LMFDB - Embedded newform 825.4.c.n.199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.181494784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 12x^{3} + 49x^{2} - 14x + 2 $ x^6 - 2x^5 + 2x^4 + 12x^3 + 49x^2 - 14*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$0.138157 + 0.138157i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.n.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.723686i q^{2} +3.00000i q^{3} +7.47628 q^{4} -2.17106 q^{6} -1.13288i q^{7} +11.2000i q^{8} -9.00000 q^{9} +11.0000 q^{11} +22.4288i q^{12} -21.8566i q^{13} +0.819851 q^{14} +51.7050 q^{16} -6.18033i q^{17} -6.51317i q^{18} +92.8393 q^{19} +3.39865 q^{21} +7.96054i q^{22} -36.7317i q^{23} -33.5999 q^{24} +15.8173 q^{26} -27.0000i q^{27} -8.46975i q^{28} -71.1375 q^{29} +186.522 q^{31} +127.018i q^{32} +33.0000i q^{33} +4.47261 q^{34} -67.2865 q^{36} +356.581i q^{37} +67.1865i q^{38} +65.5697 q^{39} +271.940 q^{41} +2.45955i q^{42} +155.780i q^{43} +82.2391 q^{44} +26.5822 q^{46} -234.566i q^{47} +155.115i q^{48} +341.717 q^{49} +18.5410 q^{51} -163.406i q^{52} +195.018i q^{53} +19.5395 q^{54} +12.6882 q^{56} +278.518i q^{57} -51.4812i q^{58} +455.930 q^{59} -441.278 q^{61} +134.983i q^{62} +10.1959i q^{63} +321.719 q^{64} -23.8816 q^{66} -133.005i q^{67} -46.2058i q^{68} +110.195 q^{69} -1041.68 q^{71} -100.800i q^{72} -160.119i q^{73} -258.053 q^{74} +694.093 q^{76} -12.4617i q^{77} +47.4519i q^{78} +761.367 q^{79} +81.0000 q^{81} +196.799i q^{82} +51.7170i q^{83} +25.4092 q^{84} -112.736 q^{86} -213.412i q^{87} +123.200i q^{88} +1075.25 q^{89} -24.7609 q^{91} -274.616i q^{92} +559.565i q^{93} +169.752 q^{94} -381.054 q^{96} +703.238i q^{97} +247.295i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} + 232 q^{14} - 170 q^{16} + 338 q^{19} - 96 q^{21} - 18 q^{24} + 334 q^{26} + 554 q^{29} - 346 q^{31} + 292 q^{34} + 126 q^{36} + 270 q^{39} + 88 q^{41} - 154 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) $ 6 * q - 14 * q^4 - 6 * q^6 - 54 * q^9 + 66 * q^11 + 232 * q^14 - 170 * q^16 + 338 * q^19 - 96 * q^21 - 18 * q^24 + 334 * q^26 + 554 * q^29 - 346 * q^31 + 292 * q^34 + 126 * q^36 + 270 * q^39 + 88 * q^41 - 154 * q^44 + 850 * q^46 + 854 * q^49 + 348 * q^51 + 54 * q^54 + 336 * q^56 + 1368 * q^59 - 2076 * q^61 + 2258 * q^64 - 66 * q^66 + 930 * q^69 - 2918 * q^71 + 3528 * q^74 + 422 * q^76 + 1012 * q^79 + 486 * q^81 + 600 * q^84 - 1054 * q^86 + 1214 * q^89 - 796 * q^91 + 3644 * q^94 - 582 * q^96 - 594 * 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 $

$2$	0.723686i	0.255862i	0.991783	+	0.127931i	$0.0408335\pi$
$2$	0.723686i	0.255862i	−0.991783	+	0.127931i	$0.959166\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	7.47628	0.934535
$5$	0	0
$5$	0	0
$6$	−2.17106	−0.147722
$7$	− 1.13288i	− 0.0611699i	−0.999532	−	0.0305850i	$-0.990263\pi$
$7$	− 1.13288i	− 0.0611699i	0.999532	−	0.0305850i	$-0.00973702\pi$
$8$	11.2000i	0.494973i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	22.4288i	0.539554i
$13$	− 21.8566i	− 0.466302i	−0.972441	−	0.233151i	$-0.925096\pi$
$13$	− 21.8566i	− 0.466302i	0.972441	−	0.233151i	$-0.0749036\pi$
$14$	0.819851	0.0156510
$15$	0	0
$16$	51.7050	0.807890
$17$	− 6.18033i	− 0.0881735i	−0.999028	−	0.0440867i	$-0.985962\pi$
$17$	− 6.18033i	− 0.0881735i	0.999028	−	0.0440867i	$-0.0140378\pi$
$18$	− 6.51317i	− 0.0852872i
$19$	92.8393	1.12099	0.560495	−	0.828158i	$-0.310611\pi$
$19$	92.8393	1.12099	0.560495	+	0.828158i	$0.310611\pi$
$20$	0	0
$21$	3.39865	0.0353165
$22$	7.96054i	0.0771452i
$23$	− 36.7317i	− 0.333004i	−0.986041	−	0.166502i	$-0.946753\pi$
$23$	− 36.7317i	− 0.333004i	0.986041	−	0.166502i	$-0.0532472\pi$
$24$	−33.5999	−0.285773
$25$	0	0
$26$	15.8173	0.119309
$27$	− 27.0000i	− 0.192450i
$28$	− 8.46975i	− 0.0571654i
$29$	−71.1375	−0.455514	−0.227757	−	0.973718i	$-0.573139\pi$
$29$	−71.1375	−0.455514	−0.227757	+	0.973718i	$0.573139\pi$
$30$	0	0
$31$	186.522	1.08065	0.540327	−	0.841455i	$-0.318301\pi$
$31$	186.522	1.08065	0.540327	+	0.841455i	$0.318301\pi$
$32$	127.018i	0.701681i
$33$	33.0000i	0.174078i
$34$	4.47261	0.0225602
$35$	0	0
$36$	−67.2865	−0.311512
$37$	356.581i	1.58437i	0.610282	+	0.792184i	$0.291056\pi$
$37$	356.581i	1.58437i	−0.610282	+	0.792184i	$0.708944\pi$
$38$	67.1865i	0.286818i
$39$	65.5697	0.269219
$40$	0	0
$41$	271.940	1.03585	0.517925	−	0.855426i	$-0.326705\pi$
$41$	271.940	1.03585	0.517925	+	0.855426i	$0.326705\pi$
$42$	2.45955i	0.00903613i
$43$	155.780i	0.552470i	0.961090	+	0.276235i	$0.0890868\pi$
$43$	155.780i	0.552470i	−0.961090	+	0.276235i	$0.910913\pi$
$44$	82.2391	0.281773
$45$	0	0
$46$	26.5822	0.0852028
$47$	− 234.566i	− 0.727978i	−0.931403	−	0.363989i	$-0.881415\pi$
$47$	− 234.566i	− 0.727978i	0.931403	−	0.363989i	$-0.118585\pi$
$48$	155.115i	0.466436i
$49$	341.717	0.996258
$50$	0	0
$51$	18.5410	0.0509070
$52$	− 163.406i	− 0.435775i
$53$	195.018i	0.505429i	0.967541	+	0.252715i	$0.0813234\pi$
$53$	195.018i	0.505429i	−0.967541	+	0.252715i	$0.918677\pi$
$54$	19.5395	0.0492406
$55$	0	0
$56$	12.6882	0.0302775
$57$	278.518i	0.647204i
$58$	− 51.4812i	− 0.116548i
$59$	455.930	1.00605	0.503025	−	0.864272i	$-0.332220\pi$
$59$	455.930	1.00605	0.503025	+	0.864272i	$0.332220\pi$
$60$	0	0
$61$	−441.278	−0.926227	−0.463114	−	0.886299i	$-0.653268\pi$
$61$	−441.278	−0.926227	−0.463114	+	0.886299i	$0.653268\pi$
$62$	134.983i	0.276498i
$63$	10.1959i	0.0203900i
$64$	321.719	0.628357
$65$	0	0
$66$	−23.8816	−0.0445398
$67$	− 133.005i	− 0.242524i	−0.992621	−	0.121262i	$-0.961306\pi$
$67$	− 133.005i	− 0.242524i	0.992621	−	0.121262i	$-0.0386941\pi$
$68$	− 46.2058i	− 0.0824012i
$69$	110.195	0.192260
$70$	0	0
$71$	−1041.68	−1.74119	−0.870596	−	0.491999i	$-0.836266\pi$
$71$	−1041.68	−1.74119	−0.870596	+	0.491999i	$0.836266\pi$
$72$	− 100.800i	− 0.164991i
$73$	− 160.119i	− 0.256719i	−0.991728	−	0.128360i	$-0.959029\pi$
$73$	− 160.119i	− 0.256719i	0.991728	−	0.128360i	$-0.0409712\pi$
$74$	−258.053	−0.405379
$75$	0	0
$76$	694.093	1.04760
$77$	− 12.4617i	− 0.0184434i
$78$	47.4519i	0.0688829i
$79$	761.367	1.08431	0.542155	−	0.840278i	$-0.317608\pi$
$79$	761.367	1.08431	0.542155	+	0.840278i	$0.317608\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	196.799i	0.265034i
$83$	51.7170i	0.0683938i	0.999415	+	0.0341969i	$0.0108873\pi$
$83$	51.7170i	0.0683938i	−0.999415	+	0.0341969i	$0.989113\pi$
$84$	25.4092	0.0330045
$85$	0	0
$86$	−112.736	−0.141356
$87$	− 213.412i	− 0.262991i
$88$	123.200i	0.149240i
$89$	1075.25	1.28064	0.640318	−	0.768110i	$-0.278802\pi$
$89$	1075.25	1.28064	0.640318	+	0.768110i	$0.278802\pi$
$90$	0	0
$91$	−24.7609	−0.0285236
$92$	− 274.616i	− 0.311203i
$93$	559.565i	0.623916i
$94$	169.752	0.186262
$95$	0	0
$96$	−381.054	−0.405116
$97$	703.238i	0.736114i	0.929803	+	0.368057i	$0.119977\pi$
$97$	703.238i	0.736114i	−0.929803	+	0.368057i	$0.880023\pi$
$98$	247.295i	0.254904i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−856.886	−0.844192	−0.422096	−	0.906551i	$-0.638705\pi$
$101$	−856.886	−0.844192	−0.422096	+	0.906551i	$0.638705\pi$
$102$	13.4178i	0.0130251i
$103$	− 805.989i	− 0.771034i	−0.922701	−	0.385517i	$-0.874023\pi$
$103$	− 805.989i	− 0.771034i	0.922701	−	0.385517i	$-0.125977\pi$
$104$	244.793	0.230807
$105$	0	0
$106$	−141.132	−0.129320
$107$	1608.55i	1.45331i	0.687004	+	0.726654i	$0.258926\pi$
$107$	1608.55i	1.45331i	−0.687004	+	0.726654i	$0.741074\pi$
$108$	− 201.860i	− 0.179851i
$109$	925.724	0.813471	0.406735	−	0.913546i	$-0.366667\pi$
$109$	925.724	0.813471	0.406735	+	0.913546i	$0.366667\pi$
$110$	0	0
$111$	−1069.74	−0.914735
$112$	− 58.5757i	− 0.0494186i
$113$	1214.81i	1.01133i	0.862731	+	0.505663i	$0.168752\pi$
$113$	1214.81i	1.01133i	−0.862731	+	0.505663i	$0.831248\pi$
$114$	−201.560	−0.165595
$115$	0	0
$116$	−531.844	−0.425693
$117$	196.709i	0.155434i
$118$	329.950i	0.257410i
$119$	−7.00159	−0.00539357
$120$	0	0
$121$	121.000	0.0909091
$122$	− 319.347i	− 0.236986i
$123$	815.819i	0.598048i
$124$	1394.49	1.00991
$125$	0	0
$126$	−7.37866	−0.00521701
$127$	− 1128.96i	− 0.788810i	−0.918937	−	0.394405i	$-0.870951\pi$
$127$	− 1128.96i	− 0.788810i	0.918937	−	0.394405i	$-0.129049\pi$
$128$	1248.97i	0.862454i
$129$	−467.340	−0.318969
$130$	0	0
$131$	−1562.32	−1.04199	−0.520993	−	0.853561i	$-0.674438\pi$
$131$	−1562.32	−1.04199	−0.520993	+	0.853561i	$0.674438\pi$
$132$	246.717i	0.162682i
$133$	− 105.176i	− 0.0685708i
$134$	96.2535	0.0620525
$135$	0	0
$136$	69.2194	0.0436435
$137$	− 44.9323i	− 0.0280206i	−0.999902	−	0.0140103i	$-0.995540\pi$
$137$	− 44.9323i	− 0.0280206i	0.999902	−	0.0140103i	$-0.00445977\pi$
$138$	79.7466i	0.0491919i
$139$	415.574	0.253587	0.126793	−	0.991929i	$-0.459532\pi$
$139$	415.574	0.253587	0.126793	+	0.991929i	$0.459532\pi$
$140$	0	0
$141$	703.698	0.420298
$142$	− 753.849i	− 0.445504i
$143$	− 240.422i	− 0.140595i
$144$	−465.345	−0.269297
$145$	0	0
$146$	115.876	0.0656846
$147$	1025.15i	0.575190i
$148$	2665.90i	1.48065i
$149$	−678.402	−0.372999	−0.186499	−	0.982455i	$-0.559714\pi$
$149$	−678.402	−0.372999	−0.186499	+	0.982455i	$0.559714\pi$
$150$	0	0
$151$	616.083	0.332027	0.166014	−	0.986123i	$-0.446910\pi$
$151$	616.083	0.332027	0.166014	+	0.986123i	$0.446910\pi$
$152$	1039.80i	0.554860i
$153$	55.6229i	0.0293912i
$154$	9.01837	0.00471896
$155$	0	0
$156$	490.217	0.251595
$157$	616.599i	0.313439i	0.987643	+	0.156720i	$0.0500919\pi$
$157$	616.599i	0.313439i	−0.987643	+	0.156720i	$0.949908\pi$
$158$	550.991i	0.277433i
$159$	−585.053	−0.291810
$160$	0	0
$161$	−41.6127	−0.0203698
$162$	58.6186i	0.0284291i
$163$	2031.15i	0.976026i	0.872836	+	0.488013i	$0.162278\pi$
$163$	2031.15i	0.976026i	−0.872836	+	0.488013i	$0.837722\pi$
$164$	2033.10	0.968038
$165$	0	0
$166$	−37.4269	−0.0174993
$167$	2114.90i	0.979973i	0.871730	+	0.489987i	$0.162998\pi$
$167$	2114.90i	0.979973i	−0.871730	+	0.489987i	$0.837002\pi$
$168$	38.0647i	0.0174807i
$169$	1719.29	0.782563
$170$	0	0
$171$	−835.554	−0.373663
$172$	1164.65i	0.516302i
$173$	− 15.5096i	− 0.00681602i	−0.999994	−	0.00340801i	$-0.998915\pi$
$173$	− 15.5096i	− 0.00681602i	0.999994	−	0.00340801i	$-0.00108481\pi$
$174$	154.444	0.0672893
$175$	0	0
$176$	568.755	0.243588
$177$	1367.79i	0.580844i
$178$	778.146i	0.327666i
$179$	−487.991	−0.203766	−0.101883	−	0.994796i	$-0.532487\pi$
$179$	−487.991	−0.203766	−0.101883	+	0.994796i	$0.532487\pi$
$180$	0	0
$181$	1248.71	0.512795	0.256397	−	0.966571i	$-0.417464\pi$
$181$	1248.71	0.512795	0.256397	+	0.966571i	$0.417464\pi$
$182$	− 17.9191i	− 0.00729810i
$183$	− 1323.83i	− 0.534758i
$184$	411.393	0.164828
$185$	0	0
$186$	−404.949	−0.159636
$187$	− 67.9836i	− 0.0265853i
$188$	− 1753.68i	− 0.680321i
$189$	−30.5878	−0.0117722
$190$	0	0
$191$	−2571.60	−0.974213	−0.487107	−	0.873342i	$-0.661948\pi$
$191$	−2571.60	−0.974213	−0.487107	+	0.873342i	$0.661948\pi$
$192$	965.156i	0.362782i
$193$	1221.80i	0.455686i	0.973698	+	0.227843i	$0.0731673\pi$
$193$	1221.80i	0.455686i	−0.973698	+	0.227843i	$0.926833\pi$
$194$	−508.924	−0.188343
$195$	0	0
$196$	2554.77	0.931038
$197$	1428.32i	0.516568i	0.966069	+	0.258284i	$0.0831571\pi$
$197$	1428.32i	0.516568i	−0.966069	+	0.258284i	$0.916843\pi$
$198$	− 71.6449i	− 0.0257151i
$199$	816.166	0.290736	0.145368	−	0.989378i	$-0.453563\pi$
$199$	816.166	0.290736	0.145368	+	0.989378i	$0.453563\pi$
$200$	0	0
$201$	399.014	0.140021
$202$	− 620.116i	− 0.215996i
$203$	80.5904i	0.0278637i
$204$	138.618	0.0475743
$205$	0	0
$206$	583.283	0.197278
$207$	330.585i	0.111001i
$208$	− 1130.09i	− 0.376721i
$209$	1021.23	0.337991
$210$	0	0
$211$	−903.360	−0.294739	−0.147369	−	0.989082i	$-0.547081\pi$
$211$	−903.360	−0.294739	−0.147369	+	0.989082i	$0.547081\pi$
$212$	1458.01i	0.472341i
$213$	− 3125.04i	− 1.00528i
$214$	−1164.08	−0.371846
$215$	0	0
$216$	302.399	0.0952576
$217$	− 211.307i	− 0.0661035i
$218$	669.934i	0.208136i
$219$	480.357	0.148217
$220$	0	0
$221$	−135.081	−0.0411154
$222$	− 774.159i	− 0.234046i
$223$	533.966i	0.160345i	0.996781	+	0.0801727i	$0.0255472\pi$
$223$	533.966i	0.160345i	−0.996781	+	0.0801727i	$0.974453\pi$
$224$	143.896	0.0429218
$225$	0	0
$226$	−879.142	−0.258760
$227$	− 696.531i	− 0.203658i	−0.994802	−	0.101829i	$-0.967531\pi$
$227$	− 696.531i	− 0.203658i	0.994802	−	0.101829i	$-0.0324695\pi$
$228$	2082.28i	0.604834i
$229$	796.794	0.229929	0.114964	−	0.993370i	$-0.463325\pi$
$229$	796.794	0.229929	0.114964	+	0.993370i	$0.463325\pi$
$230$	0	0
$231$	37.3851	0.0106483
$232$	− 796.737i	− 0.225467i
$233$	− 5657.08i	− 1.59059i	−0.606223	−	0.795295i	$-0.707316\pi$
$233$	− 5657.08i	− 1.59059i	0.606223	−	0.795295i	$-0.292684\pi$
$234$	−142.356	−0.0397696
$235$	0	0
$236$	3408.66	0.940190
$237$	2284.10i	0.626027i
$238$	− 5.06695i	− 0.00138001i
$239$	2023.06	0.547534	0.273767	−	0.961796i	$-0.411730\pi$
$239$	2023.06	0.547534	0.273767	+	0.961796i	$0.411730\pi$
$240$	0	0
$241$	−3513.86	−0.939203	−0.469601	−	0.882879i	$-0.655602\pi$
$241$	−3513.86	−0.939203	−0.469601	+	0.882879i	$0.655602\pi$
$242$	87.5660i	0.0232601i
$243$	243.000i	0.0641500i
$244$	−3299.12	−0.865592
$245$	0	0
$246$	−590.397	−0.153018
$247$	− 2029.15i	− 0.522719i
$248$	2089.03i	0.534895i
$249$	−155.151	−0.0394872
$250$	0	0
$251$	1814.32	0.456250	0.228125	−	0.973632i	$-0.426740\pi$
$251$	1814.32	0.456250	0.228125	+	0.973632i	$0.426740\pi$
$252$	76.2277i	0.0190551i
$253$	− 404.048i	− 0.100404i
$254$	817.011	0.201826
$255$	0	0
$256$	1669.89	0.407688
$257$	− 1445.46i	− 0.350839i	−0.984494	−	0.175419i	$-0.943872\pi$
$257$	− 1445.46i	− 0.350839i	0.984494	−	0.175419i	$-0.0561281\pi$
$258$	− 338.207i	− 0.0816118i
$259$	403.965	0.0969157
$260$	0	0
$261$	640.237	0.151838
$262$	− 1130.63i	− 0.266604i
$263$	− 4411.79i	− 1.03438i	−0.855870	−	0.517191i	$-0.826978\pi$
$263$	− 4411.79i	− 1.03438i	0.855870	−	0.517191i	$-0.173022\pi$
$264$	−369.599	−0.0861638
$265$	0	0
$266$	76.1144	0.0175446
$267$	3225.76i	0.739376i
$268$	− 994.379i	− 0.226647i
$269$	−3955.14	−0.896465	−0.448232	−	0.893917i	$-0.647946\pi$
$269$	−3955.14	−0.896465	−0.448232	+	0.893917i	$0.647946\pi$
$270$	0	0
$271$	6656.96	1.49218	0.746091	−	0.665844i	$-0.231928\pi$
$271$	6656.96	1.49218	0.746091	+	0.665844i	$0.231928\pi$
$272$	− 319.554i	− 0.0712345i
$273$	− 74.2828i	− 0.0164681i
$274$	32.5169	0.00716941
$275$	0	0
$276$	823.849	0.179673
$277$	− 8523.79i	− 1.84890i	−0.381304	−	0.924450i	$-0.624525\pi$
$277$	− 8523.79i	− 1.84890i	0.381304	−	0.924450i	$-0.375475\pi$
$278$	300.745i	0.0648831i
$279$	−1678.69	−0.360218
$280$	0	0
$281$	−4210.99	−0.893973	−0.446986	−	0.894541i	$-0.647503\pi$
$281$	−4210.99	−0.893973	−0.446986	+	0.894541i	$0.647503\pi$
$282$	509.256i	0.107538i
$283$	3529.66i	0.741400i	0.928753	+	0.370700i	$0.120882\pi$
$283$	3529.66i	0.741400i	−0.928753	+	0.370700i	$0.879118\pi$
$284$	−7787.88	−1.62720
$285$	0	0
$286$	173.990	0.0359729
$287$	− 308.076i	− 0.0633629i
$288$	− 1143.16i	− 0.233894i
$289$	4874.80	0.992225
$290$	0	0
$291$	−2109.71	−0.424995
$292$	− 1197.09i	− 0.239913i
$293$	− 7110.96i	− 1.41784i	−0.705290	−	0.708919i	$-0.749183\pi$
$293$	− 7110.96i	− 1.41784i	0.705290	−	0.708919i	$-0.250817\pi$
$294$	−741.886	−0.147169
$295$	0	0
$296$	−3993.70	−0.784220
$297$	− 297.000i	− 0.0580259i
$298$	− 490.950i	− 0.0954361i
$299$	−802.828	−0.155280
$300$	0	0
$301$	176.480	0.0337945
$302$	445.851i	0.0849531i
$303$	− 2570.66i	− 0.487394i
$304$	4800.25	0.905636
$305$	0	0
$306$	−40.2535	−0.00752007
$307$	− 9474.72i	− 1.76140i	−0.473671	−	0.880702i	$-0.657071\pi$
$307$	− 9474.72i	− 1.76140i	0.473671	−	0.880702i	$-0.342929\pi$
$308$	− 93.1672i	− 0.0172360i
$309$	2417.97	0.445156
$310$	0	0
$311$	−5210.73	−0.950075	−0.475038	−	0.879965i	$-0.657566\pi$
$311$	−5210.73	−0.950075	−0.475038	+	0.879965i	$0.657566\pi$
$312$	734.378i	0.133256i
$313$	1944.27i	0.351108i	0.984470	+	0.175554i	$0.0561717\pi$
$313$	1944.27i	0.351108i	−0.984470	+	0.175554i	$0.943828\pi$
$314$	−446.224	−0.0801970
$315$	0	0
$316$	5692.20	1.01333
$317$	6852.73i	1.21416i	0.794642	+	0.607079i	$0.207659\pi$
$317$	6852.73i	1.21416i	−0.794642	+	0.607079i	$0.792341\pi$
$318$	− 423.395i	− 0.0746629i
$319$	−782.512	−0.137343
$320$	0	0
$321$	−4825.64	−0.839068
$322$	− 30.1145i	− 0.00521185i
$323$	− 573.777i	− 0.0988415i
$324$	605.579	0.103837
$325$	0	0
$326$	−1469.92	−0.249727
$327$	2777.17i	0.469658i
$328$	3045.72i	0.512718i
$329$	−265.736	−0.0445304
$330$	0	0
$331$	−2019.85	−0.335411	−0.167706	−	0.985837i	$-0.553636\pi$
$331$	−2019.85	−0.335411	−0.167706	+	0.985837i	$0.553636\pi$
$332$	386.651i	0.0639163i
$333$	− 3209.23i	− 0.528123i
$334$	−1530.52	−0.250737
$335$	0	0
$336$	175.727	0.0285318
$337$	− 11738.7i	− 1.89748i	−0.316060	−	0.948739i	$-0.602360\pi$
$337$	− 11738.7i	− 1.89748i	0.316060	−	0.948739i	$-0.397640\pi$
$338$	1244.23i	0.200228i
$339$	−3644.44	−0.583890
$340$	0	0
$341$	2051.74	0.325829
$342$	− 604.679i	− 0.0956061i
$343$	− 775.704i	− 0.122111i
$344$	−1744.73	−0.273458
$345$	0	0
$346$	11.2241	0.00174396
$347$	− 12692.2i	− 1.96355i	−0.190036	−	0.981777i	$-0.560860\pi$
$347$	− 12692.2i	− 1.96355i	0.190036	−	0.981777i	$-0.439140\pi$
$348$	− 1595.53i	− 0.245774i
$349$	−12026.8	−1.84465	−0.922324	−	0.386417i	$-0.873713\pi$
$349$	−12026.8	−1.84465	−0.922324	+	0.386417i	$0.873713\pi$
$350$	0	0
$351$	−590.127	−0.0897398
$352$	1397.20i	0.211565i
$353$	− 8948.39i	− 1.34922i	−0.738174	−	0.674610i	$-0.764312\pi$
$353$	− 8948.39i	− 1.34922i	0.738174	−	0.674610i	$-0.235688\pi$
$354$	−989.850	−0.148616
$355$	0	0
$356$	8038.90	1.19680
$357$	− 21.0048i	− 0.00311398i
$358$	− 353.152i	− 0.0521360i
$359$	−3906.52	−0.574313	−0.287156	−	0.957884i	$-0.592710\pi$
$359$	−3906.52	−0.574313	−0.287156	+	0.957884i	$0.592710\pi$
$360$	0	0
$361$	1760.14	0.256617
$362$	903.673i	0.131204i
$363$	363.000i	0.0524864i
$364$	−185.120	−0.0266563
$365$	0	0
$366$	958.040	0.136824
$367$	− 4225.04i	− 0.600941i	−0.953791	−	0.300471i	$-0.902856\pi$
$367$	− 4225.04i	− 0.600941i	0.953791	−	0.300471i	$-0.0971438\pi$
$368$	− 1899.21i	− 0.269030i
$369$	−2447.46	−0.345283
$370$	0	0
$371$	220.932	0.0309171
$372$	4183.46i	0.583071i
$373$	4223.50i	0.586285i	0.956069	+	0.293143i	$0.0947011\pi$
$373$	4223.50i	0.586285i	−0.956069	+	0.293143i	$0.905299\pi$
$374$	49.1988	0.00680216
$375$	0	0
$376$	2627.13	0.360330
$377$	1554.82i	0.212407i
$378$	− 22.1360i	− 0.00301204i
$379$	748.009	0.101379	0.0506895	−	0.998714i	$-0.483858\pi$
$379$	748.009	0.101379	0.0506895	+	0.998714i	$0.483858\pi$
$380$	0	0
$381$	3386.87	0.455420
$382$	− 1861.03i	− 0.249264i
$383$	4668.07i	0.622786i	0.950281	+	0.311393i	$0.100796\pi$
$383$	4668.07i	0.622786i	−0.950281	+	0.311393i	$0.899204\pi$
$384$	−3746.90	−0.497938
$385$	0	0
$386$	−884.203	−0.116593
$387$	− 1402.02i	− 0.184157i
$388$	5257.61i	0.687924i
$389$	−6544.12	−0.852957	−0.426478	−	0.904498i	$-0.640246\pi$
$389$	−6544.12	−0.852957	−0.426478	+	0.904498i	$0.640246\pi$
$390$	0	0
$391$	−227.014	−0.0293621
$392$	3827.21i	0.493121i
$393$	− 4686.95i	− 0.601591i
$394$	−1033.66	−0.132170
$395$	0	0
$396$	−740.152	−0.0939243
$397$	− 6439.84i	− 0.814121i	−0.913401	−	0.407061i	$-0.866554\pi$
$397$	− 6439.84i	− 0.814121i	0.913401	−	0.407061i	$-0.133446\pi$
$398$	590.648i	0.0743882i
$399$	315.528	0.0395894
$400$	0	0
$401$	7727.03	0.962268	0.481134	−	0.876647i	$-0.340225\pi$
$401$	7727.03	0.962268	0.481134	+	0.876647i	$0.340225\pi$
$402$	288.761i	0.0358260i
$403$	− 4076.72i	− 0.503911i
$404$	−6406.32	−0.788927
$405$	0	0
$406$	−58.3222	−0.00712926
$407$	3922.40i	0.477705i
$408$	207.658i	0.0251976i
$409$	2539.97	0.307074	0.153537	−	0.988143i	$-0.450934\pi$
$409$	2539.97	0.307074	0.153537	+	0.988143i	$0.450934\pi$
$410$	0	0
$411$	134.797	0.0161777
$412$	− 6025.80i	− 0.720558i
$413$	− 516.515i	− 0.0615401i
$414$	−239.240	−0.0284009
$415$	0	0
$416$	2776.17	0.327195
$417$	1246.72i	0.146408i
$418$	739.052i	0.0864789i
$419$	−1170.17	−0.136436	−0.0682181	−	0.997670i	$-0.521731\pi$
$419$	−1170.17	−0.136436	−0.0682181	+	0.997670i	$0.521731\pi$
$420$	0	0
$421$	4009.01	0.464103	0.232052	−	0.972703i	$-0.425456\pi$
$421$	4009.01	0.464103	0.232052	+	0.972703i	$0.425456\pi$
$422$	− 653.749i	− 0.0754123i
$423$	2111.09i	0.242659i
$424$	−2184.19	−0.250174
$425$	0	0
$426$	2261.55	0.257212
$427$	499.916i	0.0566573i
$428$	12025.9i	1.35817i
$429$	721.267	0.0811727
$430$	0	0
$431$	−2676.70	−0.299146	−0.149573	−	0.988751i	$-0.547790\pi$
$431$	−2676.70	−0.299146	−0.149573	+	0.988751i	$0.547790\pi$
$432$	− 1396.03i	− 0.155479i
$433$	5655.83i	0.627718i	0.949470	+	0.313859i	$0.101622\pi$
$433$	5655.83i	0.627718i	−0.949470	+	0.313859i	$0.898378\pi$
$434$	152.920	0.0169133
$435$	0	0
$436$	6920.97	0.760217
$437$	− 3410.14i	− 0.373294i
$438$	347.628i	0.0379230i
$439$	6305.94	0.685572	0.342786	−	0.939414i	$-0.388629\pi$
$439$	6305.94	0.685572	0.342786	+	0.939414i	$0.388629\pi$
$440$	0	0
$441$	−3075.45	−0.332086
$442$	− 97.7560i	− 0.0105199i
$443$	− 6772.22i	− 0.726315i	−0.931728	−	0.363158i	$-0.881699\pi$
$443$	− 6772.22i	− 0.726315i	0.931728	−	0.363158i	$-0.118301\pi$
$444$	−7997.71	−0.854852
$445$	0	0
$446$	−386.424	−0.0410262
$447$	− 2035.20i	− 0.215351i
$448$	− 364.470i	− 0.0384365i
$449$	5049.16	0.530701	0.265351	−	0.964152i	$-0.414512\pi$
$449$	5049.16	0.530701	0.265351	+	0.964152i	$0.414512\pi$
$450$	0	0
$451$	2991.34	0.312321
$452$	9082.27i	0.945120i
$453$	1848.25i	0.191696i
$454$	504.070	0.0521083
$455$	0	0
$456$	−3119.39	−0.320348
$457$	− 3866.14i	− 0.395733i	−0.980229	−	0.197867i	$-0.936599\pi$
$457$	− 3866.14i	− 0.395733i	0.980229	−	0.197867i	$-0.0634013\pi$
$458$	576.629i	0.0588299i
$459$	−166.869	−0.0169690
$460$	0	0
$461$	−9108.85	−0.920263	−0.460132	−	0.887851i	$-0.652198\pi$
$461$	−9108.85	−0.920263	−0.460132	+	0.887851i	$0.652198\pi$
$462$	27.0551i	0.00272450i
$463$	− 17421.8i	− 1.74873i	−0.485273	−	0.874363i	$-0.661280\pi$
$463$	− 17421.8i	− 1.74873i	0.485273	−	0.874363i	$-0.338720\pi$
$464$	−3678.16	−0.368005
$465$	0	0
$466$	4093.95	0.406971
$467$	− 2602.36i	− 0.257864i	−0.991653	−	0.128932i	$-0.958845\pi$
$467$	− 2602.36i	− 0.257864i	0.991653	−	0.128932i	$-0.0411550\pi$
$468$	1470.65i	0.145258i
$469$	−150.679	−0.0148352
$470$	0	0
$471$	−1849.80	−0.180964
$472$	5106.40i	0.497968i
$473$	1713.58i	0.166576i
$474$	−1652.97	−0.160176
$475$	0	0
$476$	−52.3458	−0.00504047
$477$	− 1755.16i	− 0.168476i
$478$	1464.06i	0.140093i
$479$	−10751.8	−1.02560	−0.512801	−	0.858508i	$-0.671392\pi$
$479$	−10751.8	−1.02560	−0.512801	+	0.858508i	$0.671392\pi$
$480$	0	0
$481$	7793.65	0.738794
$482$	− 2542.93i	− 0.240306i
$483$	− 124.838i	− 0.0117605i
$484$	904.630	0.0849577
$485$	0	0
$486$	−175.856	−0.0164135
$487$	2115.37i	0.196831i	0.995145	+	0.0984154i	$0.0313774\pi$
$487$	2115.37i	0.196831i	−0.995145	+	0.0984154i	$0.968623\pi$
$488$	− 4942.30i	− 0.458458i
$489$	−6093.46	−0.563509
$490$	0	0
$491$	4634.30	0.425953	0.212977	−	0.977057i	$-0.431684\pi$
$491$	4634.30	0.425953	0.212977	+	0.977057i	$0.431684\pi$
$492$	6099.29i	0.558897i
$493$	439.653i	0.0401642i
$494$	1468.47	0.133744
$495$	0	0
$496$	9644.09	0.873049
$497$	1180.10i	0.106509i
$498$	− 112.281i	− 0.0101032i
$499$	−2495.43	−0.223870	−0.111935	−	0.993716i	$-0.535705\pi$
$499$	−2495.43	−0.223870	−0.111935	+	0.993716i	$0.535705\pi$
$500$	0	0
$501$	−6344.69	−0.565788
$502$	1313.00i	0.116737i
$503$	18521.0i	1.64177i	0.571092	+	0.820886i	$0.306520\pi$
$503$	18521.0i	1.64177i	−0.571092	+	0.820886i	$0.693480\pi$
$504$	−114.194	−0.0100925
$505$	0	0
$506$	292.404	0.0256896
$507$	5157.87i	0.451813i
$508$	− 8440.41i	− 0.737170i
$509$	6102.40	0.531403	0.265702	−	0.964055i	$-0.414396\pi$
$509$	6102.40	0.531403	0.265702	+	0.964055i	$0.414396\pi$
$510$	0	0
$511$	−181.396	−0.0157035
$512$	11200.2i	0.966765i
$513$	− 2506.66i	− 0.215735i
$514$	1046.06	0.0897661
$515$	0	0
$516$	−3493.96	−0.298087
$517$	− 2580.23i	− 0.219494i
$518$	292.344i	0.0247970i
$519$	46.5288	0.00393523
$520$	0	0
$521$	−2813.12	−0.236555	−0.118277	−	0.992981i	$-0.537737\pi$
$521$	−2813.12	−0.236555	−0.118277	+	0.992981i	$0.537737\pi$
$522$	463.331i	0.0388495i
$523$	1563.27i	0.130702i	0.997862	+	0.0653509i	$0.0208167\pi$
$523$	1563.27i	0.130702i	−0.997862	+	0.0653509i	$0.979183\pi$
$524$	−11680.3	−0.973772
$525$	0	0
$526$	3192.75	0.264659
$527$	− 1152.76i	− 0.0952850i
$528$	1706.26i	0.140636i
$529$	10817.8	0.889109
$530$	0	0
$531$	−4103.37	−0.335350
$532$	− 786.326i	− 0.0640818i
$533$	− 5943.67i	− 0.483019i
$534$	−2334.44	−0.189178
$535$	0	0
$536$	1489.65	0.120043
$537$	− 1463.97i	− 0.117645i
$538$	− 2862.28i	− 0.229371i
$539$	3758.88	0.300383
$540$	0	0
$541$	−5959.61	−0.473611	−0.236806	−	0.971557i	$-0.576100\pi$
$541$	−5959.61	−0.473611	−0.236806	+	0.971557i	$0.576100\pi$
$542$	4817.55i	0.381792i
$543$	3746.13i	0.296062i
$544$	785.012	0.0618697
$545$	0	0
$546$	53.7574	0.00421356
$547$	− 18084.2i	− 1.41357i	−0.707426	−	0.706787i	$-0.750144\pi$
$547$	− 18084.2i	− 1.41357i	0.707426	−	0.706787i	$-0.249856\pi$
$548$	− 335.927i	− 0.0261863i
$549$	3971.50	0.308742
$550$	0	0
$551$	−6604.35	−0.510626
$552$	1234.18i	0.0951634i
$553$	− 862.540i	− 0.0663272i
$554$	6168.55	0.473062
$555$	0	0
$556$	3106.95	0.236985
$557$	− 18129.5i	− 1.37912i	−0.724227	−	0.689561i	$-0.757803\pi$
$557$	− 18129.5i	− 1.37912i	0.724227	−	0.689561i	$-0.242197\pi$
$558$	− 1214.85i	− 0.0921659i
$559$	3404.81	0.257618
$560$	0	0
$561$	203.951	0.0153490
$562$	− 3047.43i	− 0.228733i
$563$	− 17910.2i	− 1.34072i	−0.742035	−	0.670361i	$-0.766139\pi$
$563$	− 17910.2i	− 1.34072i	0.742035	−	0.670361i	$-0.233861\pi$
$564$	5261.04	0.392784
$565$	0	0
$566$	−2554.36	−0.189696
$567$	− 91.7635i	− 0.00679666i
$568$	− 11666.8i	− 0.861843i
$569$	−24176.6	−1.78126	−0.890629	−	0.454731i	$-0.849735\pi$
$569$	−24176.6	−1.78126	−0.890629	+	0.454731i	$0.849735\pi$
$570$	0	0
$571$	−13167.8	−0.965073	−0.482537	−	0.875876i	$-0.660285\pi$
$571$	−13167.8	−0.965073	−0.482537	+	0.875876i	$0.660285\pi$
$572$	− 1797.46i	− 0.131391i
$573$	− 7714.81i	− 0.562462i
$574$	222.950	0.0162121
$575$	0	0
$576$	−2895.47	−0.209452
$577$	722.513i	0.0521293i	0.999660	+	0.0260646i	$0.00829757\pi$
$577$	722.513i	0.0521293i	−0.999660	+	0.0260646i	$0.991702\pi$
$578$	3527.83i	0.253872i
$579$	−3665.41	−0.263091
$580$	0	0
$581$	58.5893	0.00418364
$582$	− 1526.77i	− 0.108740i
$583$	2145.19i	0.152393i
$584$	1793.33	0.127069
$585$	0	0
$586$	5146.10	0.362770
$587$	17038.8i	1.19807i	0.800723	+	0.599034i	$0.204449\pi$
$587$	17038.8i	1.19807i	−0.800723	+	0.599034i	$0.795551\pi$
$588$	7664.31i	0.537535i
$589$	17316.5	1.21140
$590$	0	0
$591$	−4284.97	−0.298241
$592$	18437.0i	1.28000i
$593$	6206.74i	0.429815i	0.976634	+	0.214908i	$0.0689450\pi$
$593$	6206.74i	0.429815i	−0.976634	+	0.214908i	$0.931055\pi$
$594$	214.935	0.0148466
$595$	0	0
$596$	−5071.92	−0.348580
$597$	2448.50i	0.167856i
$598$	− 580.996i	− 0.0397302i
$599$	23050.7	1.57233	0.786164	−	0.618018i	$-0.212064\pi$
$599$	23050.7	1.57233	0.786164	+	0.618018i	$0.212064\pi$
$600$	0	0
$601$	−6323.69	−0.429199	−0.214600	−	0.976702i	$-0.568845\pi$
$601$	−6323.69	−0.429199	−0.214600	+	0.976702i	$0.568845\pi$
$602$	127.716i	0.00864673i
$603$	1197.04i	0.0808412i
$604$	4606.01	0.310291
$605$	0	0
$606$	1860.35	0.124705
$607$	− 4625.14i	− 0.309273i	−0.987971	−	0.154636i	$-0.950579\pi$
$607$	− 4625.14i	− 0.309273i	0.987971	−	0.154636i	$-0.0494206\pi$
$608$	11792.3i	0.786577i
$609$	−241.771	−0.0160871
$610$	0	0
$611$	−5126.81	−0.339457
$612$	415.853i	0.0274671i
$613$	− 28075.3i	− 1.84984i	−0.380164	−	0.924919i	$-0.624132\pi$
$613$	− 28075.3i	− 1.84984i	0.380164	−	0.924919i	$-0.375868\pi$
$614$	6856.72	0.450676
$615$	0	0
$616$	139.571	0.00912900
$617$	7264.85i	0.474023i	0.971507	+	0.237011i	$0.0761678\pi$
$617$	7264.85i	0.474023i	−0.971507	+	0.237011i	$0.923832\pi$
$618$	1749.85i	0.113898i
$619$	30583.2	1.98585	0.992927	−	0.118728i	$-0.0378815\pi$
$619$	30583.2	1.98585	0.992927	+	0.118728i	$0.0378815\pi$
$620$	0	0
$621$	−991.755	−0.0640866
$622$	− 3770.93i	− 0.243088i
$623$	− 1218.14i	− 0.0783365i
$624$	3390.28	0.217500
$625$	0	0
$626$	−1407.04	−0.0898351
$627$	3063.70i	0.195139i
$628$	4609.86i	0.292920i
$629$	2203.79	0.139699
$630$	0	0
$631$	−16679.8	−1.05232	−0.526159	−	0.850386i	$-0.676368\pi$
$631$	−16679.8	−1.05232	−0.526159	+	0.850386i	$0.676368\pi$
$632$	8527.29i	0.536705i
$633$	− 2710.08i	− 0.170167i
$634$	−4959.23	−0.310656
$635$	0	0
$636$	−4374.02	−0.272706
$637$	− 7468.75i	− 0.464557i
$638$	− 566.293i	− 0.0351407i
$639$	9375.11	0.580397
$640$	0	0
$641$	4225.31	0.260358	0.130179	−	0.991490i	$-0.458445\pi$
$641$	4225.31	0.260358	0.130179	+	0.991490i	$0.458445\pi$
$642$	− 3492.25i	− 0.214685i
$643$	23703.5i	1.45377i	0.686758	+	0.726886i	$0.259033\pi$
$643$	23703.5i	1.45377i	−0.686758	+	0.726886i	$0.740967\pi$
$644$	−311.108	−0.0190363
$645$	0	0
$646$	415.234	0.0252898
$647$	10045.1i	0.610374i	0.952292	+	0.305187i	$0.0987190\pi$
$647$	10045.1i	0.610374i	−0.952292	+	0.305187i	$0.901281\pi$
$648$	907.197i	0.0549970i
$649$	5015.23	0.303336
$650$	0	0
$651$	633.921	0.0381649
$652$	15185.5i	0.912130i
$653$	− 32314.2i	− 1.93652i	−0.249936	−	0.968262i	$-0.580410\pi$
$653$	− 32314.2i	− 1.93652i	0.249936	−	0.968262i	$-0.419590\pi$
$654$	−2009.80	−0.120167
$655$	0	0
$656$	14060.6	0.836853
$657$	1441.07i	0.0855731i
$658$	− 192.309i	− 0.0113936i
$659$	12032.4	0.711253	0.355627	−	0.934628i	$-0.384267\pi$
$659$	12032.4	0.711253	0.355627	+	0.934628i	$0.384267\pi$
$660$	0	0
$661$	12864.5	0.756988	0.378494	−	0.925604i	$-0.376442\pi$
$661$	12864.5	0.756988	0.378494	+	0.925604i	$0.376442\pi$
$662$	− 1461.74i	− 0.0858189i
$663$	− 405.242i	− 0.0237380i
$664$	−579.229	−0.0338531
$665$	0	0
$666$	2322.48	0.135126
$667$	2613.00i	0.151688i
$668$	15811.5i	0.915819i
$669$	−1601.90	−0.0925754
$670$	0	0
$671$	−4854.06	−0.279268
$672$	431.689i	0.0247809i
$673$	12007.5i	0.687751i	0.939015	+	0.343876i	$0.111740\pi$
$673$	12007.5i	0.687751i	−0.939015	+	0.343876i	$0.888260\pi$
$674$	8495.16	0.485492
$675$	0	0
$676$	12853.9	0.731332
$677$	10732.5i	0.609284i	0.952467	+	0.304642i	$0.0985368\pi$
$677$	10732.5i	0.609284i	−0.952467	+	0.304642i	$0.901463\pi$
$678$	− 2637.43i	− 0.149395i
$679$	796.687	0.0450280
$680$	0	0
$681$	2089.59	0.117582
$682$	1484.81i	0.0833672i
$683$	14126.0i	0.791383i	0.918383	+	0.395692i	$0.129495\pi$
$683$	14126.0i	0.791383i	−0.918383	+	0.395692i	$0.870505\pi$
$684$	−6246.83	−0.349201
$685$	0	0
$686$	561.366	0.0312435
$687$	2390.38i	0.132749i
$688$	8054.59i	0.446335i
$689$	4262.42	0.235682
$690$	0	0
$691$	−18467.8	−1.01671	−0.508357	−	0.861147i	$-0.669747\pi$
$691$	−18467.8	−1.01671	−0.508357	+	0.861147i	$0.669747\pi$
$692$	− 115.954i	− 0.00636981i
$693$	112.155i	0.00614781i
$694$	9185.17	0.502398
$695$	0	0
$696$	2390.21	0.130173
$697$	− 1680.68i	− 0.0913345i
$698$	− 8703.66i	− 0.471975i
$699$	16971.2	0.918327
$700$	0	0
$701$	−24876.0	−1.34030	−0.670152	−	0.742224i	$-0.733771\pi$
$701$	−24876.0	−1.34030	−0.670152	+	0.742224i	$0.733771\pi$
$702$	− 427.067i	− 0.0229610i
$703$	33104.8i	1.77606i
$704$	3538.91	0.189457
$705$	0	0
$706$	6475.83	0.345214
$707$	970.752i	0.0516391i
$708$	10226.0i	0.542819i
$709$	−21271.1	−1.12673	−0.563367	−	0.826207i	$-0.690494\pi$
$709$	−21271.1	−1.12673	−0.563367	+	0.826207i	$0.690494\pi$
$710$	0	0
$711$	−6852.31	−0.361437
$712$	12042.8i	0.633881i
$713$	− 6851.25i	− 0.359862i
$714$	15.2008	0.000796747 0
$715$	0	0
$716$	−3648.36	−0.190427
$717$	6069.17i	0.316119i
$718$	− 2827.09i	− 0.146945i
$719$	−19213.9	−0.996603	−0.498302	−	0.867004i	$-0.666043\pi$
$719$	−19213.9	−0.996603	−0.498302	+	0.867004i	$0.666043\pi$
$720$	0	0
$721$	−913.091	−0.0471641
$722$	1273.79i	0.0656585i
$723$	− 10541.6i	− 0.542249i
$724$	9335.70	0.479224
$725$	0	0
$726$	−262.698	−0.0134293
$727$	27454.7i	1.40060i	0.713846	+	0.700302i	$0.246951\pi$
$727$	27454.7i	1.40060i	−0.713846	+	0.700302i	$0.753049\pi$
$728$	− 277.322i	− 0.0141184i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	962.770	0.0487132
$732$	− 9897.35i	− 0.499750i
$733$	− 1182.16i	− 0.0595691i	−0.999556	−	0.0297846i	$-0.990518\pi$
$733$	− 1182.16i	− 0.0595691i	0.999556	−	0.0297846i	$-0.00948213\pi$
$734$	3057.61	0.153758
$735$	0	0
$736$	4665.58	0.233662
$737$	− 1463.05i	− 0.0731237i
$738$	− 1771.19i	− 0.0883447i
$739$	−25688.2	−1.27869	−0.639347	−	0.768918i	$-0.720795\pi$
$739$	−25688.2	−1.27869	−0.639347	+	0.768918i	$0.720795\pi$
$740$	0	0
$741$	6087.45	0.301792
$742$	159.886i	0.00791049i
$743$	32431.2i	1.60132i	0.599116	+	0.800662i	$0.295519\pi$
$743$	32431.2i	1.60132i	−0.599116	+	0.800662i	$0.704481\pi$
$744$	−6267.10	−0.308822
$745$	0	0
$746$	−3056.49	−0.150008
$747$	− 465.453i	− 0.0227979i
$748$	− 508.264i	− 0.0248449i
$749$	1822.29	0.0888988
$750$	0	0
$751$	−32093.5	−1.55940	−0.779700	−	0.626154i	$-0.784628\pi$
$751$	−32093.5	−1.55940	−0.779700	+	0.626154i	$0.784628\pi$
$752$	− 12128.2i	− 0.588127i
$753$	5442.96i	0.263416i
$754$	−1125.20	−0.0543467
$755$	0	0
$756$	−228.683	−0.0110015
$757$	1293.36i	0.0620977i	0.999518	+	0.0310488i	$0.00988474\pi$
$757$	1293.36i	0.0620977i	−0.999518	+	0.0310488i	$0.990115\pi$
$758$	541.323i	0.0259390i
$759$	1212.15	0.0579685
$760$	0	0
$761$	29013.1	1.38203	0.691014	−	0.722841i	$-0.257164\pi$
$761$	29013.1	1.38203	0.691014	+	0.722841i	$0.257164\pi$
$762$	2451.03i	0.116524i
$763$	− 1048.74i	− 0.0497599i
$764$	−19226.0	−0.910436
$765$	0	0
$766$	−3378.22	−0.159347
$767$	− 9965.06i	− 0.469123i
$768$	5009.67i	0.235379i
$769$	31304.1	1.46795	0.733976	−	0.679176i	$-0.237663\pi$
$769$	31304.1	1.46795	0.733976	+	0.679176i	$0.237663\pi$
$770$	0	0
$771$	4336.39	0.202557
$772$	9134.55i	0.425855i
$773$	41913.5i	1.95023i	0.221710	+	0.975113i	$0.428836\pi$
$773$	41913.5i	1.95023i	−0.221710	+	0.975113i	$0.571164\pi$
$774$	1014.62	0.0471186
$775$	0	0
$776$	−7876.24	−0.364357
$777$	1211.90i	0.0559543i
$778$	− 4735.89i	− 0.218239i
$779$	25246.7	1.16118
$780$	0	0
$781$	−11458.5	−0.524989
$782$	− 164.287i	− 0.00751263i
$783$	1920.71i	0.0876637i
$784$	17668.4	0.804867
$785$	0	0
$786$	3391.88	0.153924
$787$	6648.00i	0.301113i	0.988601	+	0.150556i	$0.0481065\pi$
$787$	6648.00i	0.301113i	−0.988601	+	0.150556i	$0.951894\pi$
$788$	10678.6i	0.482751i
$789$	13235.4	0.597201
$790$	0	0
$791$	1376.24	0.0618628
$792$	− 1108.80i	− 0.0497467i
$793$	9644.82i	0.431901i
$794$	4660.42	0.208302
$795$	0	0
$796$	6101.88	0.271703
$797$	− 14798.4i	− 0.657701i	−0.944382	−	0.328850i	$-0.893339\pi$
$797$	− 14798.4i	− 0.657701i	0.944382	−	0.328850i	$-0.106661\pi$
$798$	228.343i	0.0101294i
$799$	−1449.69	−0.0641884
$800$	0	0
$801$	−9677.28	−0.426879
$802$	5591.94i	0.246207i
$803$	− 1761.31i	− 0.0774038i
$804$	2983.14	0.130855
$805$	0	0
$806$	2950.27	0.128931
$807$	− 11865.4i	− 0.517574i
$808$	− 9597.09i	− 0.417852i
$809$	−4500.04	−0.195566	−0.0977831	−	0.995208i	$-0.531175\pi$
$809$	−4500.04	−0.195566	−0.0977831	+	0.995208i	$0.531175\pi$
$810$	0	0
$811$	−1909.49	−0.0826773	−0.0413386	−	0.999145i	$-0.513162\pi$
$811$	−1909.49	−0.0826773	−0.0413386	+	0.999145i	$0.513162\pi$
$812$	602.517i	0.0260396i
$813$	19970.9i	0.861512i
$814$	−2838.58	−0.122226
$815$	0	0
$816$	958.661	0.0411273
$817$	14462.5i	0.619313i
$818$	1838.14i	0.0785685i
$819$	222.848	0.00950788
$820$	0	0
$821$	10928.6	0.464569	0.232285	−	0.972648i	$-0.425380\pi$
$821$	10928.6	0.464569	0.232285	+	0.972648i	$0.425380\pi$
$822$	97.5507i	0.00413926i
$823$	29278.4i	1.24007i	0.784573	+	0.620036i	$0.212882\pi$
$823$	29278.4i	1.24007i	−0.784573	+	0.620036i	$0.787118\pi$
$824$	9027.05	0.381641
$825$	0	0
$826$	373.795	0.0157457
$827$	878.905i	0.0369559i	0.999829	+	0.0184779i	$0.00588205\pi$
$827$	878.905i	0.0369559i	−0.999829	+	0.0184779i	$0.994118\pi$
$828$	2471.55i	0.103734i
$829$	−16130.1	−0.675779	−0.337890	−	0.941186i	$-0.609713\pi$
$829$	−16130.1	−0.675779	−0.337890	+	0.941186i	$0.609713\pi$
$830$	0	0
$831$	25571.4	1.06746
$832$	− 7031.67i	− 0.293004i
$833$	− 2111.92i	− 0.0878436i
$834$	−902.235	−0.0374603
$835$	0	0
$836$	7635.02	0.315864
$837$	− 5036.08i	− 0.207972i
$838$	− 846.838i	− 0.0349088i
$839$	39531.8	1.62669	0.813343	−	0.581784i	$-0.197645\pi$
$839$	39531.8	1.62669	0.813343	+	0.581784i	$0.197645\pi$
$840$	0	0
$841$	−19328.5	−0.792507
$842$	2901.27i	0.118746i
$843$	− 12633.0i	− 0.516135i
$844$	−6753.77	−0.275443
$845$	0	0
$846$	−1527.77	−0.0620872
$847$	− 137.079i	− 0.00556090i
$848$	10083.4i	0.408331i
$849$	−10589.0	−0.428048
$850$	0	0
$851$	13097.8	0.527600
$852$	− 23363.7i	− 0.939467i
$853$	− 5373.68i	− 0.215699i	−0.994167	−	0.107849i	$-0.965604\pi$
$853$	− 5373.68i	− 0.215699i	0.994167	−	0.107849i	$-0.0343965\pi$
$854$	−361.782	−0.0144964
$855$	0	0
$856$	−18015.7	−0.719349
$857$	45378.4i	1.80875i	0.426740	+	0.904374i	$0.359662\pi$
$857$	45378.4i	1.80875i	−0.426740	+	0.904374i	$0.640338\pi$
$858$	521.971i	0.0207690i
$859$	−35336.8	−1.40358	−0.701791	−	0.712383i	$-0.747616\pi$
$859$	−35336.8	−1.40358	−0.701791	+	0.712383i	$0.747616\pi$
$860$	0	0
$861$	924.228	0.0365826
$862$	− 1937.09i	− 0.0765401i
$863$	− 24595.3i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$863$	− 24595.3i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$864$	3429.48	0.135039
$865$	0	0
$866$	−4093.05	−0.160609
$867$	14624.4i	0.572862i
$868$	− 1579.79i	− 0.0617760i
$869$	8375.04	0.326932
$870$	0	0
$871$	−2907.02	−0.113089
$872$	10368.1i	0.402646i
$873$	− 6329.14i	− 0.245371i
$874$	2467.87	0.0955115
$875$	0	0
$876$	3591.28	0.138514
$877$	8251.06i	0.317695i	0.987303	+	0.158848i	$0.0507778\pi$
$877$	8251.06i	0.317695i	−0.987303	+	0.158848i	$0.949222\pi$
$878$	4563.52i	0.175412i
$879$	21332.9	0.818589
$880$	0	0
$881$	−3528.36	−0.134930	−0.0674651	−	0.997722i	$-0.521491\pi$
$881$	−3528.36	−0.134930	−0.0674651	+	0.997722i	$0.521491\pi$
$882$	− 2225.66i	− 0.0849681i
$883$	35459.1i	1.35141i	0.737174	+	0.675703i	$0.236160\pi$
$883$	35459.1i	1.35141i	−0.737174	+	0.675703i	$0.763840\pi$
$884$	−1009.90	−0.0384238
$885$	0	0
$886$	4900.96	0.185836
$887$	− 32544.8i	− 1.23196i	−0.787763	−	0.615979i	$-0.788761\pi$
$887$	− 32544.8i	− 1.23196i	0.787763	−	0.615979i	$-0.211239\pi$
$888$	− 11981.1i	− 0.452770i
$889$	−1278.98	−0.0482514
$890$	0	0
$891$	891.000	0.0335013
$892$	3992.08i	0.149848i
$893$	− 21777.0i	− 0.816056i
$894$	1472.85	0.0551000
$895$	0	0
$896$	1414.93	0.0527562
$897$	− 2408.49i	− 0.0896510i
$898$	3654.01i	0.135786i
$899$	−13268.7	−0.492253
$900$	0	0
$901$	1205.27	0.0445654
$902$	2164.79i	0.0799108i
$903$	529.441i	0.0195113i
$904$	−13605.9	−0.500579
$905$	0	0
$906$	−1337.55	−0.0490477
$907$	− 20383.4i	− 0.746219i	−0.927787	−	0.373110i	$-0.878291\pi$
$907$	− 20383.4i	− 0.746219i	0.927787	−	0.373110i	$-0.121709\pi$
$908$	− 5207.46i	− 0.190326i
$909$	7711.98	0.281397
$910$	0	0
$911$	7783.29	0.283065	0.141532	−	0.989934i	$-0.454797\pi$
$911$	7783.29	0.283065	0.141532	+	0.989934i	$0.454797\pi$
$912$	14400.8i	0.522869i
$913$	568.887i	0.0206215i
$914$	2797.87	0.101253
$915$	0	0
$916$	5957.05	0.214876
$917$	1769.92i	0.0637382i
$918$	− 120.761i	− 0.00434171i
$919$	−27294.9	−0.979734	−0.489867	−	0.871797i	$-0.662955\pi$
$919$	−27294.9	−0.979734	−0.489867	+	0.871797i	$0.662955\pi$
$920$	0	0
$921$	28424.2	1.01695
$922$	− 6591.95i	− 0.235460i
$923$	22767.5i	0.811920i
$924$	279.502	0.00995122
$925$	0	0
$926$	12607.9	0.447432
$927$	7253.90i	0.257011i
$928$	− 9035.73i	− 0.319625i
$929$	46468.7	1.64111	0.820554	−	0.571568i	$-0.193665\pi$
$929$	46468.7	1.64111	0.820554	+	0.571568i	$0.193665\pi$
$930$	0	0
$931$	31724.7	1.11679
$932$	− 42293.9i	− 1.48646i
$933$	− 15632.2i	− 0.548526i
$934$	1883.29	0.0659776
$935$	0	0
$936$	−2203.14	−0.0769356
$937$	− 6922.45i	− 0.241352i	−0.992692	−	0.120676i	$-0.961494\pi$
$937$	− 6922.45i	− 0.241352i	0.992692	−	0.120676i	$-0.0385062\pi$
$938$	− 109.044i	− 0.00379575i
$939$	−5832.82	−0.202712
$940$	0	0
$941$	−23772.1	−0.823536	−0.411768	−	0.911289i	$-0.635089\pi$
$941$	−23772.1	−0.823536	−0.411768	+	0.911289i	$0.635089\pi$
$942$	− 1338.67i	− 0.0463018i
$943$	− 9988.80i	− 0.344942i
$944$	23573.8	0.812779
$945$	0	0
$946$	−1240.09	−0.0426204
$947$	− 7177.76i	− 0.246300i	−0.992388	−	0.123150i	$-0.960700\pi$
$947$	− 7177.76i	− 0.246300i	0.992388	−	0.123150i	$-0.0392996\pi$
$948$	17076.6i	0.585044i
$949$	−3499.65	−0.119709
$950$	0	0
$951$	−20558.2	−0.700994
$952$	− 78.4175i	− 0.00266967i
$953$	− 27530.2i	− 0.935772i	−0.883789	−	0.467886i	$-0.845016\pi$
$953$	− 27530.2i	− 0.935772i	0.883789	−	0.467886i	$-0.154984\pi$
$954$	1270.18	0.0431066
$955$	0	0
$956$	15124.9	0.511690
$957$	− 2347.54i	− 0.0792948i
$958$	− 7780.94i	− 0.262412i
$959$	−50.9031	−0.00171402
$960$	0	0
$961$	4999.29	0.167812
$962$	5640.15i	0.189029i
$963$	− 14476.9i	− 0.484436i
$964$	−26270.6	−0.877718
$965$	0	0
$966$	90.3435	0.00300906
$967$	55407.1i	1.84258i	0.388881	+	0.921288i	$0.372862\pi$
$967$	55407.1i	1.84258i	−0.388881	+	0.921288i	$0.627138\pi$
$968$	1355.20i	0.0449976i
$969$	1721.33	0.0570662
$970$	0	0
$971$	−55100.7	−1.82108	−0.910539	−	0.413423i	$-0.864333\pi$
$971$	−55100.7	−1.82108	−0.910539	+	0.413423i	$0.864333\pi$
$972$	1816.74i	0.0599504i
$973$	− 470.797i	− 0.0155119i
$974$	−1530.86	−0.0503615
$975$	0	0
$976$	−22816.3	−0.748290
$977$	− 44473.9i	− 1.45634i	−0.685396	−	0.728170i	$-0.740371\pi$
$977$	− 44473.9i	− 1.45634i	0.685396	−	0.728170i	$-0.259629\pi$
$978$	− 4409.75i	− 0.144180i
$979$	11827.8	0.386127
$980$	0	0
$981$	−8331.52	−0.271157
$982$	3353.78i	0.108985i
$983$	− 28003.0i	− 0.908605i	−0.890848	−	0.454302i	$-0.849889\pi$
$983$	− 28003.0i	− 0.908605i	0.890848	−	0.454302i	$-0.150111\pi$
$984$	−9137.15	−0.296018
$985$	0	0
$986$	−318.170	−0.0102765
$987$	− 797.208i	− 0.0257096i
$988$	− 15170.5i	− 0.488499i
$989$	5722.06	0.183974
$990$	0	0
$991$	−40719.5	−1.30524	−0.652622	−	0.757684i	$-0.726331\pi$
$991$	−40719.5	−1.30524	−0.652622	+	0.757684i	$0.726331\pi$
$992$	23691.6i	0.758274i
$993$	− 6059.56i	− 0.193650i
$994$	−854.022	−0.0272514
$995$	0	0
$996$	−1159.95	−0.0369021
$997$	− 18161.1i	− 0.576898i	−0.957495	−	0.288449i	$-0.906860\pi$
$997$	− 18161.1i	− 0.576898i	0.957495	−	0.288449i	$-0.0931395\pi$
$998$	− 1805.91i	− 0.0572796i
$999$	9627.70	0.304912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.n.199.4		6
5.2	odd	4		825.4.a.o.1.2	✓	3
5.3	odd	4		825.4.a.q.1.2	yes	3
5.4	even	2	inner	825.4.c.n.199.3		6
15.2	even	4		2475.4.a.y.1.2		3
15.8	even	4		2475.4.a.v.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.2	✓	3	5.2	odd	4
825.4.a.q.1.2	yes	3	5.3	odd	4
825.4.c.n.199.3		6	5.4	even	2	inner
825.4.c.n.199.4		6	1.1	even	1	trivial
2475.4.a.v.1.2		3	15.8	even	4
2475.4.a.y.1.2		3	15.2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.723686i q^{2} +3.00000i q^{3} +7.47628 q^{4} -2.17106 q^{6} -1.13288i q^{7} +11.2000i q^{8} -9.00000 q^{9} +11.0000 q^{11} +22.4288i q^{12} -21.8566i q^{13} +0.819851 q^{14} +51.7050 q^{16} -6.18033i q^{17} -6.51317i q^{18} +92.8393 q^{19} +3.39865 q^{21} +7.96054i q^{22} -36.7317i q^{23} -33.5999 q^{24} +15.8173 q^{26} -27.0000i q^{27} -8.46975i q^{28} -71.1375 q^{29} +186.522 q^{31} +127.018i q^{32} +33.0000i q^{33} +4.47261 q^{34} -67.2865 q^{36} +356.581i q^{37} +67.1865i q^{38} +65.5697 q^{39} +271.940 q^{41} +2.45955i q^{42} +155.780i q^{43} +82.2391 q^{44} +26.5822 q^{46} -234.566i q^{47} +155.115i q^{48} +341.717 q^{49} +18.5410 q^{51} -163.406i q^{52} +195.018i q^{53} +19.5395 q^{54} +12.6882 q^{56} +278.518i q^{57} -51.4812i q^{58} +455.930 q^{59} -441.278 q^{61} +134.983i q^{62} +10.1959i q^{63} +321.719 q^{64} -23.8816 q^{66} -133.005i q^{67} -46.2058i q^{68} +110.195 q^{69} -1041.68 q^{71} -100.800i q^{72} -160.119i q^{73} -258.053 q^{74} +694.093 q^{76} -12.4617i q^{77} +47.4519i q^{78} +761.367 q^{79} +81.0000 q^{81} +196.799i q^{82} +51.7170i q^{83} +25.4092 q^{84} -112.736 q^{86} -213.412i q^{87} +123.200i q^{88} +1075.25 q^{89} -24.7609 q^{91} -274.616i q^{92} +559.565i q^{93} +169.752 q^{94} -381.054 q^{96} +703.238i q^{97} +247.295i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} + 232 q^{14} - 170 q^{16} + 338 q^{19} - 96 q^{21} - 18 q^{24} + 334 q^{26} + 554 q^{29} - 346 q^{31} + 292 q^{34} + 126 q^{36} + 270 q^{39} + 88 q^{41} - 154 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) \) 6 * q - 14 * q^4 - 6 * q^6 - 54 * q^9 + 66 * q^11 + 232 * q^14 - 170 * q^16 + 338 * q^19 - 96 * q^21 - 18 * q^24 + 334 * q^26 + 554 * q^29 - 346 * q^31 + 292 * q^34 + 126 * q^36 + 270 * q^39 + 88 * q^41 - 154 * q^44 + 850 * q^46 + 854 * q^49 + 348 * q^51 + 54 * q^54 + 336 * q^56 + 1368 * q^59 - 2076 * q^61 + 2258 * q^64 - 66 * q^66 + 930 * q^69 - 2918 * q^71 + 3528 * q^74 + 422 * q^76 + 1012 * q^79 + 486 * q^81 + 600 * q^84 - 1054 * q^86 + 1214 * q^89 - 796 * q^91 + 3644 * q^94 - 582 * q^96 - 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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