$711$	−3212.62	−0.169455
$712$	−23935.0	−1.25983
$713$	−39079.8	−2.05267
$714$	5406.57	0.283383
$715$	0	0
$716$	−22036.1	−1.15018
$717$	923.679	0.0481108
$718$	9234.22	0.479969
$719$	−18959.8	−0.983421	−0.491711	−	0.870759i	$-0.663628\pi$
$719$	−18959.8	−0.983421	−0.491711	+	0.870759i	$0.663628\pi$
$720$	0	0
$721$	−14112.9	−0.728975
$722$	−4867.63	−0.250906
$723$	13591.3	0.699124
$724$	−20414.1	−1.04790
$725$	0	0
$726$	5022.13	0.256734
$727$	−20821.5	−1.06221	−0.531105	−	0.847306i	$-0.678223\pi$
$727$	−20821.5	−1.06221	−0.531105	+	0.847306i	$0.678223\pi$
$728$	−10800.0	−0.549827
$729$	729.000	0.0370370
$730$	0	0
$731$	4701.46	0.237879
$732$	−11044.4	−0.557670
$733$	−24886.4	−1.25403	−0.627013	−	0.779009i	$-0.715723\pi$
$733$	−24886.4	−1.25403	−0.627013	+	0.779009i	$0.715723\pi$
$734$	−7.72287	−0.000388360 0
$735$	0	0
$736$	24888.5	1.24647
$737$	−6810.97	−0.340414
$738$	−3017.22	−0.150495
$739$	−23608.7	−1.17518	−0.587591	−	0.809158i	$-0.699923\pi$
$739$	−23608.7	−1.17518	−0.587591	+	0.809158i	$0.699923\pi$
$740$	0	0
$741$	−7433.31	−0.368515
$742$	13159.3	0.651068
$743$	−27341.5	−1.35002	−0.675008	−	0.737810i	$-0.735860\pi$
$743$	−27341.5	−1.35002	−0.675008	+	0.737810i	$0.735860\pi$
$744$	17355.3	0.855212
$745$	0	0
$746$	17778.1	0.872521
$747$	−8526.45	−0.417626
$748$	−4103.22	−0.200573
$749$	16052.5	0.783104
$750$	0	0
$751$	−13630.9	−0.662313	−0.331156	−	0.943576i	$-0.607439\pi$
$751$	−13630.9	−0.662313	−0.331156	+	0.943576i	$0.607439\pi$
$752$	−9957.83	−0.482879
$753$	−3961.52	−0.191721
$754$	−996.997	−0.0481545
$755$	0	0
$756$	3640.09	0.175117
$757$	−30623.7	−1.47033	−0.735164	−	0.677890i	$-0.762895\pi$
$757$	−30623.7	−1.47033	−0.735164	+	0.677890i	$0.762895\pi$
$758$	10174.1	0.487518
$759$	−4785.09	−0.228837
$760$	0	0
$761$	1659.21	0.0790358	0.0395179	−	0.999219i	$-0.487418\pi$
$761$	1659.21	0.0790358	0.0395179	+	0.999219i	$0.487418\pi$
$762$	−10553.4	−0.501720
$763$	−44149.1	−2.09476
$764$	3819.06	0.180849
$765$	0	0
$766$	4590.38	0.216524
$767$	7817.79	0.368037
$768$	−8124.20	−0.381715
$769$	37688.8	1.76735	0.883676	−	0.468098i	$-0.155061\pi$
$769$	37688.8	1.76735	0.883676	+	0.468098i	$0.155061\pi$
$770$	0	0
$771$	6617.95	0.309131
$772$	−14667.8	−0.683816
$773$	−29767.0	−1.38505	−0.692526	−	0.721393i	$-0.743502\pi$
$773$	−29767.0	−1.38505	−0.692526	+	0.721393i	$0.743502\pi$
$774$	−1044.96	−0.0485276
$775$	0	0
$776$	11796.0	0.545687
$777$	26467.0	1.22201
$778$	−17448.2	−0.804047
$779$	−24161.8	−1.11128
$780$	0	0
$781$	−8240.89	−0.377570
$782$	10725.2	0.490450
$783$	783.000	0.0357371
$784$	3235.37	0.147384
$785$	0	0
$786$	−9111.79	−0.413495
$787$	−3381.10	−0.153142	−0.0765712	−	0.997064i	$-0.524397\pi$
$787$	−3381.10	−0.153142	−0.0765712	+	0.997064i	$0.524397\pi$
$788$	−29985.9	−1.35559
$789$	19891.4	0.897533
$790$	0	0
$791$	24806.5	1.11507
$792$	2125.06	0.0953416
$793$	14934.2	0.668765
$794$	13441.3	0.600772
$795$	0	0
$796$	−6684.64	−0.297652
$797$	40041.9	1.77962	0.889810	−	0.456331i	$-0.150837\pi$
$797$	40041.9	1.77962	0.889810	+	0.456331i	$0.150837\pi$
$798$	−9623.01	−0.426881
$799$	−28002.8	−1.23988
$800$	0	0
$801$	−10908.4	−0.481186
$802$	−649.325	−0.0285891
$803$	−1497.27	−0.0658003
$804$	10278.1	0.450849
$805$	0	0
$806$	−10071.5	−0.440140
$807$	14232.9	0.620846
$808$	−6283.06	−0.273561
$809$	−37278.9	−1.62010	−0.810048	−	0.586364i	$-0.800559\pi$
$809$	−37278.9	−1.62010	−0.810048	+	0.586364i	$0.800559\pi$
$810$	0	0
$811$	4707.22	0.203814	0.101907	−	0.994794i	$-0.467506\pi$
$811$	4707.22	0.203814	0.101907	+	0.994794i	$0.467506\pi$
$812$	3909.72	0.168971
$813$	−16925.5	−0.730138
$814$	6631.09	0.285528
$815$	0	0
$816$	3473.05	0.148996
$817$	−8368.01	−0.358335
$818$	−17992.7	−0.769072
$819$	−4922.11	−0.210003
$820$	0	0
$821$	−36716.3	−1.56079	−0.780394	−	0.625289i	$-0.784981\pi$
$821$	−36716.3	−1.56079	−0.780394	+	0.625289i	$0.784981\pi$
$822$	4769.60	0.202383
$823$	24668.7	1.04483	0.522415	−	0.852691i	$-0.325031\pi$
$823$	24668.7	1.04483	0.522415	+	0.852691i	$0.325031\pi$
$824$	12433.1	0.525641
$825$	0	0
$826$	10120.8	0.426327
$827$	−4400.44	−0.185028	−0.0925140	−	0.995711i	$-0.529490\pi$
$827$	−4400.44	−0.185028	−0.0925140	+	0.995711i	$0.529490\pi$
$828$	7220.97	0.303075
$829$	−25354.0	−1.06222	−0.531111	−	0.847302i	$-0.678225\pi$
$829$	−25354.0	−1.06222	−0.531111	+	0.847302i	$0.678225\pi$
$830$	0	0
$831$	16293.1	0.680148
$832$	2453.88	0.102251
$833$	9098.29	0.378436
$834$	−5626.66	−0.233615
$835$	0	0
$836$	7303.21	0.302137
$837$	7909.73	0.326643
$838$	−5908.07	−0.243545
$839$	25121.5	1.03372	0.516860	−	0.856070i	$-0.327101\pi$
$839$	25121.5	1.03372	0.516860	+	0.856070i	$0.327101\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	14034.9	0.574437
$843$	28176.0	1.15117
$844$	27995.9	1.14178
$845$	0	0
$846$	6223.97	0.252937
$847$	26630.5	1.08032
$848$	8453.21	0.342316
$849$	19753.9	0.798530
$850$	0	0
$851$	52503.6	2.11492
$852$	12436.0	0.500058
$853$	−26685.4	−1.07115	−0.535575	−	0.844488i	$-0.679905\pi$
$853$	−26685.4	−1.07115	−0.535575	+	0.844488i	$0.679905\pi$
$854$	19333.6	0.774685
$855$	0	0
$856$	−14141.9	−0.564672
$857$	−3968.30	−0.158173	−0.0790866	−	0.996868i	$-0.525200\pi$
$857$	−3968.30	−0.158173	−0.0790866	+	0.996868i	$0.525200\pi$
$858$	−1233.19	−0.0490682
$859$	−8041.36	−0.319403	−0.159702	−	0.987165i	$-0.551053\pi$
$859$	−8041.36	−0.319403	−0.159702	+	0.987165i	$0.551053\pi$
$860$	0	0
$861$	−15999.2	−0.633277
$862$	1265.41	0.0500001
$863$	−20859.3	−0.822780	−0.411390	−	0.911459i	$-0.634957\pi$
$863$	−20859.3	−0.822780	−0.411390	+	0.911459i	$0.634957\pi$
$864$	−5037.41	−0.198352
$865$	0	0
$866$	−18425.3	−0.722998
$867$	−4972.32	−0.194774
$868$	39495.3	1.54442
$869$	−4268.06	−0.166610
$870$	0	0
$871$	−13898.1	−0.540663
$872$	38894.3	1.51047
$873$	5376.06	0.208422
$874$	−19089.5	−0.738801
$875$	0	0
$876$	2259.47	0.0871466
$877$	3538.07	0.136228	0.0681141	−	0.997678i	$-0.478302\pi$
$877$	3538.07	0.136228	0.0681141	+	0.997678i	$0.478302\pi$
$878$	91.1015	0.00350174
$879$	4173.39	0.160142
$880$	0	0
$881$	37143.5	1.42043	0.710213	−	0.703987i	$-0.248599\pi$
$881$	37143.5	1.42043	0.710213	+	0.703987i	$0.248599\pi$
$882$	−2022.21	−0.0772012
$883$	−37855.2	−1.44273	−0.721364	−	0.692556i	$-0.756485\pi$
$883$	−37855.2	−1.44273	−0.721364	+	0.692556i	$0.756485\pi$
$884$	−8372.77	−0.318560
$885$	0	0
$886$	143.386	0.00543697
$887$	−6236.28	−0.236070	−0.118035	−	0.993009i	$-0.537659\pi$
$887$	−6236.28	−0.236070	−0.118035	+	0.993009i	$0.537659\pi$
$888$	−23316.8	−0.881150
$889$	−55961.0	−2.11122
$890$	0	0
$891$	968.498	0.0364151
$892$	−4522.57	−0.169761
$893$	49841.4	1.86772
$894$	−11370.4	−0.425372
$895$	0	0
$896$	−30280.0	−1.12900
$897$	−9764.16	−0.363451
$898$	−17750.3	−0.659616
$899$	8495.63	0.315178
$900$	0	0
$901$	23771.6	0.878963
$902$	−4008.46	−0.147968
$903$	−5541.04	−0.204202
$904$	−21854.0	−0.804041
$905$	0	0
$906$	7543.08	0.276603
$907$	13383.7	0.489965	0.244982	−	0.969528i	$-0.421218\pi$
$907$	13383.7	0.489965	0.244982	+	0.969528i	$0.421218\pi$
$908$	−6475.14	−0.236657
$909$	−2863.52	−0.104485
$910$	0	0
$911$	31847.2	1.15823	0.579114	−	0.815247i	$-0.303399\pi$
$911$	31847.2	1.15823	0.579114	+	0.815247i	$0.303399\pi$
$912$	−6181.59	−0.224444
$913$	−11327.6	−0.410613
$914$	6474.82	0.234319
$915$	0	0
$916$	18337.7	0.661459
$917$	−48316.5	−1.73997
$918$	−2170.77	−0.0780459
$919$	−41399.0	−1.48599	−0.742996	−	0.669296i	$-0.766596\pi$
$919$	−41399.0	−1.48599	−0.742996	+	0.669296i	$0.766596\pi$
$920$	0	0
$921$	15083.5	0.539651
$922$	−5391.50	−0.192581
$923$	−16815.9	−0.599676
$924$	4835.96	0.172177
$925$	0	0
$926$	20277.6	0.719614
$927$	5666.41	0.200765
$928$	−5410.55	−0.191390
$929$	−36789.2	−1.29926	−0.649632	−	0.760249i	$-0.725077\pi$
$929$	−36789.2	−1.29926	−0.649632	+	0.760249i	$0.725077\pi$
$930$	0	0
$931$	−16193.8	−0.570065
$932$	121.648	0.00427543
$933$	2819.96	0.0989512
$934$	−16557.0	−0.580045
$935$	0	0
$936$	4336.26	0.151426
$937$	−24811.5	−0.865055	−0.432528	−	0.901621i	$-0.642378\pi$
$937$	−24811.5	−0.865055	−0.432528	+	0.901621i	$0.642378\pi$
$938$	−17992.1	−0.626295
$939$	27084.0	0.941272
$940$	0	0
$941$	−24449.4	−0.847002	−0.423501	−	0.905896i	$-0.639199\pi$
$941$	−24449.4	−0.847002	−0.423501	+	0.905896i	$0.639199\pi$
$942$	−4162.34	−0.143966
$943$	−31738.2	−1.09601
$944$	6501.33	0.224153
$945$	0	0
$946$	−1388.26	−0.0477127
$947$	30170.8	1.03529	0.517644	−	0.855596i	$-0.326809\pi$
$947$	30170.8	1.03529	0.517644	+	0.855596i	$0.326809\pi$
$948$	6440.74	0.220660
$949$	−3055.25	−0.104507
$950$	0	0
$951$	7384.28	0.251789
$952$	−25256.7	−0.859848
$953$	−35385.1	−1.20277	−0.601383	−	0.798961i	$-0.705383\pi$
$953$	−35385.1	−1.20277	−0.601383	+	0.798961i	$0.705383\pi$
$954$	−5283.54	−0.179309
$955$	0	0
$956$	−1851.82	−0.0626485
$957$	1040.24	0.0351370
$958$	3411.04	0.115037
$959$	25291.4	0.851620
$960$	0	0
$961$	56030.4	1.88078
$962$	13531.0	0.453489
$963$	−6445.18	−0.215673
$964$	−27248.2	−0.910380
$965$	0	0
$966$	−12640.5	−0.421015
$967$	−4813.01	−0.160058	−0.0800289	−	0.996793i	$-0.525501\pi$
$967$	−4813.01	−0.160058	−0.0800289	+	0.996793i	$0.525501\pi$
$968$	−23460.9	−0.778988
$969$	−17383.5	−0.576303
$970$	0	0
$971$	6312.62	0.208632	0.104316	−	0.994544i	$-0.466735\pi$
$971$	6312.62	0.208632	0.104316	+	0.994544i	$0.466735\pi$
$972$	−1461.52	−0.0482286
$973$	−29836.1	−0.983043
$974$	7887.16	0.259467
$975$	0	0
$976$	12419.4	0.407312
$977$	−49628.0	−1.62512	−0.812560	−	0.582878i	$-0.801927\pi$
$977$	−49628.0	−1.62512	−0.812560	+	0.582878i	$0.801927\pi$
$978$	−2731.96	−0.0893234
$979$	−14492.2	−0.473106
$980$	0	0
$981$	17726.2	0.576914
$982$	−10973.6	−0.356601
$983$	41462.3	1.34531	0.672657	−	0.739955i	$-0.265153\pi$
$983$	41462.3	1.34531	0.672657	+	0.739955i	$0.265153\pi$
$984$	14094.9	0.456636
$985$	0	0
$986$	−2331.57	−0.0753066
$987$	33003.4	1.06435
$988$	14902.5	0.479870
$989$	−10992.0	−0.353411
$990$	0	0
$991$	−36195.5	−1.16023	−0.580114	−	0.814535i	$-0.696992\pi$
$991$	−36195.5	−1.16023	−0.580114	+	0.814535i	$0.696992\pi$
$992$	−54656.4	−1.74934
$993$	13052.8	0.417137
$994$	−21769.5	−0.694653
$995$	0	0
$996$	17094.0	0.543821
$997$	−8511.65	−0.270378	−0.135189	−	0.990820i	$-0.543164\pi$
$997$	−8511.65	−0.270378	−0.135189	+	0.990820i	$0.543164\pi$
$998$	−1068.48	−0.0338899
$999$	−10626.7	−0.336550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.n.1.3		7
5.4	even	2		435.4.a.i.1.5	✓	7
15.14	odd	2		1305.4.a.n.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.i.1.5	✓	7	5.4	even	2
1305.4.a.n.1.3		7	15.14	odd	2
2175.4.a.n.1.3		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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40909 q^{2} +3.00000 q^{3} -6.01448 q^{4} -4.22726 q^{6} -22.4156 q^{7} +19.7476 q^{8} +9.00000 q^{9} +11.9568 q^{11} -18.0434 q^{12} +24.3982 q^{13} +31.5855 q^{14} +20.2898 q^{16} +57.0575 q^{17} -12.6818 q^{18} -101.555 q^{19} -67.2467 q^{21} -16.8481 q^{22} -133.400 q^{23} +59.2428 q^{24} -34.3792 q^{26} +27.0000 q^{27} +134.818 q^{28} +29.0000 q^{29} +292.953 q^{31} -186.571 q^{32} +35.8703 q^{33} -80.3989 q^{34} -54.1303 q^{36} -393.581 q^{37} +143.100 q^{38} +73.1947 q^{39} +237.918 q^{41} +94.7564 q^{42} +82.3986 q^{43} -71.9137 q^{44} +187.972 q^{46} -490.781 q^{47} +60.8693 q^{48} +159.458 q^{49} +171.173 q^{51} -146.743 q^{52} +416.624 q^{53} -38.0453 q^{54} -442.654 q^{56} -304.666 q^{57} -40.8635 q^{58} +320.424 q^{59} +612.103 q^{61} -412.796 q^{62} -201.740 q^{63} +100.576 q^{64} -50.5443 q^{66} -569.634 q^{67} -343.171 q^{68} -400.199 q^{69} -689.224 q^{71} +177.728 q^{72} -125.224 q^{73} +554.589 q^{74} +610.801 q^{76} -268.018 q^{77} -103.138 q^{78} -356.958 q^{79} +81.0000 q^{81} -335.247 q^{82} -947.383 q^{83} +404.454 q^{84} -116.107 q^{86} +87.0000 q^{87} +236.117 q^{88} -1212.05 q^{89} -546.901 q^{91} +802.330 q^{92} +878.858 q^{93} +691.553 q^{94} -559.712 q^{96} +597.340 q^{97} -224.690 q^{98} +107.611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) \) 7 * q + 2 * q^2 + 21 * q^3 + 22 * q^4 + 6 * q^6 + 50 * q^7 + 33 * q^8 + 63 * q^9 + 76 * q^11 + 66 * q^12 - 30 * q^13 + 89 * q^14 + 138 * q^16 + 140 * q^17 + 18 * q^18 + 90 * q^19 + 150 * q^21 - 61 * q^22 - 34 * q^23 + 99 * q^24 - 241 * q^26 + 189 * q^27 + 57 * q^28 + 203 * q^29 + 524 * q^31 + 6 * q^32 + 228 * q^33 + 255 * q^34 + 198 * q^36 + 28 * q^37 - 222 * q^38 - 90 * q^39 + 1532 * q^41 + 267 * q^42 + 464 * q^43 + 1475 * q^44 + 72 * q^46 + 360 * q^47 + 414 * q^48 + 569 * q^49 + 420 * q^51 + 205 * q^52 - 282 * q^53 + 54 * q^54 + 1102 * q^56 + 270 * q^57 + 58 * q^58 + 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 450 * q^63 + 701 * q^64 - 183 * q^66 - 1546 * q^67 - 1801 * q^68 - 102 * q^69 + 1802 * q^71 + 297 * q^72 + 220 * q^73 + 1594 * q^74 + 1960 * q^76 - 3222 * q^77 - 723 * q^78 + 1298 * q^79 + 567 * q^81 - 856 * q^82 - 1652 * q^83 + 171 * q^84 + 7628 * q^86 + 609 * q^87 - 550 * q^88 + 2846 * q^89 - 816 * q^91 - 472 * q^92 + 1572 * q^93 + 745 * q^94 + 18 * q^96 - 1110 * q^97 + 761 * q^98 + 684 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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