LMFDB - Embedded newform 175.4.b.b.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(99,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("175.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.b (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:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.b.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +15.0000i q^{8} +23.0000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +15.0000i q^{8} +23.0000 q^{9} -8.00000 q^{11} -14.0000i q^{12} +28.0000i q^{13} -7.00000 q^{14} +41.0000 q^{16} -54.0000i q^{17} +23.0000i q^{18} +110.000 q^{19} +14.0000 q^{21} -8.00000i q^{22} +48.0000i q^{23} +30.0000 q^{24} -28.0000 q^{26} -100.000i q^{27} +49.0000i q^{28} +110.000 q^{29} +12.0000 q^{31} +161.000i q^{32} +16.0000i q^{33} +54.0000 q^{34} +161.000 q^{36} +246.000i q^{37} +110.000i q^{38} +56.0000 q^{39} +182.000 q^{41} +14.0000i q^{42} +128.000i q^{43} -56.0000 q^{44} -48.0000 q^{46} -324.000i q^{47} -82.0000i q^{48} -49.0000 q^{49} -108.000 q^{51} +196.000i q^{52} -162.000i q^{53} +100.000 q^{54} -105.000 q^{56} -220.000i q^{57} +110.000i q^{58} -810.000 q^{59} -488.000 q^{61} +12.0000i q^{62} +161.000i q^{63} +167.000 q^{64} -16.0000 q^{66} -244.000i q^{67} -378.000i q^{68} +96.0000 q^{69} -768.000 q^{71} +345.000i q^{72} -702.000i q^{73} -246.000 q^{74} +770.000 q^{76} -56.0000i q^{77} +56.0000i q^{78} -440.000 q^{79} +421.000 q^{81} +182.000i q^{82} -1302.00i q^{83} +98.0000 q^{84} -128.000 q^{86} -220.000i q^{87} -120.000i q^{88} -730.000 q^{89} -196.000 q^{91} +336.000i q^{92} -24.0000i q^{93} +324.000 q^{94} +322.000 q^{96} -294.000i q^{97} -49.0000i q^{98} -184.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} + 4 q^{6} + 46 q^{9}+O(q^{10}) $ 2 * q + 14 * q^4 + 4 * q^6 + 46 * q^9	$ 2 q + 14 q^{4} + 4 q^{6} + 46 q^{9} - 16 q^{11} - 14 q^{14} + 82 q^{16} + 220 q^{19} + 28 q^{21} + 60 q^{24} - 56 q^{26} + 220 q^{29} + 24 q^{31} + 108 q^{34} + 322 q^{36} + 112 q^{39} + 364 q^{41} - 112 q^{44} - 96 q^{46} - 98 q^{49} - 216 q^{51} + 200 q^{54} - 210 q^{56} - 1620 q^{59} - 976 q^{61} + 334 q^{64} - 32 q^{66} + 192 q^{69} - 1536 q^{71} - 492 q^{74} + 1540 q^{76} - 880 q^{79} + 842 q^{81} + 196 q^{84} - 256 q^{86} - 1460 q^{89} - 392 q^{91} + 648 q^{94} + 644 q^{96} - 368 q^{99}+O(q^{100}) $ 2 * q + 14 * q^4 + 4 * q^6 + 46 * q^9 - 16 * q^11 - 14 * q^14 + 82 * q^16 + 220 * q^19 + 28 * q^21 + 60 * q^24 - 56 * q^26 + 220 * q^29 + 24 * q^31 + 108 * q^34 + 322 * q^36 + 112 * q^39 + 364 * q^41 - 112 * q^44 - 96 * q^46 - 98 * q^49 - 216 * q^51 + 200 * q^54 - 210 * q^56 - 1620 * q^59 - 976 * q^61 + 334 * q^64 - 32 * q^66 + 192 * q^69 - 1536 * q^71 - 492 * q^74 + 1540 * q^76 - 880 * q^79 + 842 * q^81 + 196 * q^84 - 256 * q^86 - 1460 * q^89 - 392 * q^91 + 648 * q^94 + 644 * q^96 - 368 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.353553i	0.984251	+	0.176777i	$0.0565670\pi$
$2$	1.00000i	0.353553i	−0.984251	+	0.176777i	$0.943433\pi$
$3$	− 2.00000i	− 0.384900i	−0.981307	−	0.192450i	$-0.938357\pi$
$3$	− 2.00000i	− 0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	7.00000	0.875000
$5$	0	0
$5$	0	0
$6$	2.00000	0.136083
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	15.0000i	0.662913i
$9$	23.0000	0.851852
$10$	0	0
$11$	−8.00000	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$11$	−8.00000	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$12$	− 14.0000i	− 0.336788i
$13$	28.0000i	0.597369i	0.954352	+	0.298685i	$0.0965479\pi$
$13$	28.0000i	0.597369i	−0.954352	+	0.298685i	$0.903452\pi$
$14$	−7.00000	−0.133631
$15$	0	0
$16$	41.0000	0.640625
$17$	− 54.0000i	− 0.770407i	−0.922832	−	0.385204i	$-0.874131\pi$
$17$	− 54.0000i	− 0.770407i	0.922832	−	0.385204i	$-0.125869\pi$
$18$	23.0000i	0.301175i
$19$	110.000	1.32820	0.664098	−	0.747645i	$-0.268816\pi$
$19$	110.000	1.32820	0.664098	+	0.747645i	$0.268816\pi$
$20$	0	0
$21$	14.0000	0.145479
$22$	− 8.00000i	− 0.0775275i
$23$	48.0000i	0.435161i	0.976042	+	0.217580i	$0.0698164\pi$
$23$	48.0000i	0.435161i	−0.976042	+	0.217580i	$0.930184\pi$
$24$	30.0000	0.255155
$25$	0	0
$26$	−28.0000	−0.211202
$27$	− 100.000i	− 0.712778i
$28$	49.0000i	0.330719i
$29$	110.000	0.704362	0.352181	−	0.935932i	$-0.385440\pi$
$29$	110.000	0.704362	0.352181	+	0.935932i	$0.385440\pi$
$30$	0	0
$31$	12.0000	0.0695246	0.0347623	−	0.999396i	$-0.488933\pi$
$31$	12.0000	0.0695246	0.0347623	+	0.999396i	$0.488933\pi$
$32$	161.000i	0.889408i
$33$	16.0000i	0.0844013i
$34$	54.0000	0.272380
$35$	0	0
$36$	161.000	0.745370
$37$	246.000i	1.09303i	0.837449	+	0.546516i	$0.184046\pi$
$37$	246.000i	1.09303i	−0.837449	+	0.546516i	$0.815954\pi$
$38$	110.000i	0.469588i
$39$	56.0000	0.229928
$40$	0	0
$41$	182.000	0.693259	0.346630	−	0.938002i	$-0.387326\pi$
$41$	182.000	0.693259	0.346630	+	0.938002i	$0.387326\pi$
$42$	14.0000i	0.0514344i
$43$	128.000i	0.453949i	0.973901	+	0.226975i	$0.0728834\pi$
$43$	128.000i	0.453949i	−0.973901	+	0.226975i	$0.927117\pi$
$44$	−56.0000	−0.191871
$45$	0	0
$46$	−48.0000	−0.153852
$47$	− 324.000i	− 1.00554i	−0.864421	−	0.502769i	$-0.832315\pi$
$47$	− 324.000i	− 1.00554i	0.864421	−	0.502769i	$-0.167685\pi$
$48$	− 82.0000i	− 0.246577i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−108.000	−0.296530
$52$	196.000i	0.522698i
$53$	− 162.000i	− 0.419857i	−0.977717	−	0.209928i	$-0.932677\pi$
$53$	− 162.000i	− 0.419857i	0.977717	−	0.209928i	$-0.0673231\pi$
$54$	100.000	0.252005
$55$	0	0
$56$	−105.000	−0.250557
$57$	− 220.000i	− 0.511223i
$58$	110.000i	0.249029i
$59$	−810.000	−1.78734	−0.893670	−	0.448725i	$-0.851878\pi$
$59$	−810.000	−1.78734	−0.893670	+	0.448725i	$0.851878\pi$
$60$	0	0
$61$	−488.000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−488.000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	12.0000i	0.0245807i
$63$	161.000i	0.321970i
$64$	167.000	0.326172
$65$	0	0
$66$	−16.0000	−0.0298404
$67$	− 244.000i	− 0.444916i	−0.974942	−	0.222458i	$-0.928592\pi$
$67$	− 244.000i	− 0.444916i	0.974942	−	0.222458i	$-0.0714080\pi$
$68$	− 378.000i	− 0.674106i
$69$	96.0000	0.167493
$70$	0	0
$71$	−768.000	−1.28373	−0.641865	−	0.766818i	$-0.721839\pi$
$71$	−768.000	−1.28373	−0.641865	+	0.766818i	$0.721839\pi$
$72$	345.000i	0.564703i
$73$	− 702.000i	− 1.12552i	−0.826621	−	0.562759i	$-0.809740\pi$
$73$	− 702.000i	− 1.12552i	0.826621	−	0.562759i	$-0.190260\pi$
$74$	−246.000	−0.386445
$75$	0	0
$76$	770.000	1.16217
$77$	− 56.0000i	− 0.0828804i
$78$	56.0000i	0.0812917i
$79$	−440.000	−0.626631	−0.313316	−	0.949649i	$-0.601440\pi$
$79$	−440.000	−0.626631	−0.313316	+	0.949649i	$0.601440\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	182.000i	0.245104i
$83$	− 1302.00i	− 1.72184i	−0.508737	−	0.860922i	$-0.669887\pi$
$83$	− 1302.00i	− 1.72184i	0.508737	−	0.860922i	$-0.330113\pi$
$84$	98.0000	0.127294
$85$	0	0
$86$	−128.000	−0.160495
$87$	− 220.000i	− 0.271109i
$88$	− 120.000i	− 0.145364i
$89$	−730.000	−0.869436	−0.434718	−	0.900567i	$-0.643152\pi$
$89$	−730.000	−0.869436	−0.434718	+	0.900567i	$0.643152\pi$
$90$	0	0
$91$	−196.000	−0.225784
$92$	336.000i	0.380765i
$93$	− 24.0000i	− 0.0267600i
$94$	324.000	0.355511
$95$	0	0
$96$	322.000	0.342333
$97$	− 294.000i	− 0.307744i	−0.988091	−	0.153872i	$-0.950826\pi$
$97$	− 294.000i	− 0.307744i	0.988091	−	0.153872i	$-0.0491744\pi$
$98$	− 49.0000i	− 0.0505076i
$99$	−184.000	−0.186795
$100$	0	0
$101$	−688.000	−0.677808	−0.338904	−	0.940821i	$-0.610056\pi$
$101$	−688.000	−0.677808	−0.338904	+	0.940821i	$0.610056\pi$
$102$	− 108.000i	− 0.104839i
$103$	1388.00i	1.32780i	0.747820	+	0.663901i	$0.231101\pi$
$103$	1388.00i	1.32780i	−0.747820	+	0.663901i	$0.768899\pi$
$104$	−420.000	−0.396004
$105$	0	0
$106$	162.000	0.148442
$107$	− 244.000i	− 0.220452i	−0.993907	−	0.110226i	$-0.964843\pi$
$107$	− 244.000i	− 0.220452i	0.993907	−	0.110226i	$-0.0351575\pi$
$108$	− 700.000i	− 0.623681i
$109$	−90.0000	−0.0790866	−0.0395433	−	0.999218i	$-0.512590\pi$
$109$	−90.0000	−0.0790866	−0.0395433	+	0.999218i	$0.512590\pi$
$110$	0	0
$111$	492.000	0.420708
$112$	287.000i	0.242133i
$113$	1318.00i	1.09723i	0.836075	+	0.548615i	$0.184845\pi$
$113$	1318.00i	1.09723i	−0.836075	+	0.548615i	$0.815155\pi$
$114$	220.000	0.180745
$115$	0	0
$116$	770.000	0.616316
$117$	644.000i	0.508870i
$118$	− 810.000i	− 0.631920i
$119$	378.000	0.291187
$120$	0	0
$121$	−1267.00	−0.951916
$122$	− 488.000i	− 0.362143i
$123$	− 364.000i	− 0.266836i
$124$	84.0000	0.0608341
$125$	0	0
$126$	−161.000	−0.113833
$127$	1776.00i	1.24090i	0.784245	+	0.620451i	$0.213050\pi$
$127$	1776.00i	1.24090i	−0.784245	+	0.620451i	$0.786950\pi$
$128$	1455.00i	1.00473i
$129$	256.000	0.174725
$130$	0	0
$131$	−1118.00	−0.745650	−0.372825	−	0.927902i	$-0.621611\pi$
$131$	−1118.00	−0.745650	−0.372825	+	0.927902i	$0.621611\pi$
$132$	112.000i	0.0738511i
$133$	770.000i	0.502011i
$134$	244.000	0.157301
$135$	0	0
$136$	810.000	0.510713
$137$	− 2274.00i	− 1.41811i	−0.705154	−	0.709054i	$-0.749122\pi$
$137$	− 2274.00i	− 1.41811i	0.705154	−	0.709054i	$-0.250878\pi$
$138$	96.0000i	0.0592178i
$139$	210.000	0.128144	0.0640718	−	0.997945i	$-0.479591\pi$
$139$	210.000	0.128144	0.0640718	+	0.997945i	$0.479591\pi$
$140$	0	0
$141$	−648.000	−0.387032
$142$	− 768.000i	− 0.453867i
$143$	− 224.000i	− 0.130992i
$144$	943.000	0.545718
$145$	0	0
$146$	702.000	0.397931
$147$	98.0000i	0.0549857i
$148$	1722.00i	0.956402i
$149$	2010.00	1.10514	0.552569	−	0.833467i	$-0.313648\pi$
$149$	2010.00	1.10514	0.552569	+	0.833467i	$0.313648\pi$
$150$	0	0
$151$	1112.00	0.599293	0.299647	−	0.954050i	$-0.403131\pi$
$151$	1112.00	0.599293	0.299647	+	0.954050i	$0.403131\pi$
$152$	1650.00i	0.880478i
$153$	− 1242.00i	− 0.656273i
$154$	56.0000	0.0293027
$155$	0	0
$156$	392.000	0.201187
$157$	− 124.000i	− 0.0630336i	−0.999503	−	0.0315168i	$-0.989966\pi$
$157$	− 124.000i	− 0.0630336i	0.999503	−	0.0315168i	$-0.0100338\pi$
$158$	− 440.000i	− 0.221548i
$159$	−324.000	−0.161603
$160$	0	0
$161$	−336.000	−0.164475
$162$	421.000i	0.204178i
$163$	2008.00i	0.964900i	0.875924	+	0.482450i	$0.160253\pi$
$163$	2008.00i	0.964900i	−0.875924	+	0.482450i	$0.839747\pi$
$164$	1274.00	0.606602
$165$	0	0
$166$	1302.00	0.608764
$167$	− 2884.00i	− 1.33635i	−0.744004	−	0.668176i	$-0.767076\pi$
$167$	− 2884.00i	− 1.33635i	0.744004	−	0.668176i	$-0.232924\pi$
$168$	210.000i	0.0964396i
$169$	1413.00	0.643150
$170$	0	0
$171$	2530.00	1.13143
$172$	896.000i	0.397206i
$173$	2228.00i	0.979143i	0.871963	+	0.489571i	$0.162847\pi$
$173$	2228.00i	0.979143i	−0.871963	+	0.489571i	$0.837153\pi$
$174$	220.000	0.0958515
$175$	0	0
$176$	−328.000	−0.140477
$177$	1620.00i	0.687947i
$178$	− 730.000i	− 0.307392i
$179$	820.000	0.342400	0.171200	−	0.985236i	$-0.445236\pi$
$179$	820.000	0.342400	0.171200	+	0.985236i	$0.445236\pi$
$180$	0	0
$181$	3892.00	1.59829	0.799144	−	0.601140i	$-0.205287\pi$
$181$	3892.00	1.59829	0.799144	+	0.601140i	$0.205287\pi$
$182$	− 196.000i	− 0.0798268i
$183$	976.000i	0.394251i
$184$	−720.000	−0.288473
$185$	0	0
$186$	24.0000	0.00946110
$187$	432.000i	0.168936i
$188$	− 2268.00i	− 0.879845i
$189$	700.000	0.269405
$190$	0	0
$191$	−5048.00	−1.91236	−0.956179	−	0.292782i	$-0.905419\pi$
$191$	−5048.00	−1.91236	−0.956179	+	0.292782i	$0.905419\pi$
$192$	− 334.000i	− 0.125544i
$193$	− 2962.00i	− 1.10471i	−0.833608	−	0.552356i	$-0.813729\pi$
$193$	− 2962.00i	− 1.10471i	0.833608	−	0.552356i	$-0.186271\pi$
$194$	294.000	0.108804
$195$	0	0
$196$	−343.000	−0.125000
$197$	− 3334.00i	− 1.20577i	−0.797826	−	0.602887i	$-0.794017\pi$
$197$	− 3334.00i	− 1.20577i	0.797826	−	0.602887i	$-0.205983\pi$
$198$	− 184.000i	− 0.0660420i
$199$	−1860.00	−0.662572	−0.331286	−	0.943530i	$-0.607483\pi$
$199$	−1860.00	−0.662572	−0.331286	+	0.943530i	$0.607483\pi$
$200$	0	0
$201$	−488.000	−0.171248
$202$	− 688.000i	− 0.239641i
$203$	770.000i	0.266224i
$204$	−756.000	−0.259464
$205$	0	0
$206$	−1388.00	−0.469449
$207$	1104.00i	0.370692i
$208$	1148.00i	0.382690i
$209$	−880.000	−0.291248
$210$	0	0
$211$	−4268.00	−1.39252	−0.696259	−	0.717791i	$-0.745153\pi$
$211$	−4268.00	−1.39252	−0.696259	+	0.717791i	$0.745153\pi$
$212$	− 1134.00i	− 0.367375i
$213$	1536.00i	0.494108i
$214$	244.000	0.0779416
$215$	0	0
$216$	1500.00	0.472510
$217$	84.0000i	0.0262778i
$218$	− 90.0000i	− 0.0279613i
$219$	−1404.00	−0.433212
$220$	0	0
$221$	1512.00	0.460218
$222$	492.000i	0.148743i
$223$	− 5432.00i	− 1.63118i	−0.578629	−	0.815591i	$-0.696412\pi$
$223$	− 5432.00i	− 1.63118i	0.578629	−	0.815591i	$-0.303588\pi$
$224$	−1127.00	−0.336165
$225$	0	0
$226$	−1318.00	−0.387929
$227$	2046.00i	0.598228i	0.954217	+	0.299114i	$0.0966911\pi$
$227$	2046.00i	0.598228i	−0.954217	+	0.299114i	$0.903309\pi$
$228$	− 1540.00i	− 0.447320i
$229$	2980.00	0.859930	0.429965	−	0.902846i	$-0.358526\pi$
$229$	2980.00	0.859930	0.429965	+	0.902846i	$0.358526\pi$
$230$	0	0
$231$	−112.000	−0.0319007
$232$	1650.00i	0.466930i
$233$	4458.00i	1.25345i	0.779241	+	0.626724i	$0.215605\pi$
$233$	4458.00i	1.25345i	−0.779241	+	0.626724i	$0.784395\pi$
$234$	−644.000	−0.179913
$235$	0	0
$236$	−5670.00	−1.56392
$237$	880.000i	0.241190i
$238$	378.000i	0.102950i
$239$	−4440.00	−1.20167	−0.600836	−	0.799372i	$-0.705166\pi$
$239$	−4440.00	−1.20167	−0.600836	+	0.799372i	$0.705166\pi$
$240$	0	0
$241$	3302.00	0.882575	0.441287	−	0.897366i	$-0.354522\pi$
$241$	3302.00	0.882575	0.441287	+	0.897366i	$0.354522\pi$
$242$	− 1267.00i	− 0.336553i
$243$	− 3542.00i	− 0.935059i
$244$	−3416.00	−0.896258
$245$	0	0
$246$	364.000	0.0943406
$247$	3080.00i	0.793424i
$248$	180.000i	0.0460888i
$249$	−2604.00	−0.662738
$250$	0	0
$251$	1582.00	0.397829	0.198914	−	0.980017i	$-0.436258\pi$
$251$	1582.00	0.397829	0.198914	+	0.980017i	$0.436258\pi$
$252$	1127.00i	0.281724i
$253$	− 384.000i	− 0.0954224i
$254$	−1776.00	−0.438725
$255$	0	0
$256$	−119.000	−0.0290527
$257$	− 2354.00i	− 0.571356i	−0.958326	−	0.285678i	$-0.907781\pi$
$257$	− 2354.00i	− 0.571356i	0.958326	−	0.285678i	$-0.0922188\pi$
$258$	256.000i	0.0617747i
$259$	−1722.00	−0.413127
$260$	0	0
$261$	2530.00	0.600012
$262$	− 1118.00i	− 0.263627i
$263$	− 3872.00i	− 0.907824i	−0.891046	−	0.453912i	$-0.850028\pi$
$263$	− 3872.00i	− 0.907824i	0.891046	−	0.453912i	$-0.149972\pi$
$264$	−240.000	−0.0559507
$265$	0	0
$266$	−770.000	−0.177488
$267$	1460.00i	0.334646i
$268$	− 1708.00i	− 0.389301i
$269$	−180.000	−0.0407985	−0.0203992	−	0.999792i	$-0.506494\pi$
$269$	−180.000	−0.0407985	−0.0203992	+	0.999792i	$0.506494\pi$
$270$	0	0
$271$	2032.00	0.455480	0.227740	−	0.973722i	$-0.426866\pi$
$271$	2032.00	0.455480	0.227740	+	0.973722i	$0.426866\pi$
$272$	− 2214.00i	− 0.493542i
$273$	392.000i	0.0869045i
$274$	2274.00	0.501377
$275$	0	0
$276$	672.000	0.146557
$277$	5426.00i	1.17696i	0.808513	+	0.588478i	$0.200273\pi$
$277$	5426.00i	1.17696i	−0.808513	+	0.588478i	$0.799727\pi$
$278$	210.000i	0.0453056i
$279$	276.000	0.0592247
$280$	0	0
$281$	842.000	0.178753	0.0893764	−	0.995998i	$-0.471513\pi$
$281$	842.000	0.178753	0.0893764	+	0.995998i	$0.471513\pi$
$282$	− 648.000i	− 0.136836i
$283$	− 3782.00i	− 0.794405i	−0.917731	−	0.397202i	$-0.869981\pi$
$283$	− 3782.00i	− 0.794405i	0.917731	−	0.397202i	$-0.130019\pi$
$284$	−5376.00	−1.12326
$285$	0	0
$286$	224.000	0.0463126
$287$	1274.00i	0.262027i
$288$	3703.00i	0.757644i
$289$	1997.00	0.406473
$290$	0	0
$291$	−588.000	−0.118451
$292$	− 4914.00i	− 0.984829i
$293$	− 4312.00i	− 0.859760i	−0.902886	−	0.429880i	$-0.858556\pi$
$293$	− 4312.00i	− 0.859760i	0.902886	−	0.429880i	$-0.141444\pi$
$294$	−98.0000	−0.0194404
$295$	0	0
$296$	−3690.00	−0.724584
$297$	800.000i	0.156299i
$298$	2010.00i	0.390725i
$299$	−1344.00	−0.259952
$300$	0	0
$301$	−896.000	−0.171577
$302$	1112.00i	0.211882i
$303$	1376.00i	0.260888i
$304$	4510.00	0.850876
$305$	0	0
$306$	1242.00	0.232027
$307$	− 2674.00i	− 0.497112i	−0.968618	−	0.248556i	$-0.920044\pi$
$307$	− 2674.00i	− 0.497112i	0.968618	−	0.248556i	$-0.0799559\pi$
$308$	− 392.000i	− 0.0725204i
$309$	2776.00	0.511072
$310$	0	0
$311$	−3768.00	−0.687021	−0.343511	−	0.939149i	$-0.611616\pi$
$311$	−3768.00	−0.687021	−0.343511	+	0.939149i	$0.611616\pi$
$312$	840.000i	0.152422i
$313$	2438.00i	0.440268i	0.975470	+	0.220134i	$0.0706495\pi$
$313$	2438.00i	0.440268i	−0.975470	+	0.220134i	$0.929351\pi$
$314$	124.000	0.0222857
$315$	0	0
$316$	−3080.00	−0.548302
$317$	3186.00i	0.564491i	0.959342	+	0.282245i	$0.0910792\pi$
$317$	3186.00i	0.564491i	−0.959342	+	0.282245i	$0.908921\pi$
$318$	− 324.000i	− 0.0571353i
$319$	−880.000	−0.154453
$320$	0	0
$321$	−488.000	−0.0848520
$322$	− 336.000i	− 0.0581508i
$323$	− 5940.00i	− 1.02325i
$324$	2947.00	0.505316
$325$	0	0
$326$	−2008.00	−0.341144
$327$	180.000i	0.0304404i
$328$	2730.00i	0.459570i
$329$	2268.00	0.380057
$330$	0	0
$331$	8672.00	1.44005	0.720025	−	0.693949i	$-0.244131\pi$
$331$	8672.00	1.44005	0.720025	+	0.693949i	$0.244131\pi$
$332$	− 9114.00i	− 1.50661i
$333$	5658.00i	0.931101i
$334$	2884.00	0.472471
$335$	0	0
$336$	574.000	0.0931972
$337$	− 814.000i	− 0.131577i	−0.997834	−	0.0657884i	$-0.979044\pi$
$337$	− 814.000i	− 0.131577i	0.997834	−	0.0657884i	$-0.0209562\pi$
$338$	1413.00i	0.227388i
$339$	2636.00	0.422324
$340$	0	0
$341$	−96.0000	−0.0152454
$342$	2530.00i	0.400020i
$343$	− 343.000i	− 0.0539949i
$344$	−1920.00	−0.300929
$345$	0	0
$346$	−2228.00	−0.346179
$347$	− 9344.00i	− 1.44557i	−0.691074	−	0.722784i	$-0.742862\pi$
$347$	− 9344.00i	− 1.44557i	0.691074	−	0.722784i	$-0.257138\pi$
$348$	− 1540.00i	− 0.237220i
$349$	5180.00	0.794496	0.397248	−	0.917711i	$-0.369965\pi$
$349$	5180.00	0.794496	0.397248	+	0.917711i	$0.369965\pi$
$350$	0	0
$351$	2800.00	0.425792
$352$	− 1288.00i	− 0.195030i
$353$	12178.0i	1.83617i	0.396379	+	0.918087i	$0.370267\pi$
$353$	12178.0i	1.83617i	−0.396379	+	0.918087i	$0.629733\pi$
$354$	−1620.00	−0.243226
$355$	0	0
$356$	−5110.00	−0.760757
$357$	− 756.000i	− 0.112078i
$358$	820.000i	0.121057i
$359$	−440.000	−0.0646861	−0.0323431	−	0.999477i	$-0.510297\pi$
$359$	−440.000	−0.0646861	−0.0323431	+	0.999477i	$0.510297\pi$
$360$	0	0
$361$	5241.00	0.764106
$362$	3892.00i	0.565080i
$363$	2534.00i	0.366393i
$364$	−1372.00	−0.197561
$365$	0	0
$366$	−976.000	−0.139389
$367$	9816.00i	1.39616i	0.716019	+	0.698080i	$0.245962\pi$
$367$	9816.00i	1.39616i	−0.716019	+	0.698080i	$0.754038\pi$
$368$	1968.00i	0.278775i
$369$	4186.00	0.590554
$370$	0	0
$371$	1134.00	0.158691
$372$	− 168.000i	− 0.0234150i
$373$	− 442.000i	− 0.0613563i	−0.999529	−	0.0306781i	$-0.990233\pi$
$373$	− 442.000i	− 0.0613563i	0.999529	−	0.0306781i	$-0.00976669\pi$
$374$	−432.000	−0.0597278
$375$	0	0
$376$	4860.00	0.666583
$377$	3080.00i	0.420764i
$378$	700.000i	0.0952490i
$379$	3960.00	0.536706	0.268353	−	0.963321i	$-0.413521\pi$
$379$	3960.00	0.536706	0.268353	+	0.963321i	$0.413521\pi$
$380$	0	0
$381$	3552.00	0.477623
$382$	− 5048.00i	− 0.676121i
$383$	6708.00i	0.894942i	0.894298	+	0.447471i	$0.147675\pi$
$383$	6708.00i	0.894942i	−0.894298	+	0.447471i	$0.852325\pi$
$384$	2910.00	0.386720
$385$	0	0
$386$	2962.00	0.390575
$387$	2944.00i	0.386697i
$388$	− 2058.00i	− 0.269276i
$389$	13350.0	1.74003	0.870015	−	0.493025i	$-0.164109\pi$
$389$	13350.0	1.74003	0.870015	+	0.493025i	$0.164109\pi$
$390$	0	0
$391$	2592.00	0.335251
$392$	− 735.000i	− 0.0947018i
$393$	2236.00i	0.287001i
$394$	3334.00	0.426306
$395$	0	0
$396$	−1288.00	−0.163446
$397$	1356.00i	0.171425i	0.996320	+	0.0857125i	$0.0273166\pi$
$397$	1356.00i	0.171425i	−0.996320	+	0.0857125i	$0.972683\pi$
$398$	− 1860.00i	− 0.234255i
$399$	1540.00	0.193224
$400$	0	0
$401$	6222.00	0.774843	0.387421	−	0.921903i	$-0.373366\pi$
$401$	6222.00	0.774843	0.387421	+	0.921903i	$0.373366\pi$
$402$	− 488.000i	− 0.0605453i
$403$	336.000i	0.0415319i
$404$	−4816.00	−0.593082
$405$	0	0
$406$	−770.000	−0.0941243
$407$	− 1968.00i	− 0.239681i
$408$	− 1620.00i	− 0.196573i
$409$	−5150.00	−0.622619	−0.311309	−	0.950309i	$-0.600768\pi$
$409$	−5150.00	−0.622619	−0.311309	+	0.950309i	$0.600768\pi$
$410$	0	0
$411$	−4548.00	−0.545830
$412$	9716.00i	1.16183i
$413$	− 5670.00i	− 0.675551i
$414$	−1104.00	−0.131060
$415$	0	0
$416$	−4508.00	−0.531305
$417$	− 420.000i	− 0.0493225i
$418$	− 880.000i	− 0.102972i
$419$	−2310.00	−0.269334	−0.134667	−	0.990891i	$-0.542996\pi$
$419$	−2310.00	−0.269334	−0.134667	+	0.990891i	$0.542996\pi$
$420$	0	0
$421$	1262.00	0.146095	0.0730476	−	0.997328i	$-0.476727\pi$
$421$	1262.00	0.146095	0.0730476	+	0.997328i	$0.476727\pi$
$422$	− 4268.00i	− 0.492329i
$423$	− 7452.00i	− 0.856569i
$424$	2430.00	0.278328
$425$	0	0
$426$	−1536.00	−0.174694
$427$	− 3416.00i	− 0.387147i
$428$	− 1708.00i	− 0.192896i
$429$	−448.000	−0.0504188
$430$	0	0
$431$	−4488.00	−0.501576	−0.250788	−	0.968042i	$-0.580690\pi$
$431$	−4488.00	−0.501576	−0.250788	+	0.968042i	$0.580690\pi$
$432$	− 4100.00i	− 0.456623i
$433$	17038.0i	1.89098i	0.325652	+	0.945490i	$0.394416\pi$
$433$	17038.0i	1.89098i	−0.325652	+	0.945490i	$0.605584\pi$
$434$	−84.0000	−0.00929062
$435$	0	0
$436$	−630.000	−0.0692008
$437$	5280.00i	0.577979i
$438$	− 1404.00i	− 0.153164i
$439$	−16200.0	−1.76124	−0.880619	−	0.473824i	$-0.842873\pi$
$439$	−16200.0	−1.76124	−0.880619	+	0.473824i	$0.842873\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	1512.00i	0.162712i
$443$	− 8772.00i	− 0.940791i	−0.882456	−	0.470395i	$-0.844111\pi$
$443$	− 8772.00i	− 0.940791i	0.882456	−	0.470395i	$-0.155889\pi$
$444$	3444.00	0.368119
$445$	0	0
$446$	5432.00	0.576710
$447$	− 4020.00i	− 0.425368i
$448$	1169.00i	0.123281i
$449$	−2130.00	−0.223877	−0.111939	−	0.993715i	$-0.535706\pi$
$449$	−2130.00	−0.223877	−0.111939	+	0.993715i	$0.535706\pi$
$450$	0	0
$451$	−1456.00	−0.152019
$452$	9226.00i	0.960076i
$453$	− 2224.00i	− 0.230668i
$454$	−2046.00	−0.211506
$455$	0	0
$456$	3300.00	0.338896
$457$	− 10534.0i	− 1.07825i	−0.842226	−	0.539124i	$-0.818755\pi$
$457$	− 10534.0i	− 1.07825i	0.842226	−	0.539124i	$-0.181245\pi$
$458$	2980.00i	0.304031i
$459$	−5400.00	−0.549129
$460$	0	0
$461$	−9268.00	−0.936342	−0.468171	−	0.883638i	$-0.655087\pi$
$461$	−9268.00	−0.936342	−0.468171	+	0.883638i	$0.655087\pi$
$462$	− 112.000i	− 0.0112786i
$463$	− 9392.00i	− 0.942728i	−0.881939	−	0.471364i	$-0.843762\pi$
$463$	− 9392.00i	− 0.942728i	0.881939	−	0.471364i	$-0.156238\pi$
$464$	4510.00	0.451232
$465$	0	0
$466$	−4458.00	−0.443161
$467$	10806.0i	1.07075i	0.844613	+	0.535377i	$0.179830\pi$
$467$	10806.0i	1.07075i	−0.844613	+	0.535377i	$0.820170\pi$
$468$	4508.00i	0.445261i
$469$	1708.00	0.168162
$470$	0	0
$471$	−248.000	−0.0242616
$472$	− 12150.0i	− 1.18485i
$473$	− 1024.00i	− 0.0995424i
$474$	−880.000	−0.0852737
$475$	0	0
$476$	2646.00	0.254788
$477$	− 3726.00i	− 0.357656i
$478$	− 4440.00i	− 0.424855i
$479$	−4940.00	−0.471220	−0.235610	−	0.971848i	$-0.575709\pi$
$479$	−4940.00	−0.471220	−0.235610	+	0.971848i	$0.575709\pi$
$480$	0	0
$481$	−6888.00	−0.652943
$482$	3302.00i	0.312037i
$483$	672.000i	0.0633065i
$484$	−8869.00	−0.832926
$485$	0	0
$486$	3542.00	0.330593
$487$	5216.00i	0.485338i	0.970109	+	0.242669i	$0.0780229\pi$
$487$	5216.00i	0.485338i	−0.970109	+	0.242669i	$0.921977\pi$
$488$	− 7320.00i	− 0.679018i
$489$	4016.00	0.371390
$490$	0	0
$491$	4412.00	0.405521	0.202760	−	0.979228i	$-0.435009\pi$
$491$	4412.00	0.405521	0.202760	+	0.979228i	$0.435009\pi$
$492$	− 2548.00i	− 0.233481i
$493$	− 5940.00i	− 0.542645i
$494$	−3080.00	−0.280518
$495$	0	0
$496$	492.000	0.0445392
$497$	− 5376.00i	− 0.485204i
$498$	− 2604.00i	− 0.234313i
$499$	−19060.0	−1.70991	−0.854953	−	0.518706i	$-0.826414\pi$
$499$	−19060.0	−1.70991	−0.854953	+	0.518706i	$0.826414\pi$
$500$	0	0
$501$	−5768.00	−0.514362
$502$	1582.00i	0.140654i
$503$	12768.0i	1.13180i	0.824473	+	0.565902i	$0.191472\pi$
$503$	12768.0i	1.13180i	−0.824473	+	0.565902i	$0.808528\pi$
$504$	−2415.00	−0.213438
$505$	0	0
$506$	384.000	0.0337369
$507$	− 2826.00i	− 0.247548i
$508$	12432.0i	1.08579i
$509$	5500.00	0.478945	0.239473	−	0.970903i	$-0.423025\pi$
$509$	5500.00	0.478945	0.239473	+	0.970903i	$0.423025\pi$
$510$	0	0
$511$	4914.00	0.425406
$512$	11521.0i	0.994455i
$513$	− 11000.0i	− 0.946709i
$514$	2354.00	0.202005
$515$	0	0
$516$	1792.00	0.152884
$517$	2592.00i	0.220495i
$518$	− 1722.00i	− 0.146062i
$519$	4456.00	0.376872
$520$	0	0
$521$	−7338.00	−0.617051	−0.308526	−	0.951216i	$-0.599836\pi$
$521$	−7338.00	−0.617051	−0.308526	+	0.951216i	$0.599836\pi$
$522$	2530.00i	0.212136i
$523$	− 17582.0i	− 1.46999i	−0.678070	−	0.734997i	$-0.737183\pi$
$523$	− 17582.0i	− 1.46999i	0.678070	−	0.734997i	$-0.262817\pi$
$524$	−7826.00	−0.652444
$525$	0	0
$526$	3872.00	0.320964
$527$	− 648.000i	− 0.0535623i
$528$	656.000i	0.0540696i
$529$	9863.00	0.810635
$530$	0	0
$531$	−18630.0	−1.52255
$532$	5390.00i	0.439260i
$533$	5096.00i	0.414132i
$534$	−1460.00	−0.118315
$535$	0	0
$536$	3660.00	0.294940
$537$	− 1640.00i	− 0.131790i
$538$	− 180.000i	− 0.0144244i
$539$	392.000	0.0313259
$540$	0	0
$541$	−1618.00	−0.128583	−0.0642914	−	0.997931i	$-0.520479\pi$
$541$	−1618.00	−0.128583	−0.0642914	+	0.997931i	$0.520479\pi$
$542$	2032.00i	0.161037i
$543$	− 7784.00i	− 0.615181i
$544$	8694.00	0.685206
$545$	0	0
$546$	−392.000	−0.0307254
$547$	− 16144.0i	− 1.26192i	−0.775817	−	0.630958i	$-0.782662\pi$
$547$	− 16144.0i	− 1.26192i	0.775817	−	0.630958i	$-0.217338\pi$
$548$	− 15918.0i	− 1.24085i
$549$	−11224.0	−0.872548
$550$	0	0
$551$	12100.0	0.935531
$552$	1440.00i	0.111033i
$553$	− 3080.00i	− 0.236844i
$554$	−5426.00	−0.416117
$555$	0	0
$556$	1470.00	0.112126
$557$	− 4654.00i	− 0.354033i	−0.984208	−	0.177016i	$-0.943355\pi$
$557$	− 4654.00i	− 0.354033i	0.984208	−	0.177016i	$-0.0566446\pi$
$558$	276.000i	0.0209391i
$559$	−3584.00	−0.271175
$560$	0	0
$561$	864.000	0.0650234
$562$	842.000i	0.0631986i
$563$	10078.0i	0.754418i	0.926128	+	0.377209i	$0.123116\pi$
$563$	10078.0i	0.754418i	−0.926128	+	0.377209i	$0.876884\pi$
$564$	−4536.00	−0.338653
$565$	0	0
$566$	3782.00	0.280865
$567$	2947.00i	0.218276i
$568$	− 11520.0i	− 0.851001i
$569$	5930.00	0.436904	0.218452	−	0.975848i	$-0.429899\pi$
$569$	5930.00	0.436904	0.218452	+	0.975848i	$0.429899\pi$
$570$	0	0
$571$	−19048.0	−1.39603	−0.698016	−	0.716082i	$-0.745933\pi$
$571$	−19048.0	−1.39603	−0.698016	+	0.716082i	$0.745933\pi$
$572$	− 1568.00i	− 0.114618i
$573$	10096.0i	0.736067i
$574$	−1274.00	−0.0926406
$575$	0	0
$576$	3841.00	0.277850
$577$	14366.0i	1.03651i	0.855227	+	0.518253i	$0.173418\pi$
$577$	14366.0i	1.03651i	−0.855227	+	0.518253i	$0.826582\pi$
$578$	1997.00i	0.143710i
$579$	−5924.00	−0.425204
$580$	0	0
$581$	9114.00	0.650796
$582$	− 588.000i	− 0.0418787i
$583$	1296.00i	0.0920666i
$584$	10530.0	0.746121
$585$	0	0
$586$	4312.00	0.303971
$587$	3626.00i	0.254959i	0.991841	+	0.127480i	$0.0406887\pi$
$587$	3626.00i	0.254959i	−0.991841	+	0.127480i	$0.959311\pi$
$588$	686.000i	0.0481125i
$589$	1320.00	0.0923424
$590$	0	0
$591$	−6668.00	−0.464103
$592$	10086.0i	0.700223i
$593$	− 1062.00i	− 0.0735432i	−0.999324	−	0.0367716i	$-0.988293\pi$
$593$	− 1062.00i	− 0.0735432i	0.999324	−	0.0367716i	$-0.0117074\pi$
$594$	−800.000	−0.0552599
$595$	0	0
$596$	14070.0	0.966996
$597$	3720.00i	0.255024i
$598$	− 1344.00i	− 0.0919068i
$599$	10200.0	0.695761	0.347880	−	0.937539i	$-0.386902\pi$
$599$	10200.0	0.695761	0.347880	+	0.937539i	$0.386902\pi$
$600$	0	0
$601$	−25158.0	−1.70751	−0.853757	−	0.520671i	$-0.825682\pi$
$601$	−25158.0	−1.70751	−0.853757	+	0.520671i	$0.825682\pi$
$602$	− 896.000i	− 0.0606615i
$603$	− 5612.00i	− 0.379002i
$604$	7784.00	0.524382
$605$	0	0
$606$	−1376.00	−0.0922379
$607$	− 25664.0i	− 1.71609i	−0.513570	−	0.858047i	$-0.671677\pi$
$607$	− 25664.0i	− 1.71609i	0.513570	−	0.858047i	$-0.328323\pi$
$608$	17710.0i	1.18131i
$609$	1540.00	0.102470
$610$	0	0
$611$	9072.00	0.600677
$612$	− 8694.00i	− 0.574239i
$613$	19018.0i	1.25307i	0.779395	+	0.626533i	$0.215527\pi$
$613$	19018.0i	1.25307i	−0.779395	+	0.626533i	$0.784473\pi$
$614$	2674.00	0.175755
$615$	0	0
$616$	840.000	0.0549425
$617$	− 17334.0i	− 1.13102i	−0.824741	−	0.565511i	$-0.808679\pi$
$617$	− 17334.0i	− 1.13102i	0.824741	−	0.565511i	$-0.191321\pi$
$618$	2776.00i	0.180691i
$619$	−18730.0	−1.21619	−0.608096	−	0.793864i	$-0.708066\pi$
$619$	−18730.0	−1.21619	−0.608096	+	0.793864i	$0.708066\pi$
$620$	0	0
$621$	4800.00	0.310173
$622$	− 3768.00i	− 0.242899i
$623$	− 5110.00i	− 0.328616i
$624$	2296.00	0.147297
$625$	0	0
$626$	−2438.00	−0.155658
$627$	1760.00i	0.112101i
$628$	− 868.000i	− 0.0551544i
$629$	13284.0	0.842079
$630$	0	0
$631$	−6928.00	−0.437083	−0.218541	−	0.975828i	$-0.570130\pi$
$631$	−6928.00	−0.437083	−0.218541	+	0.975828i	$0.570130\pi$
$632$	− 6600.00i	− 0.415402i
$633$	8536.00i	0.535980i
$634$	−3186.00	−0.199578
$635$	0	0
$636$	−2268.00	−0.141403
$637$	− 1372.00i	− 0.0853385i
$638$	− 880.000i	− 0.0546074i
$639$	−17664.0	−1.09355
$640$	0	0
$641$	16302.0	1.00451	0.502255	−	0.864720i	$-0.332504\pi$
$641$	16302.0	1.00451	0.502255	+	0.864720i	$0.332504\pi$
$642$	− 488.000i	− 0.0299997i
$643$	4718.00i	0.289362i	0.989478	+	0.144681i	$0.0462156\pi$
$643$	4718.00i	0.289362i	−0.989478	+	0.144681i	$0.953784\pi$
$644$	−2352.00	−0.143916
$645$	0	0
$646$	5940.00	0.361774
$647$	21436.0i	1.30253i	0.758851	+	0.651264i	$0.225761\pi$
$647$	21436.0i	1.30253i	−0.758851	+	0.651264i	$0.774239\pi$
$648$	6315.00i	0.382834i
$649$	6480.00	0.391930
$650$	0	0
$651$	168.000	0.0101143
$652$	14056.0i	0.844287i
$653$	4458.00i	0.267159i	0.991038	+	0.133580i	$0.0426472\pi$
$653$	4458.00i	0.267159i	−0.991038	+	0.133580i	$0.957353\pi$
$654$	−180.000	−0.0107623
$655$	0	0
$656$	7462.00	0.444119
$657$	− 16146.0i	− 0.958775i
$658$	2268.00i	0.134371i
$659$	26640.0	1.57473	0.787365	−	0.616487i	$-0.211445\pi$
$659$	26640.0	1.57473	0.787365	+	0.616487i	$0.211445\pi$
$660$	0	0
$661$	7432.00	0.437324	0.218662	−	0.975801i	$-0.429831\pi$
$661$	7432.00	0.437324	0.218662	+	0.975801i	$0.429831\pi$
$662$	8672.00i	0.509134i
$663$	− 3024.00i	− 0.177138i
$664$	19530.0	1.14143
$665$	0	0
$666$	−5658.00	−0.329194
$667$	5280.00i	0.306510i
$668$	− 20188.0i	− 1.16931i
$669$	−10864.0	−0.627842
$670$	0	0
$671$	3904.00	0.224608
$672$	2254.00i	0.129390i
$673$	58.0000i	0.00332204i	0.999999	+	0.00166102i	$0.000528720\pi$
$673$	58.0000i	0.00332204i	−0.999999	+	0.00166102i	$0.999471\pi$
$674$	814.000	0.0465194
$675$	0	0
$676$	9891.00	0.562756
$677$	21516.0i	1.22146i	0.791840	+	0.610729i	$0.209124\pi$
$677$	21516.0i	1.22146i	−0.791840	+	0.610729i	$0.790876\pi$
$678$	2636.00i	0.149314i
$679$	2058.00	0.116316
$680$	0	0
$681$	4092.00	0.230258
$682$	− 96.0000i	− 0.00539007i
$683$	18108.0i	1.01447i	0.861808	+	0.507235i	$0.169332\pi$
$683$	18108.0i	1.01447i	−0.861808	+	0.507235i	$0.830668\pi$
$684$	17710.0	0.989998
$685$	0	0
$686$	343.000	0.0190901
$687$	− 5960.00i	− 0.330987i
$688$	5248.00i	0.290811i
$689$	4536.00	0.250810
$690$	0	0
$691$	−10078.0	−0.554827	−0.277413	−	0.960751i	$-0.589477\pi$
$691$	−10078.0	−0.554827	−0.277413	+	0.960751i	$0.589477\pi$
$692$	15596.0i	0.856750i
$693$	− 1288.00i	− 0.0706018i
$694$	9344.00	0.511086
$695$	0	0
$696$	3300.00	0.179722
$697$	− 9828.00i	− 0.534092i
$698$	5180.00i	0.280897i
$699$	8916.00	0.482452
$700$	0	0
$701$	18762.0	1.01089	0.505443	−	0.862860i	$-0.331329\pi$
$701$	18762.0	1.01089	0.505443	+	0.862860i	$0.331329\pi$
$702$	2800.00i	0.150540i
$703$	27060.0i	1.45176i
$704$	−1336.00	−0.0715233
$705$	0	0
$706$	−12178.0	−0.649186
$707$	− 4816.00i	− 0.256187i
$708$	11340.0i	0.601954i
$709$	−6810.00	−0.360726	−0.180363	−	0.983600i	$-0.557727\pi$
$709$	−6810.00	−0.360726	−0.180363	+	0.983600i	$0.557727\pi$
$710$	0	0
$711$	−10120.0	−0.533797
$712$	− 10950.0i	− 0.576360i
$713$	576.000i	0.0302544i
$714$	756.000	0.0396255
$715$	0	0
$716$	5740.00	0.299600
$717$	8880.00i	0.462524i
$718$	− 440.000i	− 0.0228700i
$719$	−4860.00	−0.252083	−0.126041	−	0.992025i	$-0.540227\pi$
$719$	−4860.00	−0.252083	−0.126041	+	0.992025i	$0.540227\pi$
$720$	0	0
$721$	−9716.00	−0.501862
$722$	5241.00i	0.270152i
$723$	− 6604.00i	− 0.339703i
$724$	27244.0	1.39850
$725$	0	0
$726$	−2534.00	−0.129539
$727$	13636.0i	0.695641i	0.937561	+	0.347821i	$0.113078\pi$
$727$	13636.0i	0.695641i	−0.937561	+	0.347821i	$0.886922\pi$
$728$	− 2940.00i	− 0.149675i
$729$	4283.00	0.217599
$730$	0	0
$731$	6912.00	0.349726
$732$	6832.00i	0.344970i
$733$	2088.00i	0.105214i	0.998615	+	0.0526071i	$0.0167531\pi$
$733$	2088.00i	0.105214i	−0.998615	+	0.0526071i	$0.983247\pi$
$734$	−9816.00	−0.493617
$735$	0	0
$736$	−7728.00	−0.387035
$737$	1952.00i	0.0975615i
$738$	4186.00i	0.208792i
$739$	5160.00	0.256852	0.128426	−	0.991719i	$-0.459008\pi$
$739$	5160.00	0.256852	0.128426	+	0.991719i	$0.459008\pi$
$740$	0	0
$741$	6160.00	0.305389
$742$	1134.00i	0.0561057i
$743$	− 28152.0i	− 1.39004i	−0.718992	−	0.695018i	$-0.755396\pi$
$743$	− 28152.0i	− 1.39004i	0.718992	−	0.695018i	$-0.244604\pi$
$744$	360.000	0.0177396
$745$	0	0
$746$	442.000	0.0216927
$747$	− 29946.0i	− 1.46676i
$748$	3024.00i	0.147819i
$749$	1708.00	0.0833230
$750$	0	0
$751$	−16808.0	−0.816688	−0.408344	−	0.912828i	$-0.633894\pi$
$751$	−16808.0	−0.816688	−0.408344	+	0.912828i	$0.633894\pi$
$752$	− 13284.0i	− 0.644172i
$753$	− 3164.00i	− 0.153124i
$754$	−3080.00	−0.148763
$755$	0	0
$756$	4900.00	0.235729
$757$	− 21674.0i	− 1.04063i	−0.853975	−	0.520314i	$-0.825815\pi$
$757$	− 21674.0i	− 1.04063i	0.853975	−	0.520314i	$-0.174185\pi$
$758$	3960.00i	0.189754i
$759$	−768.000	−0.0367281
$760$	0	0
$761$	7422.00	0.353544	0.176772	−	0.984252i	$-0.443434\pi$
$761$	7422.00	0.353544	0.176772	+	0.984252i	$0.443434\pi$
$762$	3552.00i	0.168865i
$763$	− 630.000i	− 0.0298919i
$764$	−35336.0	−1.67331
$765$	0	0
$766$	−6708.00	−0.316410
$767$	− 22680.0i	− 1.06770i
$768$	238.000i	0.0111824i
$769$	−13790.0	−0.646658	−0.323329	−	0.946287i	$-0.604802\pi$
$769$	−13790.0	−0.646658	−0.323329	+	0.946287i	$0.604802\pi$
$770$	0	0
$771$	−4708.00	−0.219915
$772$	− 20734.0i	− 0.966623i
$773$	− 6232.00i	− 0.289973i	−0.989434	−	0.144987i	$-0.953686\pi$
$773$	− 6232.00i	− 0.289973i	0.989434	−	0.144987i	$-0.0463139\pi$
$774$	−2944.00	−0.136718
$775$	0	0
$776$	4410.00	0.204007
$777$	3444.00i	0.159013i
$778$	13350.0i	0.615194i
$779$	20020.0	0.920784
$780$	0	0
$781$	6144.00	0.281498
$782$	2592.00i	0.118529i
$783$	− 11000.0i	− 0.502054i
$784$	−2009.00	−0.0915179
$785$	0	0
$786$	−2236.00	−0.101470
$787$	1766.00i	0.0799887i	0.999200	+	0.0399943i	$0.0127340\pi$
$787$	1766.00i	0.0799887i	−0.999200	+	0.0399943i	$0.987266\pi$
$788$	− 23338.0i	− 1.05505i
$789$	−7744.00	−0.349422
$790$	0	0
$791$	−9226.00	−0.414714
$792$	− 2760.00i	− 0.123829i
$793$	− 13664.0i	− 0.611883i
$794$	−1356.00	−0.0606079
$795$	0	0
$796$	−13020.0	−0.579751
$797$	− 1204.00i	− 0.0535105i	−0.999642	−	0.0267552i	$-0.991483\pi$
$797$	− 1204.00i	− 0.0535105i	0.999642	−	0.0267552i	$-0.00851748\pi$
$798$	1540.00i	0.0683150i
$799$	−17496.0	−0.774673
$800$	0	0
$801$	−16790.0	−0.740631
$802$	6222.00i	0.273948i
$803$	5616.00i	0.246805i
$804$	−3416.00	−0.149842
$805$	0	0
$806$	−336.000	−0.0146837
$807$	360.000i	0.0157033i
$808$	− 10320.0i	− 0.449327i
$809$	7050.00	0.306384	0.153192	−	0.988196i	$-0.451045\pi$
$809$	7050.00	0.306384	0.153192	+	0.988196i	$0.451045\pi$
$810$	0	0
$811$	23282.0	1.00807	0.504033	−	0.863684i	$-0.331849\pi$
$811$	23282.0	1.00807	0.504033	+	0.863684i	$0.331849\pi$
$812$	5390.00i	0.232946i
$813$	− 4064.00i	− 0.175315i
$814$	1968.00	0.0847400
$815$	0	0
$816$	−4428.00	−0.189964
$817$	14080.0i	0.602934i
$818$	− 5150.00i	− 0.220129i
$819$	−4508.00	−0.192335
$820$	0	0
$821$	10142.0	0.431131	0.215565	−	0.976489i	$-0.430841\pi$
$821$	10142.0	0.431131	0.215565	+	0.976489i	$0.430841\pi$
$822$	− 4548.00i	− 0.192980i
$823$	− 9192.00i	− 0.389323i	−0.980870	−	0.194662i	$-0.937639\pi$
$823$	− 9192.00i	− 0.389323i	0.980870	−	0.194662i	$-0.0623609\pi$
$824$	−20820.0	−0.880217
$825$	0	0
$826$	5670.00	0.238843
$827$	46716.0i	1.96430i	0.188104	+	0.982149i	$0.439766\pi$
$827$	46716.0i	1.96430i	−0.188104	+	0.982149i	$0.560234\pi$
$828$	7728.00i	0.324356i
$829$	−11240.0	−0.470906	−0.235453	−	0.971886i	$-0.575657\pi$
$829$	−11240.0	−0.470906	−0.235453	+	0.971886i	$0.575657\pi$
$830$	0	0
$831$	10852.0	0.453010
$832$	4676.00i	0.194845i
$833$	2646.00i	0.110058i
$834$	420.000	0.0174381
$835$	0	0
$836$	−6160.00	−0.254842
$837$	− 1200.00i	− 0.0495556i
$838$	− 2310.00i	− 0.0952239i
$839$	−700.000	−0.0288042	−0.0144021	−	0.999896i	$-0.504584\pi$
$839$	−700.000	−0.0288042	−0.0144021	+	0.999896i	$0.504584\pi$
$840$	0	0
$841$	−12289.0	−0.503875
$842$	1262.00i	0.0516525i
$843$	− 1684.00i	− 0.0688019i
$844$	−29876.0	−1.21845
$845$	0	0
$846$	7452.00	0.302843
$847$	− 8869.00i	− 0.359790i
$848$	− 6642.00i	− 0.268971i
$849$	−7564.00	−0.305767
$850$	0	0
$851$	−11808.0	−0.475644
$852$	10752.0i	0.432344i
$853$	− 37492.0i	− 1.50493i	−0.658635	−	0.752463i	$-0.728866\pi$
$853$	− 37492.0i	− 1.50493i	0.658635	−	0.752463i	$-0.271134\pi$
$854$	3416.00	0.136877
$855$	0	0
$856$	3660.00	0.146140
$857$	− 28894.0i	− 1.15169i	−0.817558	−	0.575846i	$-0.804673\pi$
$857$	− 28894.0i	− 1.15169i	0.817558	−	0.575846i	$-0.195327\pi$
$858$	− 448.000i	− 0.0178257i
$859$	2770.00	0.110025	0.0550123	−	0.998486i	$-0.482480\pi$
$859$	2770.00	0.110025	0.0550123	+	0.998486i	$0.482480\pi$
$860$	0	0
$861$	2548.00	0.100854
$862$	− 4488.00i	− 0.177334i
$863$	17688.0i	0.697690i	0.937180	+	0.348845i	$0.113426\pi$
$863$	17688.0i	0.697690i	−0.937180	+	0.348845i	$0.886574\pi$
$864$	16100.0	0.633950
$865$	0	0
$866$	−17038.0	−0.668562
$867$	− 3994.00i	− 0.156451i
$868$	588.000i	0.0229931i
$869$	3520.00	0.137408
$870$	0	0
$871$	6832.00	0.265779
$872$	− 1350.00i	− 0.0524275i
$873$	− 6762.00i	− 0.262152i
$874$	−5280.00	−0.204346
$875$	0	0
$876$	−9828.00	−0.379061
$877$	33566.0i	1.29241i	0.763164	+	0.646205i	$0.223645\pi$
$877$	33566.0i	1.29241i	−0.763164	+	0.646205i	$0.776355\pi$
$878$	− 16200.0i	− 0.622692i
$879$	−8624.00	−0.330922
$880$	0	0
$881$	−16758.0	−0.640853	−0.320426	−	0.947273i	$-0.603826\pi$
$881$	−16758.0	−0.640853	−0.320426	+	0.947273i	$0.603826\pi$
$882$	− 1127.00i	− 0.0430250i
$883$	11468.0i	0.437066i	0.975830	+	0.218533i	$0.0701271\pi$
$883$	11468.0i	0.437066i	−0.975830	+	0.218533i	$0.929873\pi$
$884$	10584.0	0.402691
$885$	0	0
$886$	8772.00	0.332620
$887$	50356.0i	1.90619i	0.302674	+	0.953094i	$0.402121\pi$
$887$	50356.0i	1.90619i	−0.302674	+	0.953094i	$0.597879\pi$
$888$	7380.00i	0.278893i
$889$	−12432.0	−0.469017
$890$	0	0
$891$	−3368.00	−0.126636
$892$	− 38024.0i	− 1.42728i
$893$	− 35640.0i	− 1.33555i
$894$	4020.00	0.150390
$895$	0	0
$896$	−10185.0	−0.379751
$897$	2688.00i	0.100055i
$898$	− 2130.00i	− 0.0791526i
$899$	1320.00	0.0489705
$900$	0	0
$901$	−8748.00	−0.323461
$902$	− 1456.00i	− 0.0537467i
$903$	1792.00i	0.0660399i
$904$	−19770.0	−0.727368
$905$	0	0
$906$	2224.00	0.0815535
$907$	8716.00i	0.319085i	0.987191	+	0.159542i	$0.0510019\pi$
$907$	8716.00i	0.319085i	−0.987191	+	0.159542i	$0.948998\pi$
$908$	14322.0i	0.523450i
$909$	−15824.0	−0.577392
$910$	0	0
$911$	7632.00	0.277563	0.138781	−	0.990323i	$-0.455682\pi$
$911$	7632.00	0.277563	0.138781	+	0.990323i	$0.455682\pi$
$912$	− 9020.00i	− 0.327502i
$913$	10416.0i	0.377568i
$914$	10534.0	0.381219
$915$	0	0
$916$	20860.0	0.752439
$917$	− 7826.00i	− 0.281829i
$918$	− 5400.00i	− 0.194147i
$919$	23080.0	0.828443	0.414221	−	0.910176i	$-0.364054\pi$
$919$	23080.0	0.828443	0.414221	+	0.910176i	$0.364054\pi$
$920$	0	0
$921$	−5348.00	−0.191338
$922$	− 9268.00i	− 0.331047i
$923$	− 21504.0i	− 0.766861i
$924$	−784.000	−0.0279131
$925$	0	0
$926$	9392.00	0.333305
$927$	31924.0i	1.13109i
$928$	17710.0i	0.626465i
$929$	−45110.0	−1.59312	−0.796561	−	0.604558i	$-0.793350\pi$
$929$	−45110.0	−1.59312	−0.796561	+	0.604558i	$0.793350\pi$
$930$	0	0
$931$	−5390.00	−0.189742
$932$	31206.0i	1.09677i
$933$	7536.00i	0.264435i
$934$	−10806.0	−0.378569
$935$	0	0
$936$	−9660.00	−0.337337
$937$	− 16674.0i	− 0.581340i	−0.956823	−	0.290670i	$-0.906122\pi$
$937$	− 16674.0i	− 0.581340i	0.956823	−	0.290670i	$-0.0938782\pi$
$938$	1708.00i	0.0594543i
$939$	4876.00	0.169459
$940$	0	0
$941$	43832.0	1.51847	0.759236	−	0.650815i	$-0.225573\pi$
$941$	43832.0	1.51847	0.759236	+	0.650815i	$0.225573\pi$
$942$	− 248.000i	− 0.00857779i
$943$	8736.00i	0.301679i
$944$	−33210.0	−1.14501
$945$	0	0
$946$	1024.00	0.0351936
$947$	736.000i	0.0252553i	0.999920	+	0.0126277i	$0.00401962\pi$
$947$	736.000i	0.0252553i	−0.999920	+	0.0126277i	$0.995980\pi$
$948$	6160.00i	0.211042i
$949$	19656.0	0.672351
$950$	0	0
$951$	6372.00	0.217273
$952$	5670.00i	0.193031i
$953$	38138.0i	1.29634i	0.761496	+	0.648169i	$0.224465\pi$
$953$	38138.0i	1.29634i	−0.761496	+	0.648169i	$0.775535\pi$
$954$	3726.00	0.126450
$955$	0	0
$956$	−31080.0	−1.05146
$957$	1760.00i	0.0594490i
$958$	− 4940.00i	− 0.166601i
$959$	15918.0	0.535995
$960$	0	0
$961$	−29647.0	−0.995166
$962$	− 6888.00i	− 0.230850i
$963$	− 5612.00i	− 0.187792i
$964$	23114.0	0.772253
$965$	0	0
$966$	−672.000	−0.0223822
$967$	− 26224.0i	− 0.872086i	−0.899926	−	0.436043i	$-0.856380\pi$
$967$	− 26224.0i	− 0.872086i	0.899926	−	0.436043i	$-0.143620\pi$
$968$	− 19005.0i	− 0.631037i
$969$	−11880.0	−0.393850
$970$	0	0
$971$	18762.0	0.620084	0.310042	−	0.950723i	$-0.399657\pi$
$971$	18762.0	0.620084	0.310042	+	0.950723i	$0.399657\pi$
$972$	− 24794.0i	− 0.818177i
$973$	1470.00i	0.0484337i
$974$	−5216.00	−0.171593
$975$	0	0
$976$	−20008.0	−0.656189
$977$	− 38394.0i	− 1.25725i	−0.777709	−	0.628625i	$-0.783618\pi$
$977$	− 38394.0i	− 1.25725i	0.777709	−	0.628625i	$-0.216382\pi$
$978$	4016.00i	0.131306i
$979$	5840.00	0.190651
$980$	0	0
$981$	−2070.00	−0.0673700
$982$	4412.00i	0.143373i
$983$	5388.00i	0.174822i	0.996172	+	0.0874112i	$0.0278594\pi$
$983$	5388.00i	0.174822i	−0.996172	+	0.0874112i	$0.972141\pi$
$984$	5460.00	0.176889
$985$	0	0
$986$	5940.00	0.191854
$987$	− 4536.00i	− 0.146284i
$988$	21560.0i	0.694246i
$989$	−6144.00	−0.197541
$990$	0	0
$991$	25472.0	0.816493	0.408247	−	0.912872i	$-0.366140\pi$
$991$	25472.0	0.816493	0.408247	+	0.912872i	$0.366140\pi$
$992$	1932.00i	0.0618357i
$993$	− 17344.0i	− 0.554275i
$994$	5376.00	0.171546
$995$	0	0
$996$	−18228.0	−0.579896
$997$	17096.0i	0.543065i	0.962429	+	0.271532i	$0.0875304\pi$
$997$	17096.0i	0.543065i	−0.962429	+	0.271532i	$0.912470\pi$
$998$	− 19060.0i	− 0.604543i
$999$	24600.0	0.779089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.b.99.2		2
5.2	odd	4		7.4.a.a.1.1	✓	1
5.3	odd	4		175.4.a.b.1.1		1
5.4	even	2	inner	175.4.b.b.99.1		2
15.2	even	4		63.4.a.b.1.1		1
15.8	even	4		1575.4.a.e.1.1		1
20.7	even	4		112.4.a.f.1.1		1
35.2	odd	12		49.4.c.c.18.1		2
35.12	even	12		49.4.c.b.18.1		2
35.13	even	4		1225.4.a.j.1.1		1
35.17	even	12		49.4.c.b.30.1		2
35.27	even	4		49.4.a.b.1.1		1
35.32	odd	12		49.4.c.c.30.1		2
40.27	even	4		448.4.a.e.1.1		1
40.37	odd	4		448.4.a.i.1.1		1
55.32	even	4		847.4.a.b.1.1		1
60.47	odd	4		1008.4.a.c.1.1		1
65.12	odd	4		1183.4.a.b.1.1		1
85.67	odd	4		2023.4.a.a.1.1		1
105.2	even	12		441.4.e.h.361.1		2
105.17	odd	12		441.4.e.e.226.1		2
105.32	even	12		441.4.e.h.226.1		2
105.47	odd	12		441.4.e.e.361.1		2
105.62	odd	4		441.4.a.i.1.1		1
140.27	odd	4		784.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.4.a.a.1.1	✓	1	5.2	odd	4
49.4.a.b.1.1		1	35.27	even	4
49.4.c.b.18.1		2	35.12	even	12
49.4.c.b.30.1		2	35.17	even	12
49.4.c.c.18.1		2	35.2	odd	12
49.4.c.c.30.1		2	35.32	odd	12
63.4.a.b.1.1		1	15.2	even	4
112.4.a.f.1.1		1	20.7	even	4
175.4.a.b.1.1		1	5.3	odd	4
175.4.b.b.99.1		2	5.4	even	2	inner
175.4.b.b.99.2		2	1.1	even	1	trivial
441.4.a.i.1.1		1	105.62	odd	4
441.4.e.e.226.1		2	105.17	odd	12
441.4.e.e.361.1		2	105.47	odd	12
441.4.e.h.226.1		2	105.32	even	12
441.4.e.h.361.1		2	105.2	even	12
448.4.a.e.1.1		1	40.27	even	4
448.4.a.i.1.1		1	40.37	odd	4
784.4.a.g.1.1		1	140.27	odd	4
847.4.a.b.1.1		1	55.32	even	4
1008.4.a.c.1.1		1	60.47	odd	4
1183.4.a.b.1.1		1	65.12	odd	4
1225.4.a.j.1.1		1	35.13	even	4
1575.4.a.e.1.1		1	15.8	even	4
2023.4.a.a.1.1		1	85.67	odd	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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +15.0000i q^{8} +23.0000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +15.0000i q^{8} +23.0000 q^{9} -8.00000 q^{11} -14.0000i q^{12} +28.0000i q^{13} -7.00000 q^{14} +41.0000 q^{16} -54.0000i q^{17} +23.0000i q^{18} +110.000 q^{19} +14.0000 q^{21} -8.00000i q^{22} +48.0000i q^{23} +30.0000 q^{24} -28.0000 q^{26} -100.000i q^{27} +49.0000i q^{28} +110.000 q^{29} +12.0000 q^{31} +161.000i q^{32} +16.0000i q^{33} +54.0000 q^{34} +161.000 q^{36} +246.000i q^{37} +110.000i q^{38} +56.0000 q^{39} +182.000 q^{41} +14.0000i q^{42} +128.000i q^{43} -56.0000 q^{44} -48.0000 q^{46} -324.000i q^{47} -82.0000i q^{48} -49.0000 q^{49} -108.000 q^{51} +196.000i q^{52} -162.000i q^{53} +100.000 q^{54} -105.000 q^{56} -220.000i q^{57} +110.000i q^{58} -810.000 q^{59} -488.000 q^{61} +12.0000i q^{62} +161.000i q^{63} +167.000 q^{64} -16.0000 q^{66} -244.000i q^{67} -378.000i q^{68} +96.0000 q^{69} -768.000 q^{71} +345.000i q^{72} -702.000i q^{73} -246.000 q^{74} +770.000 q^{76} -56.0000i q^{77} +56.0000i q^{78} -440.000 q^{79} +421.000 q^{81} +182.000i q^{82} -1302.00i q^{83} +98.0000 q^{84} -128.000 q^{86} -220.000i q^{87} -120.000i q^{88} -730.000 q^{89} -196.000 q^{91} +336.000i q^{92} -24.0000i q^{93} +324.000 q^{94} +322.000 q^{96} -294.000i q^{97} -49.0000i q^{98} -184.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} + 4 q^{6} + 46 q^{9}+O(q^{10}) \) 2 * q + 14 * q^4 + 4 * q^6 + 46 * q^9	\( 2 q + 14 q^{4} + 4 q^{6} + 46 q^{9} - 16 q^{11} - 14 q^{14} + 82 q^{16} + 220 q^{19} + 28 q^{21} + 60 q^{24} - 56 q^{26} + 220 q^{29} + 24 q^{31} + 108 q^{34} + 322 q^{36} + 112 q^{39} + 364 q^{41} - 112 q^{44} - 96 q^{46} - 98 q^{49} - 216 q^{51} + 200 q^{54} - 210 q^{56} - 1620 q^{59} - 976 q^{61} + 334 q^{64} - 32 q^{66} + 192 q^{69} - 1536 q^{71} - 492 q^{74} + 1540 q^{76} - 880 q^{79} + 842 q^{81} + 196 q^{84} - 256 q^{86} - 1460 q^{89} - 392 q^{91} + 648 q^{94} + 644 q^{96} - 368 q^{99}+O(q^{100}) \) 2 * q + 14 * q^4 + 4 * q^6 + 46 * q^9 - 16 * q^11 - 14 * q^14 + 82 * q^16 + 220 * q^19 + 28 * q^21 + 60 * q^24 - 56 * q^26 + 220 * q^29 + 24 * q^31 + 108 * q^34 + 322 * q^36 + 112 * q^39 + 364 * q^41 - 112 * q^44 - 96 * q^46 - 98 * q^49 - 216 * q^51 + 200 * q^54 - 210 * q^56 - 1620 * q^59 - 976 * q^61 + 334 * q^64 - 32 * q^66 + 192 * q^69 - 1536 * q^71 - 492 * q^74 + 1540 * q^76 - 880 * q^79 + 842 * q^81 + 196 * q^84 - 256 * q^86 - 1460 * q^89 - 392 * q^91 + 648 * q^94 + 644 * q^96 - 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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