LMFDB - Embedded newform 825.4.c.r.199.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,4,Mod(199,825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("825.199"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-92,0,24,0,0,-90,0,110,0,0,-110] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 83x^{8} + 2275x^{6} + 24517x^{4} + 87636x^{2} + 40000 $ x^10 + 83x^8 + 2275x^6 + 24517x^4 + 87636x^2 + 40000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$-3.91216i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.r.199.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.91216i q^{2} +3.00000i q^{3} -16.1293 q^{4} +14.7365 q^{6} -35.2335i q^{7} +39.9323i q^{8} -9.00000 q^{9} +11.0000 q^{11} -48.3879i q^{12} -26.9085i q^{13} -173.072 q^{14} +67.1195 q^{16} -125.196i q^{17} +44.2094i q^{18} +134.900 q^{19} +105.700 q^{21} -54.0337i q^{22} -79.5933i q^{23} -119.797 q^{24} -132.179 q^{26} -27.0000i q^{27} +568.290i q^{28} +259.535 q^{29} +177.293 q^{31} -10.2431i q^{32} +33.0000i q^{33} -614.980 q^{34} +145.164 q^{36} +32.7865i q^{37} -662.648i q^{38} +80.7254 q^{39} -329.261 q^{41} -519.217i q^{42} -134.125i q^{43} -177.422 q^{44} -390.975 q^{46} -419.191i q^{47} +201.359i q^{48} -898.397 q^{49} +375.587 q^{51} +434.015i q^{52} +483.902i q^{53} -132.628 q^{54} +1406.95 q^{56} +404.699i q^{57} -1274.88i q^{58} +136.948 q^{59} -623.843 q^{61} -870.890i q^{62} +317.101i q^{63} +486.640 q^{64} +162.101 q^{66} -541.799i q^{67} +2019.31i q^{68} +238.780 q^{69} -823.349 q^{71} -359.391i q^{72} -29.0696i q^{73} +161.052 q^{74} -2175.83 q^{76} -387.568i q^{77} -396.536i q^{78} -124.175 q^{79} +81.0000 q^{81} +1617.38i q^{82} +435.640i q^{83} -1704.87 q^{84} -658.841 q^{86} +778.604i q^{87} +439.256i q^{88} +281.086 q^{89} -948.079 q^{91} +1283.78i q^{92} +531.878i q^{93} -2059.13 q^{94} +30.7294 q^{96} +40.7134i q^{97} +4413.07i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9} + 110 q^{11} - 110 q^{14} + 1268 q^{16} + 674 q^{19} - 228 q^{21} - 432 q^{24} + 718 q^{26} + 306 q^{29} + 526 q^{31} - 1034 q^{34} + 828 q^{36} - 258 q^{39} - 176 q^{41}+ \cdots - 990 q^{99}+O(q^{100}) $ 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9 + 110 * q^11 - 110 * q^14 + 1268 * q^16 + 674 * q^19 - 228 * q^21 - 432 * q^24 + 718 * q^26 + 306 * q^29 + 526 * q^31 - 1034 * q^34 + 828 * q^36 - 258 * q^39 - 176 * q^41 - 1012 * q^44 + 872 * q^46 - 4762 * q^49 + 900 * q^51 - 216 * q^54 + 422 * q^56 - 820 * q^59 - 2260 * q^61 - 6340 * q^64 + 264 * q^66 + 1206 * q^69 + 2498 * q^71 + 5970 * q^74 - 5112 * q^76 - 4516 * q^79 + 810 * q^81 + 2646 * q^84 - 2870 * q^86 - 694 * q^89 - 1224 * q^91 - 1814 * q^94 + 1680 * q^96 - 990 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 4.91216i	− 1.73671i	−0.495943	−	0.868355i	$-0.665178\pi$
$2$	− 4.91216i	− 1.73671i	0.495943	−	0.868355i	$-0.334822\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−16.1293	−2.01616
$5$	0	0
$5$	0	0
$6$	14.7365	1.00269
$7$	− 35.2335i	− 1.90243i	−0.308532	−	0.951214i	$-0.599838\pi$
$7$	− 35.2335i	− 1.90243i	0.308532	−	0.951214i	$-0.400162\pi$
$8$	39.9323i	1.76478i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	− 48.3879i	− 1.16403i
$13$	− 26.9085i	− 0.574082i	−0.957918	−	0.287041i	$-0.907328\pi$
$13$	− 26.9085i	− 0.574082i	0.957918	−	0.287041i	$-0.0926716\pi$
$14$	−173.072	−3.30397
$15$	0	0
$16$	67.1195	1.04874
$17$	− 125.196i	− 1.78614i	−0.449918	−	0.893070i	$-0.648547\pi$
$17$	− 125.196i	− 1.78614i	0.449918	−	0.893070i	$-0.351453\pi$
$18$	44.2094i	0.578903i
$19$	134.900	1.62885	0.814423	−	0.580272i	$-0.197054\pi$
$19$	134.900	1.62885	0.814423	+	0.580272i	$0.197054\pi$
$20$	0	0
$21$	105.700	1.09837
$22$	− 54.0337i	− 0.523638i
$23$	− 79.5933i	− 0.721580i	−0.932647	−	0.360790i	$-0.882507\pi$
$23$	− 79.5933i	− 0.721580i	0.932647	−	0.360790i	$-0.117493\pi$
$24$	−119.797	−1.01889
$25$	0	0
$26$	−132.179	−0.997014
$27$	− 27.0000i	− 0.192450i
$28$	568.290i	3.83560i
$29$	259.535	1.66188	0.830938	−	0.556365i	$-0.187804\pi$
$29$	259.535	1.66188	0.830938	+	0.556365i	$0.187804\pi$
$30$	0	0
$31$	177.293	1.02718	0.513592	−	0.858034i	$-0.328314\pi$
$31$	177.293	1.02718	0.513592	+	0.858034i	$0.328314\pi$
$32$	− 10.2431i	− 0.0565858i
$33$	33.0000i	0.174078i
$34$	−614.980	−3.10201
$35$	0	0
$36$	145.164	0.672054
$37$	32.7865i	0.145677i	0.997344	+	0.0728387i	$0.0232058\pi$
$37$	32.7865i	0.145677i	−0.997344	+	0.0728387i	$0.976794\pi$
$38$	− 662.648i	− 2.82883i
$39$	80.7254	0.331447
$40$	0	0
$41$	−329.261	−1.25419	−0.627096	−	0.778942i	$-0.715757\pi$
$41$	−329.261	−1.25419	−0.627096	+	0.778942i	$0.715757\pi$
$42$	− 519.217i	− 1.90755i
$43$	− 134.125i	− 0.475670i	−0.971306	−	0.237835i	$-0.923562\pi$
$43$	− 134.125i	− 0.475670i	0.971306	−	0.237835i	$-0.0764377\pi$
$44$	−177.422	−0.607895
$45$	0	0
$46$	−390.975	−1.25318
$47$	− 419.191i	− 1.30096i	−0.759522	−	0.650481i	$-0.774567\pi$
$47$	− 419.191i	− 1.30096i	0.759522	−	0.650481i	$-0.225433\pi$
$48$	201.359i	0.605492i
$49$	−898.397	−2.61923
$50$	0	0
$51$	375.587	1.03123
$52$	434.015i	1.15744i
$53$	483.902i	1.25413i	0.778966	+	0.627066i	$0.215744\pi$
$53$	483.902i	1.25413i	−0.778966	+	0.627066i	$0.784256\pi$
$54$	−132.628	−0.334230
$55$	0	0
$56$	1406.95	3.35736
$57$	404.699i	0.940415i
$58$	− 1274.88i	− 2.88620i
$59$	136.948	0.302187	0.151094	−	0.988519i	$-0.451720\pi$
$59$	136.948	0.302187	0.151094	+	0.988519i	$0.451720\pi$
$60$	0	0
$61$	−623.843	−1.30942	−0.654712	−	0.755878i	$-0.727210\pi$
$61$	−623.843	−1.30942	−0.654712	+	0.755878i	$0.727210\pi$
$62$	− 870.890i	− 1.78392i
$63$	317.101i	0.634143i
$64$	486.640	0.950470
$65$	0	0
$66$	162.101	0.302322
$67$	− 541.799i	− 0.987929i	−0.869482	−	0.493965i	$-0.835547\pi$
$67$	− 541.799i	− 0.987929i	0.869482	−	0.493965i	$-0.164453\pi$
$68$	2019.31i	3.60114i
$69$	238.780	0.416604
$70$	0	0
$71$	−823.349	−1.37625	−0.688124	−	0.725594i	$-0.741565\pi$
$71$	−823.349	−1.37625	−0.688124	+	0.725594i	$0.741565\pi$
$72$	− 359.391i	− 0.588259i
$73$	− 29.0696i	− 0.0466074i	−0.999728	−	0.0233037i	$-0.992582\pi$
$73$	− 29.0696i	− 0.0466074i	0.999728	−	0.0233037i	$-0.00741847\pi$
$74$	161.052	0.252999
$75$	0	0
$76$	−2175.83	−3.28402
$77$	− 387.568i	− 0.573604i
$78$	− 396.536i	− 0.575626i
$79$	−124.175	−0.176845	−0.0884225	−	0.996083i	$-0.528183\pi$
$79$	−124.175	−0.176845	−0.0884225	+	0.996083i	$0.528183\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	1617.38i	2.17817i
$83$	435.640i	0.576117i	0.957613	+	0.288059i	$0.0930098\pi$
$83$	435.640i	0.576117i	−0.957613	+	0.288059i	$0.906990\pi$
$84$	−1704.87	−2.21448
$85$	0	0
$86$	−658.841	−0.826101
$87$	778.604i	0.959484i
$88$	439.256i	0.532100i
$89$	281.086	0.334775	0.167388	−	0.985891i	$-0.446467\pi$
$89$	281.086	0.334775	0.167388	+	0.985891i	$0.446467\pi$
$90$	0	0
$91$	−948.079	−1.09215
$92$	1283.78i	1.45482i
$93$	531.878i	0.593045i
$94$	−2059.13	−2.25939
$95$	0	0
$96$	30.7294	0.0326698
$97$	40.7134i	0.0426167i	0.999773	+	0.0213084i	$0.00678318\pi$
$97$	40.7134i	0.0426167i	−0.999773	+	0.0213084i	$0.993217\pi$
$98$	4413.07i	4.54885i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	261.609	0.257733	0.128867	−	0.991662i	$-0.458866\pi$
$101$	261.609	0.257733	0.128867	+	0.991662i	$0.458866\pi$
$102$	− 1844.94i	− 1.79094i
$103$	1131.24i	1.08218i	0.840964	+	0.541092i	$0.181989\pi$
$103$	1131.24i	1.08218i	−0.840964	+	0.541092i	$0.818011\pi$
$104$	1074.52	1.01313
$105$	0	0
$106$	2377.00	2.17806
$107$	− 66.9927i	− 0.0605274i	−0.999542	−	0.0302637i	$-0.990365\pi$
$107$	− 66.9927i	− 0.0605274i	0.999542	−	0.0302637i	$-0.00963470\pi$
$108$	435.491i	0.388010i
$109$	459.733	0.403985	0.201993	−	0.979387i	$-0.435258\pi$
$109$	459.733	0.403985	0.201993	+	0.979387i	$0.435258\pi$
$110$	0	0
$111$	−98.3595	−0.0841069
$112$	− 2364.85i	− 1.99516i
$113$	815.607i	0.678990i	0.940608	+	0.339495i	$0.110256\pi$
$113$	815.607i	0.678990i	−0.940608	+	0.339495i	$0.889744\pi$
$114$	1987.94	1.63323
$115$	0	0
$116$	−4186.11	−3.35061
$117$	242.176i	0.191361i
$118$	− 672.708i	− 0.524812i
$119$	−4411.07	−3.39800
$120$	0	0
$121$	121.000	0.0909091
$122$	3064.41i	2.27409i
$123$	− 987.782i	− 0.724108i
$124$	−2859.61	−2.07097
$125$	0	0
$126$	1557.65	1.10132
$127$	1764.07i	1.23257i	0.787524	+	0.616284i	$0.211362\pi$
$127$	1764.07i	1.23257i	−0.787524	+	0.616284i	$0.788638\pi$
$128$	− 2472.40i	− 1.70728i
$129$	402.374	0.274628
$130$	0	0
$131$	−452.913	−0.302070	−0.151035	−	0.988528i	$-0.548261\pi$
$131$	−452.913	−0.302070	−0.151035	+	0.988528i	$0.548261\pi$
$132$	− 532.266i	− 0.350968i
$133$	− 4752.98i	− 3.09876i
$134$	−2661.40	−1.71575
$135$	0	0
$136$	4999.35	3.15214
$137$	412.254i	0.257089i	0.991704	+	0.128545i	$0.0410306\pi$
$137$	412.254i	0.257089i	−0.991704	+	0.128545i	$0.958969\pi$
$138$	− 1172.92i	− 0.723521i
$139$	97.1910	0.0593067	0.0296534	−	0.999560i	$-0.490560\pi$
$139$	97.1910	0.0593067	0.0296534	+	0.999560i	$0.490560\pi$
$140$	0	0
$141$	1257.57	0.751111
$142$	4044.42i	2.39014i
$143$	− 295.993i	− 0.173092i
$144$	−604.076	−0.349581
$145$	0	0
$146$	−142.794	−0.0809435
$147$	− 2695.19i	− 1.51221i
$148$	− 528.823i	− 0.293709i
$149$	−621.210	−0.341554	−0.170777	−	0.985310i	$-0.554628\pi$
$149$	−621.210	−0.341554	−0.170777	+	0.985310i	$0.554628\pi$
$150$	0	0
$151$	3084.29	1.66222	0.831111	−	0.556106i	$-0.187705\pi$
$151$	3084.29	1.66222	0.831111	+	0.556106i	$0.187705\pi$
$152$	5386.85i	2.87455i
$153$	1126.76i	0.595380i
$154$	−1903.80	−0.996183
$155$	0	0
$156$	−1302.04	−0.668249
$157$	− 74.0945i	− 0.0376649i	−0.999823	−	0.0188324i	$-0.994005\pi$
$157$	− 74.0945i	− 0.0376649i	0.999823	−	0.0188324i	$-0.00599490\pi$
$158$	609.966i	0.307128i
$159$	−1451.70	−0.724073
$160$	0	0
$161$	−2804.35	−1.37275
$162$	− 397.885i	− 0.192968i
$163$	− 3093.36i	− 1.48645i	−0.669043	−	0.743224i	$-0.733296\pi$
$163$	− 3093.36i	− 1.48645i	0.669043	−	0.743224i	$-0.266704\pi$
$164$	5310.74	2.52865
$165$	0	0
$166$	2139.93	1.00055
$167$	3125.34i	1.44818i	0.689704	+	0.724091i	$0.257741\pi$
$167$	3125.34i	1.44818i	−0.689704	+	0.724091i	$0.742259\pi$
$168$	4220.86i	1.93837i
$169$	1472.93	0.670430
$170$	0	0
$171$	−1214.10	−0.542949
$172$	2163.33i	0.959027i
$173$	1967.48i	0.864650i	0.901718	+	0.432325i	$0.142307\pi$
$173$	1967.48i	0.864650i	−0.901718	+	0.432325i	$0.857693\pi$
$174$	3824.63	1.66635
$175$	0	0
$176$	738.315	0.316208
$177$	410.843i	0.174468i
$178$	− 1380.74i	− 0.581408i
$179$	480.759	0.200746	0.100373	−	0.994950i	$-0.467996\pi$
$179$	480.759	0.200746	0.100373	+	0.994950i	$0.467996\pi$
$180$	0	0
$181$	−2903.38	−1.19230	−0.596151	−	0.802872i	$-0.703304\pi$
$181$	−2903.38	−1.19230	−0.596151	+	0.802872i	$0.703304\pi$
$182$	4657.11i	1.89675i
$183$	− 1871.53i	− 0.755997i
$184$	3178.34	1.27343
$185$	0	0
$186$	2612.67	1.02995
$187$	− 1377.15i	− 0.538541i
$188$	6761.25i	2.62295i
$189$	−951.303	−0.366122
$190$	0	0
$191$	1292.54	0.489659	0.244829	−	0.969566i	$-0.421268\pi$
$191$	1292.54	0.489659	0.244829	+	0.969566i	$0.421268\pi$
$192$	1459.92i	0.548754i
$193$	− 1959.13i	− 0.730679i	−0.930874	−	0.365340i	$-0.880953\pi$
$193$	− 1959.13i	− 0.730679i	0.930874	−	0.365340i	$-0.119047\pi$
$194$	199.991	0.0740129
$195$	0	0
$196$	14490.5	5.28079
$197$	− 2572.74i	− 0.930458i	−0.885190	−	0.465229i	$-0.845972\pi$
$197$	− 2572.74i	− 0.930458i	0.885190	−	0.465229i	$-0.154028\pi$
$198$	486.304i	0.174546i
$199$	5316.19	1.89374	0.946871	−	0.321613i	$-0.104225\pi$
$199$	5316.19	1.89374	0.946871	+	0.321613i	$0.104225\pi$
$200$	0	0
$201$	1625.40	0.570381
$202$	− 1285.06i	− 0.447608i
$203$	− 9144.31i	− 3.16160i
$204$	−6057.94	−2.07912
$205$	0	0
$206$	5556.85	1.87944
$207$	716.339i	0.240527i
$208$	− 1806.08i	− 0.602065i
$209$	1483.89	0.491116
$210$	0	0
$211$	397.505	0.129694	0.0648468	−	0.997895i	$-0.479344\pi$
$211$	397.505	0.129694	0.0648468	+	0.997895i	$0.479344\pi$
$212$	− 7804.99i	− 2.52853i
$213$	− 2470.05i	− 0.794577i
$214$	−329.079	−0.105118
$215$	0	0
$216$	1078.17	0.339631
$217$	− 6246.64i	− 1.95415i
$218$	− 2258.28i	− 0.701605i
$219$	87.2088	0.0269088
$220$	0	0
$221$	−3368.82	−1.02539
$222$	483.157i	0.146069i
$223$	− 5295.04i	− 1.59005i	−0.606573	−	0.795027i	$-0.707456\pi$
$223$	− 5295.04i	− 1.59005i	0.606573	−	0.795027i	$-0.292544\pi$
$224$	−360.901	−0.107650
$225$	0	0
$226$	4006.39	1.17921
$227$	1936.60i	0.566240i	0.959084	+	0.283120i	$0.0913695\pi$
$227$	1936.60i	0.566240i	−0.959084	+	0.283120i	$0.908630\pi$
$228$	− 6527.50i	− 1.89603i
$229$	−6379.34	−1.84087	−0.920433	−	0.390899i	$-0.872164\pi$
$229$	−6379.34	−1.84087	−0.920433	+	0.390899i	$0.872164\pi$
$230$	0	0
$231$	1162.70	0.331170
$232$	10363.8i	2.93284i
$233$	2992.48i	0.841391i	0.907202	+	0.420696i	$0.138214\pi$
$233$	2992.48i	0.841391i	−0.907202	+	0.420696i	$0.861786\pi$
$234$	1189.61	0.332338
$235$	0	0
$236$	−2208.87	−0.609258
$237$	− 372.524i	− 0.102101i
$238$	21667.9i	5.90134i
$239$	−2929.63	−0.792895	−0.396447	−	0.918057i	$-0.629757\pi$
$239$	−2929.63	−0.792895	−0.396447	+	0.918057i	$0.629757\pi$
$240$	0	0
$241$	1972.74	0.527283	0.263641	−	0.964621i	$-0.415076\pi$
$241$	1972.74	0.527283	0.263641	+	0.964621i	$0.415076\pi$
$242$	− 594.371i	− 0.157883i
$243$	243.000i	0.0641500i
$244$	10062.1	2.64001
$245$	0	0
$246$	−4852.14	−1.25757
$247$	− 3629.94i	− 0.935092i
$248$	7079.71i	1.81275i
$249$	−1306.92	−0.332621
$250$	0	0
$251$	5653.16	1.42161	0.710806	−	0.703389i	$-0.248331\pi$
$251$	5653.16	1.42161	0.710806	+	0.703389i	$0.248331\pi$
$252$	− 5114.61i	− 1.27853i
$253$	− 875.526i	− 0.217565i
$254$	8665.40	2.14061
$255$	0	0
$256$	−8251.69	−2.01457
$257$	4607.50i	1.11832i	0.829060	+	0.559160i	$0.188876\pi$
$257$	4607.50i	1.11832i	−0.829060	+	0.559160i	$0.811124\pi$
$258$	− 1976.52i	− 0.476949i
$259$	1155.18	0.277141
$260$	0	0
$261$	−2335.81	−0.553959
$262$	2224.78i	0.524609i
$263$	2208.31i	0.517757i	0.965910	+	0.258878i	$0.0833529\pi$
$263$	2208.31i	0.517757i	−0.965910	+	0.258878i	$0.916647\pi$
$264$	−1317.77	−0.307208
$265$	0	0
$266$	−23347.4	−5.38165
$267$	843.257i	0.193283i
$268$	8738.83i	1.99182i
$269$	−2804.20	−0.635594	−0.317797	−	0.948159i	$-0.602943\pi$
$269$	−2804.20	−0.635594	−0.317797	+	0.948159i	$0.602943\pi$
$270$	0	0
$271$	103.286	0.0231519	0.0115759	−	0.999933i	$-0.496315\pi$
$271$	103.286	0.0231519	0.0115759	+	0.999933i	$0.496315\pi$
$272$	− 8403.06i	− 1.87320i
$273$	− 2844.24i	− 0.630553i
$274$	2025.06	0.446490
$275$	0	0
$276$	−3851.35	−0.839942
$277$	1692.10i	0.367034i	0.983017	+	0.183517i	$0.0587482\pi$
$277$	1692.10i	0.367034i	−0.983017	+	0.183517i	$0.941252\pi$
$278$	− 477.418i	− 0.102999i
$279$	−1595.64	−0.342395
$280$	0	0
$281$	−912.903	−0.193805	−0.0969025	−	0.995294i	$-0.530894\pi$
$281$	−912.903	−0.193805	−0.0969025	+	0.995294i	$0.530894\pi$
$282$	− 6177.39i	− 1.30446i
$283$	− 7244.39i	− 1.52168i	−0.648942	−	0.760838i	$-0.724788\pi$
$283$	− 7244.39i	− 1.52168i	0.648942	−	0.760838i	$-0.275212\pi$
$284$	13280.0	2.77474
$285$	0	0
$286$	−1453.97	−0.300611
$287$	11601.0i	2.38601i
$288$	92.1882i	0.0188619i
$289$	−10760.9	−2.19029
$290$	0	0
$291$	−122.140	−0.0246048
$292$	468.872i	0.0939680i
$293$	1621.05i	0.323218i	0.986855	+	0.161609i	$0.0516683\pi$
$293$	1621.05i	0.323218i	−0.986855	+	0.161609i	$0.948332\pi$
$294$	−13239.2	−2.62628
$295$	0	0
$296$	−1309.24	−0.257088
$297$	− 297.000i	− 0.0580259i
$298$	3051.48i	0.593180i
$299$	−2141.73	−0.414246
$300$	0	0
$301$	−4725.67	−0.904928
$302$	− 15150.5i	− 2.88680i
$303$	784.827i	0.148802i
$304$	9054.39	1.70824
$305$	0	0
$306$	5534.82	1.03400
$307$	6476.38i	1.20399i	0.798498	+	0.601997i	$0.205628\pi$
$307$	6476.38i	1.20399i	−0.798498	+	0.601997i	$0.794372\pi$
$308$	6251.20i	1.15648i
$309$	−3393.73	−0.624799
$310$	0	0
$311$	−6046.93	−1.10254	−0.551270	−	0.834327i	$-0.685857\pi$
$311$	−6046.93	−1.10254	−0.551270	+	0.834327i	$0.685857\pi$
$312$	3223.55i	0.584929i
$313$	− 372.777i	− 0.0673181i	−0.999433	−	0.0336591i	$-0.989284\pi$
$313$	− 372.777i	− 0.0673181i	0.999433	−	0.0336591i	$-0.0107160\pi$
$314$	−363.964	−0.0654129
$315$	0	0
$316$	2002.85	0.356548
$317$	− 1621.21i	− 0.287244i	−0.989633	−	0.143622i	$-0.954125\pi$
$317$	− 1621.21i	− 0.287244i	0.989633	−	0.143622i	$-0.0458749\pi$
$318$	7131.00i	1.25751i
$319$	2854.88	0.501074
$320$	0	0
$321$	200.978	0.0349455
$322$	13775.4i	2.38408i
$323$	− 16888.8i	− 2.90935i
$324$	−1306.47	−0.224018
$325$	0	0
$326$	−15195.1	−2.58153
$327$	1379.20i	0.233241i
$328$	− 13148.1i	− 2.21337i
$329$	−14769.5	−2.47499
$330$	0	0
$331$	−7393.98	−1.22782	−0.613912	−	0.789374i	$-0.710405\pi$
$331$	−7393.98	−1.22782	−0.613912	+	0.789374i	$0.710405\pi$
$332$	− 7026.57i	− 1.16154i
$333$	− 295.078i	− 0.0485591i
$334$	15352.2	2.51507
$335$	0	0
$336$	7094.56	1.15190
$337$	− 8244.56i	− 1.33267i	−0.745653	−	0.666335i	$-0.767862\pi$
$337$	− 8244.56i	− 1.33267i	0.745653	−	0.666335i	$-0.232138\pi$
$338$	− 7235.28i	− 1.16434i
$339$	−2446.82	−0.392015
$340$	0	0
$341$	1950.22	0.309708
$342$	5963.83i	0.942944i
$343$	19568.5i	3.08047i
$344$	5355.91	0.839451
$345$	0	0
$346$	9664.55	1.50165
$347$	− 448.062i	− 0.0693177i	−0.999399	−	0.0346589i	$-0.988966\pi$
$347$	− 448.062i	− 0.0693177i	0.999399	−	0.0346589i	$-0.0110345\pi$
$348$	− 12558.3i	− 1.93447i
$349$	9302.21	1.42675	0.713375	−	0.700782i	$-0.247166\pi$
$349$	9302.21	1.42675	0.713375	+	0.700782i	$0.247166\pi$
$350$	0	0
$351$	−726.529	−0.110482
$352$	− 112.674i	− 0.0170613i
$353$	5968.07i	0.899854i	0.893065	+	0.449927i	$0.148550\pi$
$353$	5968.07i	0.899854i	−0.893065	+	0.449927i	$0.851450\pi$
$354$	2018.12	0.303000
$355$	0	0
$356$	−4533.71	−0.674961
$357$	− 13233.2i	− 1.96184i
$358$	− 2361.56i	− 0.348638i
$359$	−11159.5	−1.64061	−0.820304	−	0.571928i	$-0.806196\pi$
$359$	−11159.5	−1.64061	−0.820304	+	0.571928i	$0.806196\pi$
$360$	0	0
$361$	11338.9	1.65314
$362$	14261.9i	2.07068i
$363$	363.000i	0.0524864i
$364$	15291.8	2.20195
$365$	0	0
$366$	−9193.24	−1.31295
$367$	− 6527.44i	− 0.928419i	−0.885725	−	0.464210i	$-0.846338\pi$
$367$	− 6527.44i	− 0.928419i	0.885725	−	0.464210i	$-0.153662\pi$
$368$	− 5342.26i	− 0.756752i
$369$	2963.35	0.418064
$370$	0	0
$371$	17049.5	2.38590
$372$	− 8578.82i	− 1.19567i
$373$	− 3994.42i	− 0.554486i	−0.960800	−	0.277243i	$-0.910579\pi$
$373$	− 3994.42i	− 0.554486i	0.960800	−	0.277243i	$-0.0894208\pi$
$374$	−6764.78	−0.935290
$375$	0	0
$376$	16739.3	2.29591
$377$	− 6983.69i	− 0.954053i
$378$	4672.95i	0.635848i
$379$	7654.69	1.03745	0.518727	−	0.854940i	$-0.326406\pi$
$379$	7654.69	1.03745	0.518727	+	0.854940i	$0.326406\pi$
$380$	0	0
$381$	−5292.21	−0.711623
$382$	− 6349.16i	− 0.850395i
$383$	804.633i	0.107349i	0.998558	+	0.0536747i	$0.0170934\pi$
$383$	804.633i	0.107349i	−0.998558	+	0.0536747i	$0.982907\pi$
$384$	7417.20	0.985696
$385$	0	0
$386$	−9623.54	−1.26898
$387$	1207.12i	0.158557i
$388$	− 656.678i	− 0.0859222i
$389$	7381.51	0.962101	0.481051	−	0.876693i	$-0.340255\pi$
$389$	7381.51	0.962101	0.481051	+	0.876693i	$0.340255\pi$
$390$	0	0
$391$	−9964.72	−1.28884
$392$	− 35875.1i	− 4.62236i
$393$	− 1358.74i	− 0.174400i
$394$	−12637.7	−1.61593
$395$	0	0
$396$	1596.80	0.202632
$397$	− 9024.46i	− 1.14087i	−0.821343	−	0.570434i	$-0.806775\pi$
$397$	− 9024.46i	− 1.14087i	0.821343	−	0.570434i	$-0.193225\pi$
$398$	− 26114.0i	− 3.28888i
$399$	14258.9	1.78907
$400$	0	0
$401$	13125.8	1.63459	0.817294	−	0.576220i	$-0.195473\pi$
$401$	13125.8	1.63459	0.817294	+	0.576220i	$0.195473\pi$
$402$	− 7984.20i	− 0.990587i
$403$	− 4770.68i	− 0.589689i
$404$	−4219.57	−0.519632
$405$	0	0
$406$	−44918.3	−5.49078
$407$	360.651i	0.0439234i
$408$	14998.0i	1.81989i
$409$	−852.781	−0.103099	−0.0515493	−	0.998670i	$-0.516416\pi$
$409$	−852.781	−0.103099	−0.0515493	+	0.998670i	$0.516416\pi$
$410$	0	0
$411$	−1236.76	−0.148431
$412$	− 18246.2i	− 2.18185i
$413$	− 4825.14i	− 0.574890i
$414$	3518.77	0.417725
$415$	0	0
$416$	−275.627	−0.0324849
$417$	291.573i	0.0342408i
$418$	− 7289.12i	− 0.852925i
$419$	−2245.97	−0.261868	−0.130934	−	0.991391i	$-0.541798\pi$
$419$	−2245.97	−0.261868	−0.130934	+	0.991391i	$0.541798\pi$
$420$	0	0
$421$	13811.3	1.59887	0.799433	−	0.600755i	$-0.205133\pi$
$421$	13811.3	1.59887	0.799433	+	0.600755i	$0.205133\pi$
$422$	− 1952.61i	− 0.225240i
$423$	3772.72i	0.433654i
$424$	−19323.3	−2.21326
$425$	0	0
$426$	−12133.3	−1.37995
$427$	21980.1i	2.49109i
$428$	1080.54i	0.122033i
$429$	887.980	0.0999349
$430$	0	0
$431$	13390.9	1.49656	0.748279	−	0.663384i	$-0.230880\pi$
$431$	13390.9	1.49656	0.748279	+	0.663384i	$0.230880\pi$
$432$	− 1812.23i	− 0.201831i
$433$	5783.14i	0.641848i	0.947105	+	0.320924i	$0.103993\pi$
$433$	5783.14i	0.641848i	−0.947105	+	0.320924i	$0.896007\pi$
$434$	−30684.5	−3.39378
$435$	0	0
$436$	−7415.16	−0.814499
$437$	− 10737.1i	− 1.17534i
$438$	− 428.383i	− 0.0467328i
$439$	2811.36	0.305647	0.152823	−	0.988254i	$-0.451163\pi$
$439$	2811.36	0.305647	0.152823	+	0.988254i	$0.451163\pi$
$440$	0	0
$441$	8085.57	0.873077
$442$	16548.2i	1.78081i
$443$	13468.2i	1.44445i	0.691656	+	0.722227i	$0.256881\pi$
$443$	13468.2i	1.44445i	−0.691656	+	0.722227i	$0.743119\pi$
$444$	1586.47	0.169573
$445$	0	0
$446$	−26010.1	−2.76146
$447$	− 1863.63i	− 0.197196i
$448$	− 17146.0i	− 1.80820i
$449$	2672.15	0.280861	0.140431	−	0.990091i	$-0.455151\pi$
$449$	2672.15	0.280861	0.140431	+	0.990091i	$0.455151\pi$
$450$	0	0
$451$	−3621.87	−0.378153
$452$	− 13155.2i	− 1.36895i
$453$	9252.86i	0.959685i
$454$	9512.88	0.983395
$455$	0	0
$456$	−16160.6	−1.65962
$457$	− 11334.8i	− 1.16021i	−0.814541	−	0.580106i	$-0.803011\pi$
$457$	− 11334.8i	− 1.16021i	0.814541	−	0.580106i	$-0.196989\pi$
$458$	31336.3i	3.19705i
$459$	−3380.28	−0.343743
$460$	0	0
$461$	−15225.8	−1.53826	−0.769130	−	0.639092i	$-0.779310\pi$
$461$	−15225.8	−1.53826	−0.769130	+	0.639092i	$0.779310\pi$
$462$	− 5711.39i	− 0.575147i
$463$	− 3312.04i	− 0.332448i	−0.986088	−	0.166224i	$-0.946842\pi$
$463$	− 3312.04i	− 0.332448i	0.986088	−	0.166224i	$-0.0531575\pi$
$464$	17419.9	1.74288
$465$	0	0
$466$	14699.6	1.46125
$467$	− 2521.58i	− 0.249861i	−0.992166	−	0.124930i	$-0.960129\pi$
$467$	− 2521.58i	− 0.249861i	0.992166	−	0.124930i	$-0.0398708\pi$
$468$	− 3906.13i	− 0.385814i
$469$	−19089.5	−1.87946
$470$	0	0
$471$	222.283	0.0217458
$472$	5468.63i	0.533293i
$473$	− 1475.37i	− 0.143420i
$474$	−1829.90	−0.177321
$475$	0	0
$476$	71147.4	6.85092
$477$	− 4355.11i	− 0.418044i
$478$	14390.8i	1.37703i
$479$	18533.3	1.76786	0.883931	−	0.467617i	$-0.154887\pi$
$479$	18533.3	1.76786	0.883931	+	0.467617i	$0.154887\pi$
$480$	0	0
$481$	882.235	0.0836308
$482$	− 9690.39i	− 0.915737i
$483$	− 8413.04i	− 0.792560i
$484$	−1951.64	−0.183287
$485$	0	0
$486$	1193.65	0.111410
$487$	881.844i	0.0820538i	0.999158	+	0.0410269i	$0.0130629\pi$
$487$	881.844i	0.0820538i	−0.999158	+	0.0410269i	$0.986937\pi$
$488$	− 24911.5i	− 2.31084i
$489$	9280.09	0.858201
$490$	0	0
$491$	11624.6	1.06846	0.534229	−	0.845340i	$-0.320602\pi$
$491$	11624.6	1.06846	0.534229	+	0.845340i	$0.320602\pi$
$492$	15932.2i	1.45992i
$493$	− 32492.6i	− 2.96834i
$494$	−17830.8	−1.62398
$495$	0	0
$496$	11899.8	1.07725
$497$	29009.4i	2.61821i
$498$	6419.80i	0.577667i
$499$	−10331.4	−0.926850	−0.463425	−	0.886136i	$-0.653380\pi$
$499$	−10331.4	−0.926850	−0.463425	+	0.886136i	$0.653380\pi$
$500$	0	0
$501$	−9376.03	−0.836108
$502$	− 27769.2i	− 2.46893i
$503$	5079.34i	0.450252i	0.974330	+	0.225126i	$0.0722794\pi$
$503$	5079.34i	0.450252i	−0.974330	+	0.225126i	$0.927721\pi$
$504$	−12662.6	−1.11912
$505$	0	0
$506$	−4300.72	−0.377847
$507$	4418.80i	0.387073i
$508$	− 28453.2i	− 2.48505i
$509$	8113.34	0.706517	0.353259	−	0.935526i	$-0.385074\pi$
$509$	8113.34	0.706517	0.353259	+	0.935526i	$0.385074\pi$
$510$	0	0
$511$	−1024.22	−0.0886672
$512$	20754.4i	1.79145i
$513$	− 3642.29i	− 0.313472i
$514$	22632.8	1.94220
$515$	0	0
$516$	−6490.00	−0.553695
$517$	− 4611.10i	− 0.392255i
$518$	− 5674.43i	− 0.481313i
$519$	−5902.43	−0.499206
$520$	0	0
$521$	−14935.1	−1.25589	−0.627945	−	0.778258i	$-0.716104\pi$
$521$	−14935.1	−1.25589	−0.627945	+	0.778258i	$0.716104\pi$
$522$	11473.9i	0.962065i
$523$	− 13732.8i	− 1.14817i	−0.818796	−	0.574085i	$-0.805358\pi$
$523$	− 13732.8i	− 1.14817i	0.818796	−	0.574085i	$-0.194642\pi$
$524$	7305.17	0.609022
$525$	0	0
$526$	10847.5	0.899193
$527$	− 22196.3i	− 1.83470i
$528$	2214.94i	0.182563i
$529$	5831.91	0.479322
$530$	0	0
$531$	−1232.53	−0.100729
$532$	76662.1i	6.24760i
$533$	8859.90i	0.720009i
$534$	4142.21	0.335676
$535$	0	0
$536$	21635.3	1.74347
$537$	1442.28i	0.115901i
$538$	13774.7i	1.10384i
$539$	−9882.36	−0.789728
$540$	0	0
$541$	−11621.5	−0.923563	−0.461781	−	0.886994i	$-0.652790\pi$
$541$	−11621.5	−0.923563	−0.461781	+	0.886994i	$0.652790\pi$
$542$	− 507.356i	− 0.0402081i
$543$	− 8710.15i	− 0.688376i
$544$	−1282.39	−0.101070
$545$	0	0
$546$	−13971.3	−1.09509
$547$	10009.3i	0.782387i	0.920308	+	0.391194i	$0.127938\pi$
$547$	10009.3i	0.782387i	−0.920308	+	0.391194i	$0.872062\pi$
$548$	− 6649.36i	− 0.518333i
$549$	5614.59	0.436475
$550$	0	0
$551$	35011.1	2.70694
$552$	9535.03i	0.735214i
$553$	4375.11i	0.336435i
$554$	8311.86	0.637431
$555$	0	0
$556$	−1567.62	−0.119572
$557$	10887.5i	0.828218i	0.910227	+	0.414109i	$0.135907\pi$
$557$	10887.5i	0.828218i	−0.910227	+	0.414109i	$0.864093\pi$
$558$	7838.01i	0.594641i
$559$	−3609.09	−0.273074
$560$	0	0
$561$	4131.45	0.310927
$562$	4484.32i	0.336583i
$563$	5715.30i	0.427835i	0.976852	+	0.213918i	$0.0686225\pi$
$563$	5715.30i	0.427835i	−0.976852	+	0.213918i	$0.931378\pi$
$564$	−20283.7	−1.51436
$565$	0	0
$566$	−35585.6	−2.64271
$567$	− 2853.91i	− 0.211381i
$568$	− 32878.2i	− 2.42877i
$569$	8049.22	0.593042	0.296521	−	0.955026i	$-0.404173\pi$
$569$	8049.22	0.593042	0.296521	+	0.955026i	$0.404173\pi$
$570$	0	0
$571$	−879.157	−0.0644336	−0.0322168	−	0.999481i	$-0.510257\pi$
$571$	−879.157	−0.0644336	−0.0322168	+	0.999481i	$0.510257\pi$
$572$	4774.16i	0.348982i
$573$	3877.62i	0.282705i
$574$	56985.9	4.14381
$575$	0	0
$576$	−4379.76	−0.316823
$577$	6362.10i	0.459026i	0.973306	+	0.229513i	$0.0737133\pi$
$577$	6362.10i	0.459026i	−0.973306	+	0.229513i	$0.926287\pi$
$578$	52859.3i	3.80390i
$579$	5877.38	0.421858
$580$	0	0
$581$	15349.1	1.09602
$582$	599.972i	0.0427314i
$583$	5322.92i	0.378135i
$584$	1160.82	0.0822516
$585$	0	0
$586$	7962.86	0.561336
$587$	− 14637.2i	− 1.02921i	−0.857429	−	0.514603i	$-0.827939\pi$
$587$	− 14637.2i	− 1.02921i	0.857429	−	0.514603i	$-0.172061\pi$
$588$	43471.5i	3.04887i
$589$	23916.7	1.67313
$590$	0	0
$591$	7718.22	0.537200
$592$	2200.61i	0.152778i
$593$	13800.7i	0.955696i	0.878442	+	0.477848i	$0.158583\pi$
$593$	13800.7i	0.955696i	−0.878442	+	0.477848i	$0.841417\pi$
$594$	−1458.91	−0.100774
$595$	0	0
$596$	10019.7	0.688628
$597$	15948.6i	1.09335i
$598$	10520.5i	0.719426i
$599$	−21401.7	−1.45985	−0.729923	−	0.683529i	$-0.760444\pi$
$599$	−21401.7	−1.45985	−0.729923	+	0.683529i	$0.760444\pi$
$600$	0	0
$601$	−3688.78	−0.250363	−0.125182	−	0.992134i	$-0.539951\pi$
$601$	−3688.78	−0.250363	−0.125182	+	0.992134i	$0.539951\pi$
$602$	23213.2i	1.57160i
$603$	4876.19i	0.329310i
$604$	−49747.3	−3.35131
$605$	0	0
$606$	3855.19	0.258427
$607$	5086.89i	0.340149i	0.985431	+	0.170075i	$0.0544008\pi$
$607$	5086.89i	0.340149i	−0.985431	+	0.170075i	$0.945599\pi$
$608$	− 1381.79i	− 0.0921696i
$609$	27432.9	1.82535
$610$	0	0
$611$	−11279.8	−0.746860
$612$	− 18173.8i	− 1.20038i
$613$	111.568i	0.00735106i	0.999993	+	0.00367553i	$0.00116996\pi$
$613$	111.568i	0.00735106i	−0.999993	+	0.00367553i	$0.998830\pi$
$614$	31813.0	2.09099
$615$	0	0
$616$	15476.5	1.01228
$617$	− 19134.1i	− 1.24848i	−0.781234	−	0.624238i	$-0.785410\pi$
$617$	− 19134.1i	− 1.24848i	0.781234	−	0.624238i	$-0.214590\pi$
$618$	16670.6i	1.08509i
$619$	−23743.6	−1.54174	−0.770870	−	0.636992i	$-0.780178\pi$
$619$	−23743.6	−1.54174	−0.770870	+	0.636992i	$0.780178\pi$
$620$	0	0
$621$	−2149.02	−0.138868
$622$	29703.5i	1.91479i
$623$	− 9903.62i	− 0.636886i
$624$	5418.25	0.347602
$625$	0	0
$626$	−1831.14	−0.116912
$627$	4451.68i	0.283546i
$628$	1195.09i	0.0759384i
$629$	4104.72	0.260200
$630$	0	0
$631$	2758.25	0.174016	0.0870081	−	0.996208i	$-0.472269\pi$
$631$	2758.25	0.174016	0.0870081	+	0.996208i	$0.472269\pi$
$632$	− 4958.59i	− 0.312092i
$633$	1192.51i	0.0748787i
$634$	−7963.64	−0.498859
$635$	0	0
$636$	23415.0	1.45985
$637$	24174.5i	1.50366i
$638$	− 14023.6i	− 0.870221i
$639$	7410.14	0.458749
$640$	0	0
$641$	−754.409	−0.0464857	−0.0232429	−	0.999730i	$-0.507399\pi$
$641$	−754.409	−0.0464857	−0.0232429	+	0.999730i	$0.507399\pi$
$642$	− 987.236i	− 0.0606902i
$643$	14511.8i	0.890029i	0.895523	+	0.445014i	$0.146801\pi$
$643$	14511.8i	0.890029i	−0.895523	+	0.445014i	$0.853199\pi$
$644$	45232.1	2.76769
$645$	0	0
$646$	−82960.5	−5.05269
$647$	20611.1i	1.25241i	0.779660	+	0.626204i	$0.215392\pi$
$647$	20611.1i	1.25241i	−0.779660	+	0.626204i	$0.784608\pi$
$648$	3234.52i	0.196086i
$649$	1506.42	0.0911129
$650$	0	0
$651$	18739.9	1.12823
$652$	49893.7i	2.99692i
$653$	− 16724.6i	− 1.00227i	−0.865368	−	0.501137i	$-0.832915\pi$
$653$	− 16724.6i	− 1.00227i	0.865368	−	0.501137i	$-0.167085\pi$
$654$	6774.84	0.405072
$655$	0	0
$656$	−22099.8	−1.31532
$657$	261.626i	0.0155358i
$658$	72550.3i	4.29834i
$659$	−17369.2	−1.02672	−0.513359	−	0.858174i	$-0.671599\pi$
$659$	−17369.2	−1.02672	−0.513359	+	0.858174i	$0.671599\pi$
$660$	0	0
$661$	26491.3	1.55884	0.779419	−	0.626503i	$-0.215514\pi$
$661$	26491.3	1.55884	0.779419	+	0.626503i	$0.215514\pi$
$662$	36320.4i	2.13237i
$663$	− 10106.5i	− 0.592010i
$664$	−17396.1	−1.01672
$665$	0	0
$666$	−1449.47	−0.0843331
$667$	− 20657.2i	− 1.19918i
$668$	− 50409.6i	− 2.91977i
$669$	15885.1	0.918019
$670$	0	0
$671$	−6862.27	−0.394806
$672$	− 1082.70i	− 0.0621520i
$673$	− 12136.5i	− 0.695139i	−0.937654	−	0.347569i	$-0.887007\pi$
$673$	− 12136.5i	− 0.695139i	0.937654	−	0.347569i	$-0.112993\pi$
$674$	−40498.6	−2.31446
$675$	0	0
$676$	−23757.4	−1.35169
$677$	− 15583.9i	− 0.884691i	−0.896845	−	0.442346i	$-0.854146\pi$
$677$	− 15583.9i	− 0.884691i	0.896845	−	0.442346i	$-0.145854\pi$
$678$	12019.2i	0.680816i
$679$	1434.47	0.0810752
$680$	0	0
$681$	−5809.80	−0.326919
$682$	− 9579.79i	− 0.537873i
$683$	10388.8i	0.582016i	0.956721	+	0.291008i	$0.0939906\pi$
$683$	10388.8i	0.582016i	−0.956721	+	0.291008i	$0.906009\pi$
$684$	19582.5	1.09467
$685$	0	0
$686$	96123.8	5.34989
$687$	− 19138.0i	− 1.06282i
$688$	− 9002.38i	− 0.498855i
$689$	13021.1	0.719975
$690$	0	0
$691$	−773.591	−0.0425887	−0.0212943	−	0.999773i	$-0.506779\pi$
$691$	−773.591	−0.0425887	−0.0212943	+	0.999773i	$0.506779\pi$
$692$	− 31734.0i	− 1.74327i
$693$	3488.11i	0.191201i
$694$	−2200.95	−0.120385
$695$	0	0
$696$	−31091.5	−1.69327
$697$	41222.0i	2.24016i
$698$	− 45693.9i	− 2.47785i
$699$	−8977.45	−0.485777
$700$	0	0
$701$	13704.5	0.738391	0.369196	−	0.929352i	$-0.379633\pi$
$701$	13704.5	0.738391	0.369196	+	0.929352i	$0.379633\pi$
$702$	3568.82i	0.191875i
$703$	4422.88i	0.237286i
$704$	5353.04	0.286577
$705$	0	0
$706$	29316.1	1.56278
$707$	− 9217.39i	− 0.490319i
$708$	− 6626.60i	− 0.351755i
$709$	−25131.5	−1.33122	−0.665608	−	0.746302i	$-0.731828\pi$
$709$	−25131.5	−1.33122	−0.665608	+	0.746302i	$0.731828\pi$
$710$	0	0
$711$	1117.57	0.0589483
$712$	11224.4i	0.590804i
$713$	− 14111.3i	− 0.741196i
$714$	−65003.6	−3.40714
$715$	0	0
$716$	−7754.29	−0.404737
$717$	− 8788.88i	− 0.457778i
$718$	54817.4i	2.84926i
$719$	−25944.0	−1.34568	−0.672842	−	0.739786i	$-0.734926\pi$
$719$	−25944.0	−1.34568	−0.672842	+	0.739786i	$0.734926\pi$
$720$	0	0
$721$	39857.7	2.05878
$722$	− 55698.4i	− 2.87102i
$723$	5918.21i	0.304427i
$724$	46829.5	2.40387
$725$	0	0
$726$	1783.11	0.0911536
$727$	− 32366.9i	− 1.65120i	−0.564255	−	0.825601i	$-0.690836\pi$
$727$	− 32366.9i	− 1.65120i	0.564255	−	0.825601i	$-0.309164\pi$
$728$	− 37859.0i	− 1.92740i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−16791.8	−0.849613
$732$	30186.4i	1.52421i
$733$	− 962.782i	− 0.0485145i	−0.999706	−	0.0242573i	$-0.992278\pi$
$733$	− 962.782i	− 0.0485145i	0.999706	−	0.0242573i	$-0.00772209\pi$
$734$	−32063.8	−1.61239
$735$	0	0
$736$	−815.284	−0.0408312
$737$	− 5959.79i	− 0.297872i
$738$	− 14556.4i	− 0.726056i
$739$	25242.1	1.25649	0.628244	−	0.778016i	$-0.283774\pi$
$739$	25242.1	1.25649	0.628244	+	0.778016i	$0.283774\pi$
$740$	0	0
$741$	10889.8	0.539875
$742$	− 83750.0i	− 4.14361i
$743$	24365.5i	1.20307i	0.798846	+	0.601535i	$0.205444\pi$
$743$	24365.5i	1.20307i	−0.798846	+	0.601535i	$0.794556\pi$
$744$	−21239.1	−1.04659
$745$	0	0
$746$	−19621.2	−0.962982
$747$	− 3920.76i	− 0.192039i
$748$	22212.5i	1.08579i
$749$	−2360.38	−0.115149
$750$	0	0
$751$	−25453.0	−1.23674	−0.618370	−	0.785887i	$-0.712207\pi$
$751$	−25453.0	−1.23674	−0.618370	+	0.785887i	$0.712207\pi$
$752$	− 28135.9i	− 1.36438i
$753$	16959.5i	0.820768i
$754$	−34305.0	−1.65691
$755$	0	0
$756$	15343.8	0.738162
$757$	− 21934.8i	− 1.05315i	−0.850129	−	0.526575i	$-0.823476\pi$
$757$	− 21934.8i	− 1.05315i	0.850129	−	0.526575i	$-0.176524\pi$
$758$	− 37601.0i	− 1.80176i
$759$	2626.58	0.125611
$760$	0	0
$761$	8162.88	0.388836	0.194418	−	0.980919i	$-0.437718\pi$
$761$	8162.88	0.388836	0.194418	+	0.980919i	$0.437718\pi$
$762$	25996.2i	1.23588i
$763$	− 16198.0i	− 0.768553i
$764$	−20847.7	−0.987231
$765$	0	0
$766$	3952.48	0.186435
$767$	− 3685.05i	− 0.173480i
$768$	− 24755.1i	− 1.16311i
$769$	35471.5	1.66337	0.831687	−	0.555244i	$-0.187375\pi$
$769$	35471.5	1.66337	0.831687	+	0.555244i	$0.187375\pi$
$770$	0	0
$771$	−13822.5	−0.645662
$772$	31599.3i	1.47317i
$773$	− 2119.31i	− 0.0986109i	−0.998784	−	0.0493055i	$-0.984299\pi$
$773$	− 2119.31i	− 0.0986109i	0.998784	−	0.0493055i	$-0.0157008\pi$
$774$	5929.57	0.275367
$775$	0	0
$776$	−1625.78	−0.0752090
$777$	3465.54i	0.160007i
$778$	− 36259.1i	− 1.67089i
$779$	−44417.1	−2.04289
$780$	0	0
$781$	−9056.84	−0.414954
$782$	48948.3i	2.23835i
$783$	− 7007.44i	− 0.319828i
$784$	−60300.0	−2.74690
$785$	0	0
$786$	−6674.34	−0.302883
$787$	21787.8i	0.986851i	0.869788	+	0.493425i	$0.164255\pi$
$787$	21787.8i	0.986851i	−0.869788	+	0.493425i	$0.835745\pi$
$788$	41496.5i	1.87595i
$789$	−6624.92	−0.298927
$790$	0	0
$791$	28736.6	1.29173
$792$	− 3953.30i	− 0.177367i
$793$	16786.7i	0.751717i
$794$	−44329.6	−1.98136
$795$	0	0
$796$	−85746.3	−3.81809
$797$	− 16775.1i	− 0.745552i	−0.927921	−	0.372776i	$-0.878406\pi$
$797$	− 16775.1i	− 0.745552i	0.927921	−	0.372776i	$-0.121594\pi$
$798$	− 70042.1i	− 3.10710i
$799$	−52480.8	−2.32370
$800$	0	0
$801$	−2529.77	−0.111592
$802$	− 64475.9i	− 2.83881i
$803$	− 319.766i	− 0.0140527i
$804$	−26216.5	−1.14998
$805$	0	0
$806$	−23434.3	−1.02412
$807$	− 8412.59i	− 0.366961i
$808$	10446.7i	0.454842i
$809$	−42770.0	−1.85873	−0.929366	−	0.369159i	$-0.879646\pi$
$809$	−42770.0	−1.85873	−0.929366	+	0.369159i	$0.879646\pi$
$810$	0	0
$811$	22532.5	0.975614	0.487807	−	0.872951i	$-0.337797\pi$
$811$	22532.5	0.975614	0.487807	+	0.872951i	$0.337797\pi$
$812$	147491.i	6.37429i
$813$	309.857i	0.0133667i
$814$	1771.58	0.0762822
$815$	0	0
$816$	25209.2	1.08149
$817$	− 18093.3i	− 0.774793i
$818$	4188.99i	0.179052i
$819$	8532.71	0.364050
$820$	0	0
$821$	34394.6	1.46210	0.731048	−	0.682326i	$-0.239032\pi$
$821$	34394.6	1.46210	0.731048	+	0.682326i	$0.239032\pi$
$822$	6075.17i	0.257781i
$823$	3038.76i	0.128705i	0.997927	+	0.0643526i	$0.0204982\pi$
$823$	3038.76i	0.128705i	−0.997927	+	0.0643526i	$0.979502\pi$
$824$	−45173.2	−1.90981
$825$	0	0
$826$	−23701.8	−0.998416
$827$	2767.63i	0.116372i	0.998306	+	0.0581861i	$0.0185317\pi$
$827$	2767.63i	0.116372i	−0.998306	+	0.0581861i	$0.981468\pi$
$828$	− 11554.0i	− 0.484940i
$829$	16593.5	0.695196	0.347598	−	0.937644i	$-0.386997\pi$
$829$	16593.5	0.695196	0.347598	+	0.937644i	$0.386997\pi$
$830$	0	0
$831$	−5076.30	−0.211907
$832$	− 13094.8i	− 0.545648i
$833$	112475.i	4.67831i
$834$	1432.25	0.0594663
$835$	0	0
$836$	−23934.2	−0.990168
$837$	− 4786.91i	− 0.197682i
$838$	11032.5i	0.454789i
$839$	11891.4	0.489315	0.244657	−	0.969610i	$-0.421325\pi$
$839$	11891.4	0.489315	0.244657	+	0.969610i	$0.421325\pi$
$840$	0	0
$841$	42969.3	1.76183
$842$	− 67843.4i	− 2.77677i
$843$	− 2738.71i	− 0.111893i
$844$	−6411.47	−0.261483
$845$	0	0
$846$	18532.2	0.753132
$847$	− 4263.25i	− 0.172948i
$848$	32479.2i	1.31526i
$849$	21733.2	0.878540
$850$	0	0
$851$	2609.58	0.105118
$852$	39840.1i	1.60199i
$853$	− 44431.4i	− 1.78347i	−0.452556	−	0.891736i	$-0.649488\pi$
$853$	− 44431.4i	− 1.78347i	0.452556	−	0.891736i	$-0.350512\pi$
$854$	107970.	4.32629
$855$	0	0
$856$	2675.17	0.106817
$857$	4728.48i	0.188473i	0.995550	+	0.0942367i	$0.0300411\pi$
$857$	4728.48i	0.188473i	−0.995550	+	0.0942367i	$0.969959\pi$
$858$	− 4361.90i	− 0.173558i
$859$	12199.4	0.484560	0.242280	−	0.970206i	$-0.422105\pi$
$859$	12199.4	0.484560	0.242280	+	0.970206i	$0.422105\pi$
$860$	0	0
$861$	−34803.0	−1.37756
$862$	− 65778.1i	− 2.59909i
$863$	− 10519.8i	− 0.414947i	−0.978241	−	0.207474i	$-0.933476\pi$
$863$	− 10519.8i	− 0.414947i	0.978241	−	0.207474i	$-0.0665241\pi$
$864$	−276.565	−0.0108899
$865$	0	0
$866$	28407.7	1.11470
$867$	− 32282.7i	− 1.26457i
$868$	100754.i	3.93987i
$869$	−1365.92	−0.0533208
$870$	0	0
$871$	−14579.0	−0.567153
$872$	18358.2i	0.712944i
$873$	− 366.421i	− 0.0142056i
$874$	−52742.3	−2.04123
$875$	0	0
$876$	−1406.62	−0.0542525
$877$	− 6620.91i	− 0.254928i	−0.991843	−	0.127464i	$-0.959316\pi$
$877$	− 6620.91i	− 0.254928i	0.991843	−	0.127464i	$-0.0406838\pi$
$878$	− 13809.8i	− 0.530819i
$879$	−4863.16	−0.186610
$880$	0	0
$881$	23166.2	0.885913	0.442957	−	0.896543i	$-0.353930\pi$
$881$	23166.2	0.885913	0.442957	+	0.896543i	$0.353930\pi$
$882$	− 39717.6i	− 1.51628i
$883$	24131.3i	0.919686i	0.888000	+	0.459843i	$0.152094\pi$
$883$	24131.3i	0.919686i	−0.888000	+	0.459843i	$0.847906\pi$
$884$	54336.7	2.06735
$885$	0	0
$886$	66157.9	2.50860
$887$	− 35611.7i	− 1.34806i	−0.738706	−	0.674028i	$-0.764563\pi$
$887$	− 35611.7i	− 1.34806i	0.738706	−	0.674028i	$-0.235437\pi$
$888$	− 3927.72i	− 0.148430i
$889$	62154.3	2.34487
$890$	0	0
$891$	891.000	0.0335013
$892$	85405.2i	3.20581i
$893$	− 56548.6i	− 2.11907i
$894$	−9154.45	−0.342473
$895$	0	0
$896$	−87111.2	−3.24797
$897$	− 6425.20i	− 0.239165i
$898$	− 13126.0i	− 0.487774i
$899$	46013.6	1.70705
$900$	0	0
$901$	60582.3	2.24005
$902$	17791.2i	0.656742i
$903$	− 14177.0i	− 0.522460i
$904$	−32569.1	−1.19826
$905$	0	0
$906$	45451.5	1.66669
$907$	− 841.317i	− 0.0307999i	−0.999881	−	0.0153999i	$-0.995098\pi$
$907$	− 841.317i	− 0.0307999i	0.999881	−	0.0153999i	$-0.00490214\pi$
$908$	− 31235.9i	− 1.14163i
$909$	−2354.48	−0.0859111
$910$	0	0
$911$	−47484.6	−1.72693	−0.863466	−	0.504406i	$-0.831711\pi$
$911$	−47484.6	−1.72693	−0.863466	+	0.504406i	$0.831711\pi$
$912$	27163.2i	0.986253i
$913$	4792.04i	0.173706i
$914$	−55678.1	−2.01495
$915$	0	0
$916$	102894.	3.71148
$917$	15957.7i	0.574667i
$918$	16604.5i	0.596981i
$919$	−19432.0	−0.697500	−0.348750	−	0.937216i	$-0.613394\pi$
$919$	−19432.0	−0.697500	−0.348750	+	0.937216i	$0.613394\pi$
$920$	0	0
$921$	−19429.1	−0.695127
$922$	74791.7i	2.67151i
$923$	22155.1i	0.790079i
$924$	−18753.6	−0.667692
$925$	0	0
$926$	−16269.3	−0.577366
$927$	− 10181.2i	− 0.360728i
$928$	− 2658.45i	− 0.0940386i
$929$	−18727.8	−0.661399	−0.330699	−	0.943736i	$-0.607285\pi$
$929$	−18727.8	−0.661399	−0.330699	+	0.943736i	$0.607285\pi$
$930$	0	0
$931$	−121193.	−4.26633
$932$	− 48266.6i	− 1.69638i
$933$	− 18140.8i	− 0.636552i
$934$	−12386.4	−0.433936
$935$	0	0
$936$	−9670.66	−0.337709
$937$	− 54002.9i	− 1.88282i	−0.337271	−	0.941408i	$-0.609504\pi$
$937$	− 54002.9i	− 1.88282i	0.337271	−	0.941408i	$-0.390496\pi$
$938$	93770.4i	3.26408i
$939$	1118.33	0.0388661
$940$	0	0
$941$	6293.49	0.218025	0.109013	−	0.994040i	$-0.465231\pi$
$941$	6293.49	0.218025	0.109013	+	0.994040i	$0.465231\pi$
$942$	− 1091.89i	− 0.0377662i
$943$	26206.9i	0.905000i
$944$	9191.86	0.316917
$945$	0	0
$946$	−7247.25	−0.249079
$947$	50524.8i	1.73372i	0.498548	+	0.866862i	$0.333867\pi$
$947$	50524.8i	1.73372i	−0.498548	+	0.866862i	$0.666133\pi$
$948$	6008.55i	0.205853i
$949$	−782.219	−0.0267565
$950$	0	0
$951$	4863.63	0.165840
$952$	− 176144.i	− 5.99671i
$953$	10331.6i	0.351180i	0.984463	+	0.175590i	$0.0561833\pi$
$953$	10331.6i	0.351180i	−0.984463	+	0.175590i	$0.943817\pi$
$954$	−21393.0	−0.726021
$955$	0	0
$956$	47252.8	1.59860
$957$	8564.65i	0.289295i
$958$	− 91038.3i	− 3.07026i
$959$	14525.1	0.489094
$960$	0	0
$961$	1641.75	0.0551088
$962$	− 4333.67i	− 0.145242i
$963$	602.934i	0.0201758i
$964$	−31818.8	−1.06309
$965$	0	0
$966$	−41326.2	−1.37645
$967$	− 36951.3i	− 1.22883i	−0.788985	−	0.614413i	$-0.789393\pi$
$967$	− 36951.3i	− 1.22883i	0.788985	−	0.614413i	$-0.210607\pi$
$968$	4831.81i	0.160434i
$969$	50666.4	1.67971
$970$	0	0
$971$	45936.7	1.51821	0.759104	−	0.650969i	$-0.225637\pi$
$971$	45936.7	1.51821	0.759104	+	0.650969i	$0.225637\pi$
$972$	− 3919.42i	− 0.129337i
$973$	− 3424.38i	− 0.112827i
$974$	4331.76	0.142504
$975$	0	0
$976$	−41872.0	−1.37325
$977$	26083.4i	0.854126i	0.904222	+	0.427063i	$0.140452\pi$
$977$	26083.4i	0.854126i	−0.904222	+	0.427063i	$0.859548\pi$
$978$	− 45585.2i	− 1.49045i
$979$	3091.94	0.100939
$980$	0	0
$981$	−4137.59	−0.134662
$982$	− 57102.0i	− 1.85560i
$983$	47213.0i	1.53190i	0.642898	+	0.765952i	$0.277732\pi$
$983$	47213.0i	1.53190i	−0.642898	+	0.765952i	$0.722268\pi$
$984$	39444.4	1.27789
$985$	0	0
$986$	−159609.	−5.15515
$987$	− 44308.6i	− 1.42894i
$988$	58548.4i	1.88530i
$989$	−10675.4	−0.343234
$990$	0	0
$991$	15853.2	0.508168	0.254084	−	0.967182i	$-0.418226\pi$
$991$	15853.2	0.508168	0.254084	+	0.967182i	$0.418226\pi$
$992$	− 1816.03i	− 0.0581241i
$993$	− 22181.9i	− 0.708885i
$994$	142499.	4.54707
$995$	0	0
$996$	21079.7	0.670618
$997$	14205.9i	0.451259i	0.974213	+	0.225630i	$0.0724439\pi$
$997$	14205.9i	0.451259i	−0.974213	+	0.225630i	$0.927556\pi$
$998$	50749.6i	1.60967i
$999$	885.235	0.0280356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.r.199.3		10
5.2	odd	4		825.4.a.z.1.4	yes	5
5.3	odd	4		825.4.a.w.1.2	✓	5
5.4	even	2	inner	825.4.c.r.199.8		10
15.2	even	4		2475.4.a.bf.1.2		5
15.8	even	4		2475.4.a.bm.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.w.1.2	✓	5	5.3	odd	4
825.4.a.z.1.4	yes	5	5.2	odd	4
825.4.c.r.199.3		10	1.1	even	1	trivial
825.4.c.r.199.8		10	5.4	even	2	inner
2475.4.a.bf.1.2		5	15.2	even	4
2475.4.a.bm.1.4		5	15.8	even	4

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.91216i q^{2} +3.00000i q^{3} -16.1293 q^{4} +14.7365 q^{6} -35.2335i q^{7} +39.9323i q^{8} -9.00000 q^{9} +11.0000 q^{11} -48.3879i q^{12} -26.9085i q^{13} -173.072 q^{14} +67.1195 q^{16} -125.196i q^{17} +44.2094i q^{18} +134.900 q^{19} +105.700 q^{21} -54.0337i q^{22} -79.5933i q^{23} -119.797 q^{24} -132.179 q^{26} -27.0000i q^{27} +568.290i q^{28} +259.535 q^{29} +177.293 q^{31} -10.2431i q^{32} +33.0000i q^{33} -614.980 q^{34} +145.164 q^{36} +32.7865i q^{37} -662.648i q^{38} +80.7254 q^{39} -329.261 q^{41} -519.217i q^{42} -134.125i q^{43} -177.422 q^{44} -390.975 q^{46} -419.191i q^{47} +201.359i q^{48} -898.397 q^{49} +375.587 q^{51} +434.015i q^{52} +483.902i q^{53} -132.628 q^{54} +1406.95 q^{56} +404.699i q^{57} -1274.88i q^{58} +136.948 q^{59} -623.843 q^{61} -870.890i q^{62} +317.101i q^{63} +486.640 q^{64} +162.101 q^{66} -541.799i q^{67} +2019.31i q^{68} +238.780 q^{69} -823.349 q^{71} -359.391i q^{72} -29.0696i q^{73} +161.052 q^{74} -2175.83 q^{76} -387.568i q^{77} -396.536i q^{78} -124.175 q^{79} +81.0000 q^{81} +1617.38i q^{82} +435.640i q^{83} -1704.87 q^{84} -658.841 q^{86} +778.604i q^{87} +439.256i q^{88} +281.086 q^{89} -948.079 q^{91} +1283.78i q^{92} +531.878i q^{93} -2059.13 q^{94} +30.7294 q^{96} +40.7134i q^{97} +4413.07i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9} + 110 q^{11} - 110 q^{14} + 1268 q^{16} + 674 q^{19} - 228 q^{21} - 432 q^{24} + 718 q^{26} + 306 q^{29} + 526 q^{31} - 1034 q^{34} + 828 q^{36} - 258 q^{39} - 176 q^{41}+ \cdots - 990 q^{99}+O(q^{100}) \) 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9 + 110 * q^11 - 110 * q^14 + 1268 * q^16 + 674 * q^19 - 228 * q^21 - 432 * q^24 + 718 * q^26 + 306 * q^29 + 526 * q^31 - 1034 * q^34 + 828 * q^36 - 258 * q^39 - 176 * q^41 - 1012 * q^44 + 872 * q^46 - 4762 * q^49 + 900 * q^51 - 216 * q^54 + 422 * q^56 - 820 * q^59 - 2260 * q^61 - 6340 * q^64 + 264 * q^66 + 1206 * q^69 + 2498 * q^71 + 5970 * q^74 - 5112 * q^76 - 4516 * q^79 + 810 * q^81 + 2646 * q^84 - 2870 * q^86 - 694 * q^89 - 1224 * q^91 - 1814 * q^94 + 1680 * q^96 - 990 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more