LMFDB - Embedded newform 1617.2.c.b.538.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(538,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1617.538");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.c (of order $2$, degree $1$, 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:	$12.9118100068$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			538.18
Character	$\chi$	$=$	1617.538
Dual form			1617.2.c.b.538.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.03215i q^{2} +1.00000i q^{3} +0.934667 q^{4} +3.55499i q^{5} +1.03215 q^{6} -3.02902i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.03215i q^{2} +1.00000i q^{3} +0.934667 q^{4} +3.55499i q^{5} +1.03215 q^{6} -3.02902i q^{8} -1.00000 q^{9} +3.66928 q^{10} +(0.693101 - 3.24339i) q^{11} +0.934667i q^{12} +4.92468 q^{13} -3.55499 q^{15} -1.25706 q^{16} +5.73278 q^{17} +1.03215i q^{18} -2.80728 q^{19} +3.32274i q^{20} +(-3.34767 - 0.715384i) q^{22} +7.37952 q^{23} +3.02902 q^{24} -7.63797 q^{25} -5.08301i q^{26} -1.00000i q^{27} +9.68475i q^{29} +3.66928i q^{30} -7.44346i q^{31} -4.76056i q^{32} +(3.24339 + 0.693101i) q^{33} -5.91709i q^{34} -0.934667 q^{36} +0.0589861 q^{37} +2.89754i q^{38} +4.92468i q^{39} +10.7681 q^{40} -8.35421 q^{41} -6.43860i q^{43} +(0.647819 - 3.03150i) q^{44} -3.55499i q^{45} -7.61677i q^{46} +8.08175i q^{47} -1.25706i q^{48} +7.88352i q^{50} +5.73278i q^{51} +4.60294 q^{52} -3.55680 q^{53} -1.03215 q^{54} +(11.5302 + 2.46397i) q^{55} -2.80728i q^{57} +9.99611 q^{58} +9.88146i q^{59} -3.32274 q^{60} -6.54379 q^{61} -7.68276 q^{62} -7.42773 q^{64} +17.5072i q^{65} +(0.715384 - 3.34767i) q^{66} +5.87720 q^{67} +5.35824 q^{68} +7.37952i q^{69} +13.2728 q^{71} +3.02902i q^{72} +4.35922 q^{73} -0.0608825i q^{74} -7.63797i q^{75} -2.62388 q^{76} +5.08301 q^{78} +6.11303i q^{79} -4.46885i q^{80} +1.00000 q^{81} +8.62279i q^{82} -0.336875 q^{83} +20.3800i q^{85} -6.64559 q^{86} -9.68475 q^{87} +(-9.82429 - 2.09941i) q^{88} +3.49186i q^{89} -3.66928 q^{90} +6.89740 q^{92} +7.44346 q^{93} +8.34157 q^{94} -9.97987i q^{95} +4.76056 q^{96} +10.9128i q^{97} +(-0.693101 + 3.24339i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) $ 48 * q - 64 * q^4 - 48 * q^9	$ 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) $ 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1617\mathbb{Z}\right)^\times$.

$n$	$199$	$442$	$1079$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		−	1.03215i		−	0.729840i	−0.931039	−	0.364920i	$-0.881096\pi$
$2$		−	1.03215i		−	0.729840i	0.931039	−	0.364920i	$-0.118904\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	0.934667			0.467334
$5$			3.55499i			1.58984i	0.606714	+	0.794920i	$0.292487\pi$
$5$			3.55499i			1.58984i	−0.606714	+	0.794920i	$0.707513\pi$
$6$	1.03215			0.421373
$7$		0			0
$7$		0			0
$8$		−	3.02902i		−	1.07092i
$9$	−1.00000			−0.333333
$10$	3.66928			1.16033
$11$	0.693101	−	3.24339i	0.208978	−	0.977920i
$11$	0.693101	−	3.24339i	0.208978	−	0.977920i
$12$			0.934667i			0.269815i
$13$	4.92468			1.36586			0.682930	−	0.730483i	$-0.260705\pi$
$13$	4.92468			1.36586			0.682930	+	0.730483i	$0.260705\pi$
$14$		0			0
$15$	−3.55499			−0.917895
$16$	−1.25706			−0.314266
$17$	5.73278			1.39040			0.695202	−	0.718815i	$-0.255315\pi$
$17$	5.73278			1.39040			0.695202	+	0.718815i	$0.255315\pi$
$18$			1.03215i			0.243280i
$19$	−2.80728			−0.644035			−0.322017	−	0.946734i	$-0.604361\pi$
$19$	−2.80728			−0.644035			−0.322017	+	0.946734i	$0.604361\pi$
$20$			3.32274i			0.742986i
$21$		0			0
$22$	−3.34767	−	0.715384i	−0.713725	−	0.152520i
$23$	7.37952			1.53874			0.769368	−	0.638805i	$-0.220571\pi$
$23$	7.37952			1.53874			0.769368	+	0.638805i	$0.220571\pi$
$24$	3.02902			0.618295
$25$	−7.63797			−1.52759
$26$		−	5.08301i		−	0.996860i
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$			9.68475i			1.79841i	0.437524	+	0.899207i	$0.355855\pi$
$29$			9.68475i			1.79841i	−0.437524	+	0.899207i	$0.644145\pi$
$30$			3.66928i			0.669916i
$31$		−	7.44346i		−	1.33688i	−0.743764	−	0.668442i	$-0.766961\pi$
$31$		−	7.44346i		−	1.33688i	0.743764	−	0.668442i	$-0.233039\pi$
$32$		−	4.76056i		−	0.841555i
$33$	3.24339	+	0.693101i	0.564603	+	0.120653i
$34$		−	5.91709i		−	1.01477i
$35$		0			0
$36$	−0.934667			−0.155778
$37$	0.0589861			0.00969725			0.00484863	−	0.999988i	$-0.498457\pi$
$37$	0.0589861			0.00969725			0.00484863	+	0.999988i	$0.498457\pi$
$38$			2.89754i			0.470042i
$39$			4.92468i			0.788580i
$40$	10.7681			1.70259
$41$	−8.35421			−1.30471			−0.652354	−	0.757914i	$-0.726219\pi$
$41$	−8.35421			−1.30471			−0.652354	+	0.757914i	$0.726219\pi$
$42$		0			0
$43$		−	6.43860i		−	0.981877i	−0.871194	−	0.490938i	$-0.836654\pi$
$43$		−	6.43860i		−	0.981877i	0.871194	−	0.490938i	$-0.163346\pi$
$44$	0.647819	−	3.03150i	0.0976624	−	0.457015i
$45$		−	3.55499i		−	0.529947i
$46$		−	7.61677i		−	1.12303i
$47$			8.08175i			1.17884i	0.807825	+	0.589422i	$0.200644\pi$
$47$			8.08175i			1.17884i	−0.807825	+	0.589422i	$0.799356\pi$
$48$		−	1.25706i		−	0.181441i
$49$		0			0
$50$			7.88352i			1.11490i
$51$			5.73278i			0.802750i
$52$	4.60294			0.638313
$53$	−3.55680			−0.488564			−0.244282	−	0.969704i	$-0.578552\pi$
$53$	−3.55680			−0.488564			−0.244282	+	0.969704i	$0.578552\pi$
$54$	−1.03215			−0.140458
$55$	11.5302	+	2.46397i	1.55474	+	0.332241i
$56$		0			0
$57$		−	2.80728i		−	0.371834i
$58$	9.99611			1.31255
$59$			9.88146i			1.28646i	0.765674	+	0.643228i	$0.222405\pi$
$59$			9.88146i			1.28646i	−0.765674	+	0.643228i	$0.777595\pi$
$60$	−3.32274			−0.428963
$61$	−6.54379			−0.837846			−0.418923	−	0.908022i	$-0.637592\pi$
$61$	−6.54379			−0.837846			−0.418923	+	0.908022i	$0.637592\pi$
$62$	−7.68276			−0.975712
$63$		0			0
$64$	−7.42773			−0.928466
$65$			17.5072i			2.17150i
$66$	0.715384	−	3.34767i	0.0880577	−	0.412070i
$67$	5.87720			0.718014			0.359007	−	0.933335i	$-0.383115\pi$
$67$	5.87720			0.718014			0.359007	+	0.933335i	$0.383115\pi$
$68$	5.35824			0.649782
$69$			7.37952i			0.888390i
$70$		0			0
$71$	13.2728			1.57519			0.787597	−	0.616190i	$-0.211325\pi$
$71$	13.2728			1.57519			0.787597	+	0.616190i	$0.211325\pi$
$72$			3.02902i			0.356973i
$73$	4.35922			0.510209			0.255104	−	0.966914i	$-0.417890\pi$
$73$	4.35922			0.510209			0.255104	+	0.966914i	$0.417890\pi$
$74$		−	0.0608825i		−	0.00707744i
$75$		−	7.63797i		−	0.881957i
$76$	−2.62388			−0.300979
$77$		0			0
$78$	5.08301			0.575537
$79$			6.11303i			0.687770i	0.939012	+	0.343885i	$0.111743\pi$
$79$			6.11303i			0.687770i	−0.939012	+	0.343885i	$0.888257\pi$
$80$		−	4.46885i		−	0.499632i
$81$	1.00000			0.111111
$82$			8.62279i			0.952228i
$83$	−0.336875			−0.0369768			−0.0184884	−	0.999829i	$-0.505885\pi$
$83$	−0.336875			−0.0369768			−0.0184884	+	0.999829i	$0.505885\pi$
$84$		0			0
$85$			20.3800i			2.21052i
$86$	−6.64559			−0.716613
$87$	−9.68475			−1.03831
$88$	−9.82429	−	2.09941i	−1.04727	−	0.223798i
$89$			3.49186i			0.370137i	0.982726	+	0.185068i	$0.0592506\pi$
$89$			3.49186i			0.370137i	−0.982726	+	0.185068i	$0.940749\pi$
$90$	−3.66928			−0.386776
$91$		0			0
$92$	6.89740			0.719104
$93$	7.44346			0.771851
$94$	8.34157			0.860367
$95$		−	9.97987i		−	1.02391i
$96$	4.76056			0.485872
$97$			10.9128i			1.10802i	0.832509	+	0.554011i	$0.186904\pi$
$97$			10.9128i			1.10802i	−0.832509	+	0.554011i	$0.813096\pi$
$98$		0			0
$99$	−0.693101	+	3.24339i	−0.0696593	+	0.325973i
$100$	−7.13896			−0.713896
$101$	5.43207			0.540511			0.270255	−	0.962789i	$-0.412892\pi$
$101$	5.43207			0.540511			0.270255	+	0.962789i	$0.412892\pi$
$102$	5.91709			0.585879
$103$		−	0.722394i		−	0.0711796i	−0.999366	−	0.0355898i	$-0.988669\pi$
$103$		−	0.722394i		−	0.0711796i	0.999366	−	0.0355898i	$-0.0113310\pi$
$104$		−	14.9169i		−	1.46273i
$105$		0			0
$106$			3.67115i			0.356573i
$107$			9.52833i			0.921139i	0.887624	+	0.460569i	$0.152355\pi$
$107$			9.52833i			0.921139i	−0.887624	+	0.460569i	$0.847645\pi$
$108$		−	0.934667i		−	0.0899384i
$109$		−	9.47473i		−	0.907514i	−0.891125	−	0.453757i	$-0.850083\pi$
$109$		−	9.47473i		−	0.907514i	0.891125	−	0.453757i	$-0.149917\pi$
$110$	2.54318	−	11.9009i	0.242483	−	1.13471i
$111$			0.0589861i			0.00559871i
$112$		0			0
$113$	−14.5978			−1.37325			−0.686624	−	0.727012i	$-0.740908\pi$
$113$	−14.5978			−1.37325			−0.686624	+	0.727012i	$0.740908\pi$
$114$	−2.89754			−0.271379
$115$			26.2341i			2.44635i
$116$			9.05202i			0.840459i
$117$	−4.92468			−0.455287
$118$	10.1991			0.938907
$119$		0			0
$120$			10.7681i			0.982991i
$121$	−10.0392	−	4.49600i	−0.912657	−	0.408727i
$122$			6.75417i			0.611494i
$123$		−	8.35421i		−	0.753274i
$124$		−	6.95716i		−	0.624771i
$125$		−	9.37795i		−	0.838790i
$126$		0			0
$127$		−	9.65213i		−	0.856488i	−0.903663	−	0.428244i	$-0.859132\pi$
$127$		−	9.65213i		−	0.856488i	0.903663	−	0.428244i	$-0.140868\pi$
$128$		−	1.85458i		−	0.163924i
$129$	6.43860			0.566887
$130$	18.0700			1.58485
$131$	1.06713			0.0932359			0.0466179	−	0.998913i	$-0.485156\pi$
$131$	1.06713			0.0932359			0.0466179	+	0.998913i	$0.485156\pi$
$132$	3.03150	+	0.647819i	0.263858	+	0.0563854i
$133$		0			0
$134$		−	6.06615i		−	0.524035i
$135$	3.55499			0.305965
$136$		−	17.3647i		−	1.48901i
$137$	−6.31864			−0.539838			−0.269919	−	0.962883i	$-0.586997\pi$
$137$	−6.31864			−0.539838			−0.269919	+	0.962883i	$0.586997\pi$
$138$	7.61677			0.648383
$139$	4.18721			0.355155			0.177577	−	0.984107i	$-0.443174\pi$
$139$	4.18721			0.355155			0.177577	+	0.984107i	$0.443174\pi$
$140$		0			0
$141$	−8.08175			−0.680606
$142$		−	13.6995i		−	1.14964i
$143$	3.41330	−	15.9727i	0.285435	−	1.33570i
$144$	1.25706			0.104755
$145$	−34.4292			−2.85919
$146$		−	4.49937i		−	0.372371i
$147$		0			0
$148$	0.0551324			0.00453185
$149$			4.33826i			0.355404i	0.984084	+	0.177702i	$0.0568663\pi$
$149$			4.33826i			0.355404i	−0.984084	+	0.177702i	$0.943134\pi$
$150$	−7.88352			−0.643687
$151$		−	2.74724i		−	0.223567i	−0.993733	−	0.111784i	$-0.964344\pi$
$151$		−	2.74724i		−	0.223567i	0.993733	−	0.111784i	$-0.0356564\pi$
$152$			8.50330i			0.689709i
$153$	−5.73278			−0.463468
$154$		0			0
$155$	26.4614			2.12543
$156$			4.60294i			0.368530i
$157$		−	18.8232i		−	1.50226i	−0.660155	−	0.751129i	$-0.729510\pi$
$157$		−	18.8232i		−	1.50226i	0.660155	−	0.751129i	$-0.270490\pi$
$158$	6.30956			0.501962
$159$		−	3.55680i		−	0.282072i
$160$	16.9237			1.33794
$161$		0			0
$162$		−	1.03215i		−	0.0810933i
$163$	−8.55245			−0.669880			−0.334940	−	0.942240i	$-0.608716\pi$
$163$	−8.55245			−0.669880			−0.334940	+	0.942240i	$0.608716\pi$
$164$	−7.80841			−0.609734
$165$	−2.46397	+	11.5302i	−0.191820	+	0.897628i
$166$			0.347705i			0.0269872i
$167$	16.0911			1.24516			0.622582	−	0.782554i	$-0.286084\pi$
$167$	16.0911			1.24516			0.622582	+	0.782554i	$0.286084\pi$
$168$		0			0
$169$	11.2525			0.865576
$170$	21.0352			1.61333
$171$	2.80728			0.214678
$172$		−	6.01795i		−	0.458864i
$173$	−14.2046			−1.07996			−0.539979	−	0.841678i	$-0.681568\pi$
$173$	−14.2046			−1.07996			−0.539979	+	0.841678i	$0.681568\pi$
$174$			9.99611i			0.757803i
$175$		0			0
$176$	−0.871271	+	4.07715i	−0.0656745	+	0.307327i
$177$	−9.88146			−0.742736
$178$	3.60412			0.270140
$179$	−2.92970			−0.218976			−0.109488	−	0.993988i	$-0.534921\pi$
$179$	−2.92970			−0.218976			−0.109488	+	0.993988i	$0.534921\pi$
$180$		−	3.32274i		−	0.247662i
$181$		−	3.88933i		−	0.289092i	−0.989498	−	0.144546i	$-0.953828\pi$
$181$		−	3.88933i		−	0.289092i	0.989498	−	0.144546i	$-0.0461721\pi$
$182$		0			0
$183$		−	6.54379i		−	0.483731i
$184$		−	22.3527i		−	1.64786i
$185$			0.209695i			0.0154171i
$186$		−	7.68276i		−	0.563327i
$187$	3.97340	−	18.5937i	0.290563	−	1.35970i
$188$			7.55375i			0.550913i
$189$		0			0
$190$	−10.3007			−0.747293
$191$	−11.5840			−0.838190			−0.419095	−	0.907942i	$-0.637653\pi$
$191$	−11.5840			−0.838190			−0.419095	+	0.907942i	$0.637653\pi$
$192$		−	7.42773i		−	0.536050i
$193$		−	1.33967i		−	0.0964316i	−0.998837	−	0.0482158i	$-0.984646\pi$
$193$		−	1.33967i		−	0.0964316i	0.998837	−	0.0482158i	$-0.0153535\pi$
$194$	11.2636			0.808679
$195$	−17.5072			−1.25372
$196$		0			0
$197$		−	13.2114i		−	0.941270i	−0.882328	−	0.470635i	$-0.844025\pi$
$197$		−	13.2114i		−	0.941270i	0.882328	−	0.470635i	$-0.155975\pi$
$198$	3.34767	+	0.715384i	0.237908	+	0.0508401i
$199$		−	6.26044i		−	0.443791i	−0.975070	−	0.221895i	$-0.928776\pi$
$199$		−	6.26044i		−	0.443791i	0.975070	−	0.221895i	$-0.0712243\pi$
$200$			23.1355i			1.63593i
$201$			5.87720i			0.414546i
$202$		−	5.60670i		−	0.394486i
$203$		0			0
$204$			5.35824i			0.375152i
$205$		−	29.6992i		−	2.07428i
$206$	−0.745619			−0.0519497
$207$	−7.37952			−0.512912
$208$	−6.19063			−0.429243
$209$	−1.94573	+	9.10513i	−0.134589	+	0.629815i
$210$		0			0
$211$		−	3.25114i		−	0.223818i	−0.993718	−	0.111909i	$-0.964303\pi$
$211$		−	3.25114i		−	0.223818i	0.993718	−	0.111909i	$-0.0356965\pi$
$212$	−3.32442			−0.228322
$213$			13.2728i			0.909439i
$214$	9.83466			0.672284
$215$	22.8892			1.56103
$216$	−3.02902			−0.206098
$217$		0			0
$218$	−9.77933			−0.662340
$219$			4.35922i			0.294569i
$220$	10.7769	+	2.30299i	0.726581	+	0.155268i
$221$	28.2321			1.89910
$222$	0.0608825			0.00408616
$223$		−	4.33127i		−	0.290043i	−0.989428	−	0.145022i	$-0.953675\pi$
$223$		−	4.33127i		−	0.290043i	0.989428	−	0.145022i	$-0.0463251\pi$
$224$		0			0
$225$	7.63797			0.509198
$226$			15.0671i			1.00225i
$227$	−2.07969			−0.138034			−0.0690169	−	0.997615i	$-0.521986\pi$
$227$	−2.07969			−0.138034			−0.0690169	+	0.997615i	$0.521986\pi$
$228$		−	2.62388i		−	0.173770i
$229$		−	2.93787i		−	0.194140i	−0.995278	−	0.0970699i	$-0.969053\pi$
$229$		−	2.93787i		−	0.194140i	0.995278	−	0.0970699i	$-0.0309470\pi$
$230$	27.0776			1.78544
$231$		0			0
$232$	29.3353			1.92595
$233$			5.55073i			0.363641i	0.983332	+	0.181820i	$0.0581989\pi$
$233$			5.55073i			0.363641i	−0.983332	+	0.181820i	$0.941801\pi$
$234$			5.08301i			0.332287i
$235$	−28.7305			−1.87417
$236$			9.23588i			0.601205i
$237$	−6.11303			−0.397084
$238$		0			0
$239$		−	23.2458i		−	1.50365i	−0.659365	−	0.751823i	$-0.729175\pi$
$239$		−	23.2458i		−	1.50365i	0.659365	−	0.751823i	$-0.270825\pi$
$240$	4.46885			0.288463
$241$	13.5925			0.875571			0.437786	−	0.899079i	$-0.355763\pi$
$241$	13.5925			0.875571			0.437786	+	0.899079i	$0.355763\pi$
$242$	−4.64054	+	10.3620i	−0.298305	+	0.666093i
$243$			1.00000i			0.0641500i
$244$	−6.11627			−0.391554
$245$		0			0
$246$	−8.62279			−0.549769
$247$	−13.8250			−0.879662
$248$	−22.5464			−1.43169
$249$		−	0.336875i		−	0.0213486i
$250$	−9.67945			−0.612182
$251$		−	5.97859i		−	0.377365i	−0.982038	−	0.188683i	$-0.939578\pi$
$251$		−	5.97859i		−	0.377365i	0.982038	−	0.188683i	$-0.0604218\pi$
$252$		0			0
$253$	5.11475	−	23.9347i	0.321562	−	1.50476i
$254$	−9.96244			−0.625099
$255$	−20.3800			−1.27624
$256$	−16.7697			−1.04810
$257$		−	6.41238i		−	0.399994i	−0.979797	−	0.199997i	$-0.935907\pi$
$257$		−	6.41238i		−	0.399994i	0.979797	−	0.199997i	$-0.0640932\pi$
$258$		−	6.64559i		−	0.413737i
$259$		0			0
$260$			16.3634i			1.01482i
$261$		−	9.68475i		−	0.599471i
$262$		−	1.10144i		−	0.0680473i
$263$			12.3087i			0.758990i	0.925194	+	0.379495i	$0.123902\pi$
$263$			12.3087i			0.758990i	−0.925194	+	0.379495i	$0.876098\pi$
$264$	2.09941	−	9.82429i	0.129210	−	0.604643i
$265$		−	12.6444i		−	0.776738i
$266$		0			0
$267$	−3.49186			−0.213698
$268$	5.49322			0.335552
$269$			16.1799i			0.986504i	0.869887	+	0.493252i	$0.164192\pi$
$269$			16.1799i			0.986504i	−0.869887	+	0.493252i	$0.835808\pi$
$270$		−	3.66928i		−	0.223305i
$271$	−21.1208			−1.28300			−0.641498	−	0.767125i	$-0.721687\pi$
$271$	−21.1208			−1.28300			−0.641498	+	0.767125i	$0.721687\pi$
$272$	−7.20646			−0.436956
$273$		0			0
$274$			6.52178i			0.393995i
$275$	−5.29388	+	24.7729i	−0.319233	+	1.49386i
$276$			6.89740i			0.415175i
$277$			20.4296i			1.22749i	0.789503	+	0.613747i	$0.210338\pi$
$277$			20.4296i			1.22749i	−0.789503	+	0.613747i	$0.789662\pi$
$278$		−	4.32183i		−	0.259206i
$279$			7.44346i			0.445628i
$280$		0			0
$281$		−	31.2575i		−	1.86467i	−0.361599	−	0.932334i	$-0.617769\pi$
$281$		−	31.2575i		−	1.86467i	0.361599	−	0.932334i	$-0.382231\pi$
$282$			8.34157i			0.496733i
$283$	−12.8756			−0.765375			−0.382688	−	0.923878i	$-0.625001\pi$
$283$	−12.8756			−0.765375			−0.382688	+	0.923878i	$0.625001\pi$
$284$	12.4057			0.736141
$285$	9.97987			0.591156
$286$	−16.4862	−	3.52304i	−0.974849	−	0.208322i
$287$		0			0
$288$			4.76056i			0.280518i
$289$	15.8648			0.933222
$290$			35.5361i			2.08675i
$291$	−10.9128			−0.639717
$292$	4.07443			0.238438
$293$	−30.0461			−1.75531			−0.877656	−	0.479290i	$-0.840894\pi$
$293$	−30.0461			−1.75531			−0.877656	+	0.479290i	$0.840894\pi$
$294$		0			0
$295$	−35.1285			−2.04526
$296$		−	0.178670i		−	0.0103850i
$297$	−3.24339	−	0.693101i	−0.188201	−	0.0402178i
$298$	4.47773			0.259388
$299$	36.3418			2.10170
$300$		−	7.13896i		−	0.412168i
$301$		0			0
$302$	−2.83556			−0.163168
$303$			5.43207i			0.312064i
$304$	3.52893			0.202398
$305$		−	23.2631i		−	1.33204i
$306$			5.91709i			0.338257i
$307$	−29.2380			−1.66870			−0.834351	−	0.551233i	$-0.814157\pi$
$307$	−29.2380			−1.66870			−0.834351	+	0.551233i	$0.814157\pi$
$308$		0			0
$309$	0.722394			0.0410956
$310$		−	27.3122i		−	1.55123i
$311$		−	2.45932i		−	0.139455i	−0.997566	−	0.0697276i	$-0.977787\pi$
$311$		−	2.45932i		−	0.139455i	0.997566	−	0.0697276i	$-0.0222130\pi$
$312$	14.9169			0.844505
$313$			29.5027i			1.66759i	0.552074	+	0.833795i	$0.313837\pi$
$313$			29.5027i			1.66759i	−0.552074	+	0.833795i	$0.686163\pi$
$314$	−19.4284			−1.09641
$315$		0			0
$316$			5.71365i			0.321418i
$317$	14.7399			0.827878			0.413939	−	0.910305i	$-0.364153\pi$
$317$	14.7399			0.827878			0.413939	+	0.910305i	$0.364153\pi$
$318$	−3.67115			−0.205868
$319$	31.4115	+	6.71251i	1.75871	+	0.375828i
$320$		−	26.4055i		−	1.47611i
$321$	−9.52833			−0.531820
$322$		0			0
$323$	−16.0935			−0.895468
$324$	0.934667			0.0519260
$325$	−37.6146			−2.08648
$326$			8.82741i			0.488905i
$327$	9.47473			0.523954
$328$			25.3050i			1.39724i
$329$		0			0
$330$	11.9009	+	2.54318i	0.655125	+	0.139998i
$331$	14.1350			0.776933			0.388466	−	0.921463i	$-0.373005\pi$
$331$	14.1350			0.776933			0.388466	+	0.921463i	$0.373005\pi$
$332$	−0.314866			−0.0172805
$333$	−0.0589861			−0.00323242
$334$		−	16.6084i		−	0.908771i
$335$			20.8934i			1.14153i
$336$		0			0
$337$		−	6.90415i		−	0.376093i	−0.982160	−	0.188047i	$-0.939784\pi$
$337$		−	6.90415i		−	0.376093i	0.982160	−	0.188047i	$-0.0602156\pi$
$338$		−	11.6142i		−	0.631732i
$339$		−	14.5978i		−	0.792846i
$340$			19.0485i			1.03305i
$341$	−24.1421	−	5.15907i	−1.30737	−	0.279379i
$342$		−	2.89754i		−	0.156681i
$343$		0			0
$344$	−19.5026			−1.05151
$345$	−26.2341			−1.41240
$346$			14.6613i			0.788197i
$347$			11.5873i			0.622039i	0.950404	+	0.311019i	$0.100670\pi$
$347$			11.5873i			0.622039i	−0.950404	+	0.311019i	$0.899330\pi$
$348$	−9.05202			−0.485239
$349$	8.67091			0.464143			0.232072	−	0.972699i	$-0.425450\pi$
$349$	8.67091			0.464143			0.232072	+	0.972699i	$0.425450\pi$
$350$		0			0
$351$		−	4.92468i		−	0.262860i
$352$	−15.4404	−	3.29955i	−0.822974	−	0.175866i
$353$		−	23.9862i		−	1.27666i	−0.769764	−	0.638329i	$-0.779626\pi$
$353$		−	23.9862i		−	1.27666i	0.769764	−	0.638329i	$-0.220374\pi$
$354$			10.1991i			0.542078i
$355$			47.1848i			2.50431i
$356$			3.26373i			0.172977i
$357$		0			0
$358$			3.02388i			0.159817i
$359$		−	30.4151i		−	1.60525i	−0.596485	−	0.802624i	$-0.703437\pi$
$359$		−	30.4151i		−	1.60525i	0.596485	−	0.802624i	$-0.296563\pi$
$360$	−10.7681			−0.567530
$361$	−11.1192			−0.585219
$362$	−4.01437			−0.210991
$363$	4.49600	−	10.0392i	0.235979	−	0.526923i
$364$		0			0
$365$			15.4970i			0.811151i
$366$	−6.75417			−0.353046
$367$		−	6.74897i		−	0.352293i	−0.984364	−	0.176147i	$-0.943637\pi$
$367$		−	6.74897i		−	0.352293i	0.984364	−	0.176147i	$-0.0563633\pi$
$368$	−9.27652			−0.483572
$369$	8.35421			0.434903
$370$	0.216437			0.0112520
$371$		0			0
$372$	6.95716			0.360712
$373$		−	11.9964i		−	0.621152i	−0.950549	−	0.310576i	$-0.899478\pi$
$373$		−	11.9964i		−	0.621152i	0.950549	−	0.310576i	$-0.100522\pi$
$374$	−19.1914	−	4.10114i	−0.992366	−	0.212065i
$375$	9.37795			0.484275
$376$	24.4797			1.26245
$377$			47.6943i			2.45638i
$378$		0			0
$379$	−35.8538			−1.84168			−0.920842	−	0.389937i	$-0.872497\pi$
$379$	−35.8538			−1.84168			−0.920842	+	0.389937i	$0.872497\pi$
$380$		−	9.32786i		−	0.478509i
$381$	9.65213			0.494494
$382$			11.9564i			0.611745i
$383$			5.56647i			0.284433i	0.989836	+	0.142217i	$0.0454229\pi$
$383$			5.56647i			0.284433i	−0.989836	+	0.142217i	$0.954577\pi$
$384$	1.85458			0.0946413
$385$		0			0
$386$	−1.38274			−0.0703797
$387$			6.43860i			0.327292i
$388$			10.1998i			0.517816i
$389$	6.14741			0.311686			0.155843	−	0.987782i	$-0.450191\pi$
$389$	6.14741			0.311686			0.155843	+	0.987782i	$0.450191\pi$
$390$			18.0700i			0.915013i
$391$	42.3052			2.13947
$392$		0			0
$393$			1.06713i			0.0538298i
$394$	−13.6361			−0.686976
$395$	−21.7318			−1.09344
$396$	−0.647819	+	3.03150i	−0.0325541	+	0.152338i
$397$		−	3.13183i		−	0.157182i	−0.996907	−	0.0785911i	$-0.974958\pi$
$397$		−	3.13183i		−	0.157182i	0.996907	−	0.0785911i	$-0.0250421\pi$
$398$	−6.46171			−0.323896
$399$		0			0
$400$	9.60140			0.480070
$401$	30.9439			1.54526			0.772631	−	0.634855i	$-0.218940\pi$
$401$	30.9439			1.54526			0.772631	+	0.634855i	$0.218940\pi$
$402$	6.06615			0.302552
$403$		−	36.6567i		−	1.82600i
$404$	5.07717			0.252599
$405$			3.55499i			0.176649i
$406$		0			0
$407$	0.0408833	−	0.191315i	0.00202651	−	0.00948314i
$408$	17.3647			0.859680
$409$	9.44327			0.466939			0.233470	−	0.972364i	$-0.424992\pi$
$409$	9.44327			0.466939			0.233470	+	0.972364i	$0.424992\pi$
$410$	−30.6540			−1.51389
$411$		−	6.31864i		−	0.311676i
$412$		−	0.675199i		−	0.0332646i
$413$		0			0
$414$			7.61677i			0.374344i
$415$		−	1.19759i		−	0.0587873i
$416$		−	23.4442i		−	1.14945i
$417$			4.18721i			0.205049i
$418$	9.39785	+	2.00828i	0.459664	+	0.0982284i
$419$		−	10.1531i		−	0.496013i	−0.968758	−	0.248007i	$-0.920225\pi$
$419$		−	10.1531i		−	0.496013i	0.968758	−	0.248007i	$-0.0797755\pi$
$420$		0			0
$421$	18.0096			0.877732			0.438866	−	0.898553i	$-0.355380\pi$
$421$	18.0096			0.877732			0.438866	+	0.898553i	$0.355380\pi$
$422$	−3.35567			−0.163351
$423$		−	8.08175i		−	0.392948i
$424$			10.7736i			0.523212i
$425$	−43.7868			−2.12397
$426$	13.6995			0.663745
$427$		0			0
$428$			8.90582i			0.430479i
$429$	15.9727	+	3.41330i	0.771169	+	0.164796i
$430$		−	23.6250i		−	1.13930i
$431$		−	8.11136i		−	0.390711i	−0.980733	−	0.195355i	$-0.937414\pi$
$431$		−	8.11136i		−	0.390711i	0.980733	−	0.195355i	$-0.0625860\pi$
$432$			1.25706i			0.0604804i
$433$		−	18.7818i		−	0.902597i	−0.892373	−	0.451299i	$-0.850961\pi$
$433$		−	18.7818i		−	0.902597i	0.892373	−	0.451299i	$-0.149039\pi$
$434$		0			0
$435$		−	34.4292i		−	1.65075i
$436$		−	8.85572i		−	0.424112i
$437$	−20.7164			−0.991000
$438$	4.49937			0.214988
$439$	−19.8364			−0.946739			−0.473370	−	0.880864i	$-0.656962\pi$
$439$	−19.8364			−0.946739			−0.473370	+	0.880864i	$0.656962\pi$
$440$	7.46340	−	34.9253i	0.355804	−	1.66500i
$441$		0			0
$442$		−	29.1398i		−	1.38604i
$443$	17.6583			0.838972			0.419486	−	0.907762i	$-0.362210\pi$
$443$	17.6583			0.838972			0.419486	+	0.907762i	$0.362210\pi$
$444$			0.0551324i			0.00261647i
$445$	−12.4135			−0.588458
$446$	−4.47051			−0.211685
$447$	−4.33826			−0.205192
$448$		0			0
$449$	25.8667			1.22073			0.610363	−	0.792122i	$-0.291024\pi$
$449$	25.8667			1.22073			0.610363	+	0.792122i	$0.291024\pi$
$450$		−	7.88352i		−	0.371633i
$451$	−5.79031	+	27.0960i	−0.272655	+	1.27590i
$452$	−13.6441			−0.641765
$453$	2.74724			0.129077
$454$			2.14655i			0.100743i
$455$		0			0
$456$	−8.50330			−0.398204
$457$		−	18.3825i		−	0.859899i	−0.902853	−	0.429949i	$-0.858531\pi$
$457$		−	18.3825i		−	0.859899i	0.902853	−	0.429949i	$-0.141469\pi$
$458$	−3.03232			−0.141691
$459$		−	5.73278i		−	0.267583i
$460$			24.5202i			1.14326i
$461$	−35.7261			−1.66393			−0.831964	−	0.554830i	$-0.812783\pi$
$461$	−35.7261			−1.66393			−0.831964	+	0.554830i	$0.812783\pi$
$462$		0			0
$463$	15.0201			0.698045			0.349023	−	0.937114i	$-0.386514\pi$
$463$	15.0201			0.698045			0.349023	+	0.937114i	$0.386514\pi$
$464$		−	12.1743i		−	0.565179i
$465$			26.4614i			1.22712i
$466$	5.72919			0.265399
$467$		−	0.682589i		−	0.0315865i	−0.999875	−	0.0157932i	$-0.994973\pi$
$467$		−	0.682589i		−	0.0315865i	0.999875	−	0.0157932i	$-0.00502735\pi$
$468$	−4.60294			−0.212771
$469$		0			0
$470$			29.6542i			1.36785i
$471$	18.8232			0.867329
$472$	29.9311			1.37769
$473$	−20.8829	−	4.46260i	−0.960197	−	0.205190i
$474$			6.30956i			0.289808i
$475$	21.4419			0.983824
$476$		0			0
$477$	3.55680			0.162855
$478$	−23.9931			−1.09742
$479$	26.4593			1.20896			0.604478	−	0.796622i	$-0.293382\pi$
$479$	26.4593			1.20896			0.604478	+	0.796622i	$0.293382\pi$
$480$			16.9237i			0.772459i
$481$	0.290488			0.0132451
$482$		−	14.0295i		−	0.639027i
$483$		0			0
$484$	−9.38333	−	4.20226i	−0.426515	−	0.191012i
$485$	−38.7948			−1.76158
$486$	1.03215			0.0468193
$487$	−9.23092			−0.418293			−0.209146	−	0.977884i	$-0.567069\pi$
$487$	−9.23092			−0.418293			−0.209146	+	0.977884i	$0.567069\pi$
$488$			19.8212i			0.897265i
$489$		−	8.55245i		−	0.386755i
$490$		0			0
$491$			7.16558i			0.323378i	0.986842	+	0.161689i	$0.0516942\pi$
$491$			7.16558i			0.323378i	−0.986842	+	0.161689i	$0.948306\pi$
$492$		−	7.80841i		−	0.352030i
$493$			55.5206i			2.50052i
$494$			14.2694i			0.642012i
$495$	−11.5302	−	2.46397i	−0.518246	−	0.110747i
$496$			9.35689i			0.420137i
$497$		0			0
$498$	−0.347705			−0.0155811
$499$	16.1520			0.723064			0.361532	−	0.932360i	$-0.382254\pi$
$499$	16.1520			0.723064			0.361532	+	0.932360i	$0.382254\pi$
$500$		−	8.76527i		−	0.391995i
$501$			16.0911i			0.718896i
$502$	−6.17080			−0.275416
$503$	−7.03825			−0.313820			−0.156910	−	0.987613i	$-0.550153\pi$
$503$	−7.03825			−0.313820			−0.156910	+	0.987613i	$0.550153\pi$
$504$		0			0
$505$			19.3109i			0.859326i
$506$	−24.7042	−	5.27919i	−1.09824	−	0.234689i
$507$			11.2525i			0.499740i
$508$		−	9.02153i		−	0.400266i
$509$			31.0208i			1.37497i	0.726198	+	0.687486i	$0.241286\pi$
$509$			31.0208i			1.37497i	−0.726198	+	0.687486i	$0.758714\pi$
$510$			21.0352i			0.931454i
$511$		0			0
$512$			13.5996i			0.601025i
$513$			2.80728i			0.123945i
$514$	−6.61854			−0.291931
$515$	2.56811			0.113164
$516$	6.01795			0.264925
$517$	26.2123	+	5.60147i	1.15282	+	0.246352i
$518$		0			0
$519$		−	14.2046i		−	0.623514i
$520$	53.0296			2.32550
$521$		−	9.81384i		−	0.429952i	−0.976619	−	0.214976i	$-0.931033\pi$
$521$		−	9.81384i		−	0.429952i	0.976619	−	0.214976i	$-0.0689674\pi$
$522$	−9.99611			−0.437518
$523$	−20.7562			−0.907605			−0.453802	−	0.891102i	$-0.649933\pi$
$523$	−20.7562			−0.907605			−0.453802	+	0.891102i	$0.649933\pi$
$524$	0.997415			0.0435723
$525$		0			0
$526$	12.7045			0.553941
$527$		−	42.6717i		−	1.85881i
$528$	−4.07715	−	0.871271i	−0.177435	−	0.0379172i
$529$	31.4574			1.36771
$530$	−13.0509			−0.566895
$531$		−	9.88146i		−	0.428819i
$532$		0			0
$533$	−41.1418			−1.78205
$534$			3.60412i			0.155966i
$535$	−33.8731			−1.46446
$536$		−	17.8021i		−	0.768935i
$537$		−	2.92970i		−	0.126426i
$538$	16.7000			0.719990
$539$		0			0
$540$	3.32274			0.142988
$541$			26.6672i			1.14651i	0.819377	+	0.573256i	$0.194320\pi$
$541$			26.6672i			1.14651i	−0.819377	+	0.573256i	$0.805680\pi$
$542$			21.7998i			0.936382i
$543$	3.88933			0.166907
$544$		−	27.2912i		−	1.17010i
$545$	33.6826			1.44280
$546$		0			0
$547$		−	2.75176i		−	0.117657i	−0.998268	−	0.0588283i	$-0.981264\pi$
$547$		−	2.75176i		−	0.117657i	0.998268	−	0.0588283i	$-0.0187365\pi$
$548$	−5.90583			−0.252284
$549$	6.54379			0.279282
$550$	25.5694	+	5.46408i	1.09028	+	0.232989i
$551$		−	27.1878i		−	1.15824i
$552$	22.3527			0.951394
$553$		0			0
$554$	21.0864			0.895873
$555$	−0.209695			−0.00890106
$556$	3.91365			0.165976
$557$			3.15847i			0.133829i	0.997759	+	0.0669144i	$0.0213154\pi$
$557$			3.15847i			0.133829i	−0.997759	+	0.0669144i	$0.978685\pi$
$558$	7.68276			0.325237
$559$		−	31.7080i		−	1.34111i
$560$		0			0
$561$	18.5937	+	3.97340i	0.785025	+	0.167757i
$562$	−32.2624			−1.36091
$563$	−28.7892			−1.21332			−0.606659	−	0.794962i	$-0.707491\pi$
$563$	−28.7892			−1.21332			−0.606659	+	0.794962i	$0.707491\pi$
$564$	−7.55375			−0.318070
$565$		−	51.8952i		−	2.18325i
$566$			13.2896i			0.558601i
$567$		0			0
$568$		−	40.2036i		−	1.68691i
$569$		−	12.8534i		−	0.538842i	−0.963023	−	0.269421i	$-0.913168\pi$
$569$		−	12.8534i		−	0.538842i	0.963023	−	0.269421i	$-0.0868323\pi$
$570$		−	10.3007i		−	0.431450i
$571$			47.1019i			1.97115i	0.169225	+	0.985577i	$0.445873\pi$
$571$			47.1019i			1.97115i	−0.169225	+	0.985577i	$0.554127\pi$
$572$	3.19030	−	14.9291i	0.133393	−	0.624219i
$573$		−	11.5840i		−	0.483930i
$574$		0			0
$575$	−56.3646			−2.35056
$576$	7.42773			0.309489
$577$		−	0.387059i		−	0.0161135i	−0.999968	−	0.00805674i	$-0.997435\pi$
$577$		−	0.387059i		−	0.0161135i	0.999968	−	0.00805674i	$-0.00256457\pi$
$578$		−	16.3748i		−	0.681103i
$579$	1.33967			0.0556748
$580$	−32.1799			−1.33620
$581$		0			0
$582$			11.2636i			0.466891i
$583$	−2.46522	+	11.5361i	−0.102099	+	0.477776i
$584$		−	13.2042i		−	0.546392i
$585$		−	17.5072i		−	0.723834i
$586$			31.0121i			1.28110i
$587$			4.12417i			0.170223i	0.996371	+	0.0851113i	$0.0271246\pi$
$587$			4.12417i			0.170223i	−0.996371	+	0.0851113i	$0.972875\pi$
$588$		0			0
$589$			20.8959i			0.861000i
$590$			36.2579i			1.49271i
$591$	13.2114			0.543442
$592$	−0.0741492			−0.00304751
$593$	16.3866			0.672916			0.336458	−	0.941699i	$-0.390771\pi$
$593$	16.3866			0.672916			0.336458	+	0.941699i	$0.390771\pi$
$594$	−0.715384	+	3.34767i	−0.0293526	+	0.137357i
$595$		0			0
$596$			4.05483i			0.166092i
$597$	6.26044			0.256223
$598$		−	37.5102i		−	1.53390i
$599$	−21.5077			−0.878781			−0.439390	−	0.898296i	$-0.644805\pi$
$599$	−21.5077			−0.878781			−0.439390	+	0.898296i	$0.644805\pi$
$600$	−23.1355			−0.944504
$601$	−21.0378			−0.858150			−0.429075	−	0.903269i	$-0.641160\pi$
$601$	−21.0378			−0.858150			−0.429075	+	0.903269i	$0.641160\pi$
$602$		0			0
$603$	−5.87720			−0.239338
$604$		−	2.56776i		−	0.104481i
$605$	15.9832	−	35.6894i	0.649811	−	1.45098i
$606$	5.60670			0.227757
$607$	5.88685			0.238940			0.119470	−	0.992838i	$-0.461880\pi$
$607$	5.88685			0.238940			0.119470	+	0.992838i	$0.461880\pi$
$608$			13.3642i			0.541991i
$609$		0			0
$610$	−24.0110			−0.972178
$611$			39.8000i			1.61014i
$612$	−5.35824			−0.216594
$613$		−	36.3517i		−	1.46823i	−0.679024	−	0.734116i	$-0.737597\pi$
$613$		−	36.3517i		−	1.46823i	0.679024	−	0.734116i	$-0.262403\pi$
$614$			30.1780i			1.21789i
$615$	29.6992			1.19759
$616$		0			0
$617$	−34.1281			−1.37394			−0.686972	−	0.726684i	$-0.741061\pi$
$617$	−34.1281			−1.37394			−0.686972	+	0.726684i	$0.741061\pi$
$618$		−	0.745619i		−	0.0299932i
$619$		−	21.6187i		−	0.868929i	−0.900689	−	0.434465i	$-0.856938\pi$
$619$		−	21.6187i		−	0.868929i	0.900689	−	0.434465i	$-0.143062\pi$
$620$	24.7326			0.993287
$621$		−	7.37952i		−	0.296130i
$622$	−2.53838			−0.101780
$623$		0			0
$624$		−	6.19063i		−	0.247824i
$625$	−4.85129			−0.194051
$626$	30.4512			1.21707
$627$	−9.10513	−	1.94573i	−0.363624	−	0.0777050i
$628$		−	17.5935i		−	0.702056i
$629$	0.338154			0.0134831
$630$		0			0
$631$	21.6861			0.863311			0.431656	−	0.902039i	$-0.357930\pi$
$631$	21.6861			0.863311			0.431656	+	0.902039i	$0.357930\pi$
$632$	18.5165			0.736545
$633$	3.25114			0.129221
$634$		−	15.2138i		−	0.604218i
$635$	34.3132			1.36168
$636$		−	3.32442i		−	0.131822i
$637$		0			0
$638$	6.92831	−	32.4213i	0.274295	−	1.28357i
$639$	−13.2728			−0.525065
$640$	6.59303			0.260612
$641$	30.9477			1.22236			0.611181	−	0.791491i	$-0.290695\pi$
$641$	30.9477			1.22236			0.611181	+	0.791491i	$0.290695\pi$
$642$			9.83466i			0.388143i
$643$			6.12149i			0.241408i	0.992689	+	0.120704i	$0.0385152\pi$
$643$			6.12149i			0.241408i	−0.992689	+	0.120704i	$0.961485\pi$
$644$		0			0
$645$			22.8892i			0.901260i
$646$			16.6109i			0.653549i
$647$			42.5032i			1.67097i	0.549512	+	0.835486i	$0.314814\pi$
$647$			42.5032i			1.67097i	−0.549512	+	0.835486i	$0.685186\pi$
$648$		−	3.02902i		−	0.118991i
$649$	32.0495	+	6.84885i	1.25805	+	0.268841i
$650$			38.8238i			1.52280i
$651$		0			0
$652$	−7.99370			−0.313057
$653$	−45.9359			−1.79761			−0.898805	−	0.438348i	$-0.855564\pi$
$653$	−45.9359			−1.79761			−0.898805	+	0.438348i	$0.855564\pi$
$654$		−	9.77933i		−	0.382402i
$655$			3.79365i			0.148230i
$656$	10.5018			0.410025
$657$	−4.35922			−0.170070
$658$		0			0
$659$			26.1835i			1.01997i	0.860185	+	0.509983i	$0.170348\pi$
$659$			26.1835i			1.01997i	−0.860185	+	0.509983i	$0.829652\pi$
$660$	−2.30299	+	10.7769i	−0.0896438	+	0.419492i
$661$			1.27192i			0.0494720i	0.999694	+	0.0247360i	$0.00787451\pi$
$661$			1.27192i			0.0494720i	−0.999694	+	0.0247360i	$0.992125\pi$
$662$		−	14.5895i		−	0.567036i
$663$			28.2321i			1.09644i
$664$			1.02040i			0.0395992i
$665$		0			0
$666$			0.0608825i			0.00235915i
$667$			71.4689i			2.76729i
$668$	15.0398			0.581907
$669$	4.33127			0.167456
$670$	21.5651			0.833133
$671$	−4.53551	+	21.2241i	−0.175091	+	0.819347i
$672$		0			0
$673$			41.9428i			1.61677i	0.588651	+	0.808387i	$0.299659\pi$
$673$			41.9428i			1.61677i	−0.588651	+	0.808387i	$0.700341\pi$
$674$	−7.12612			−0.274488
$675$			7.63797i			0.293986i
$676$	10.5173			0.404513
$677$	−49.0501			−1.88515			−0.942574	−	0.333999i	$-0.891602\pi$
$677$	−49.0501			−1.88515			−0.942574	+	0.333999i	$0.891602\pi$
$678$	−15.0671			−0.578650
$679$		0			0
$680$	61.7313			2.36729
$681$		−	2.07969i		−	0.0796938i
$682$	−5.32493	+	24.9182i	−0.203902	+	0.954168i
$683$	38.4421			1.47095			0.735474	−	0.677553i	$-0.236960\pi$
$683$	38.4421			1.47095			0.735474	+	0.677553i	$0.236960\pi$
$684$	2.62388			0.100326
$685$		−	22.4627i		−	0.858256i
$686$		0			0
$687$	2.93787			0.112087
$688$			8.09371i			0.308570i
$689$	−17.5161			−0.667310
$690$			27.0776i			1.03083i
$691$		−	48.9326i		−	1.86148i	−0.365676	−	0.930742i	$-0.619162\pi$
$691$		−	48.9326i		−	1.86148i	0.365676	−	0.930742i	$-0.380838\pi$
$692$	−13.2766			−0.504701
$693$		0			0
$694$	11.9598			0.453989
$695$			14.8855i			0.564640i
$696$			29.3353i			1.11195i
$697$	−47.8929			−1.81407
$698$		−	8.94968i		−	0.338750i
$699$	−5.55073			−0.209948
$700$		0			0
$701$			8.38544i			0.316714i	0.987382	+	0.158357i	$0.0506196\pi$
$701$			8.38544i			0.316714i	−0.987382	+	0.158357i	$0.949380\pi$
$702$	−5.08301			−0.191846
$703$	−0.165591			−0.00624537
$704$	−5.14817	+	24.0911i	−0.194029	+	0.907966i
$705$		−	28.7305i		−	1.08205i
$706$	−24.7574			−0.931756
$707$		0			0
$708$	−9.23588			−0.347106
$709$	37.1614			1.39563			0.697813	−	0.716280i	$-0.254157\pi$
$709$	37.1614			1.39563			0.697813	+	0.716280i	$0.254157\pi$
$710$	48.7018			1.82774
$711$		−	6.11303i		−	0.229257i
$712$	10.5769			0.396386
$713$		−	54.9292i		−	2.05711i
$714$		0			0
$715$	56.7828	+	12.1343i	2.12356	+	0.453796i
$716$	−2.73829			−0.102335
$717$	23.2458			0.868130
$718$	−31.3929			−1.17157
$719$		−	19.8934i		−	0.741899i	−0.928653	−	0.370950i	$-0.879032\pi$
$719$		−	19.8934i		−	0.741899i	0.928653	−	0.370950i	$-0.120968\pi$
$720$			4.46885i			0.166544i
$721$		0			0
$722$			11.4766i			0.427116i
$723$			13.5925i			0.505511i
$724$		−	3.63523i		−	0.135102i
$725$		−	73.9718i		−	2.74724i
$726$	−10.3620	−	4.64054i	−0.384569	−	0.172227i
$727$			31.6817i			1.17501i	0.809220	+	0.587505i	$0.199890\pi$
$727$			31.6817i			1.17501i	−0.809220	+	0.587505i	$0.800110\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	15.9952			0.592010
$731$		−	36.9111i		−	1.36520i
$732$		−	6.11627i		−	0.226064i
$733$	34.7777			1.28455			0.642273	−	0.766476i	$-0.277992\pi$
$733$	34.7777			1.28455			0.642273	+	0.766476i	$0.277992\pi$
$734$	−6.96595			−0.257118
$735$		0			0
$736$		−	35.1306i		−	1.29493i
$737$	4.07349	−	19.0621i	0.150049	−	0.702160i
$738$		−	8.62279i		−	0.317409i
$739$			45.0616i			1.65762i	0.559532	+	0.828809i	$0.310981\pi$
$739$			45.0616i			1.65762i	−0.559532	+	0.828809i	$0.689019\pi$
$740$			0.195995i			0.00720492i
$741$		−	13.8250i		−	0.507873i
$742$		0			0
$743$		−	25.9489i		−	0.951975i	−0.879452	−	0.475987i	$-0.842091\pi$
$743$		−	25.9489i		−	0.951975i	0.879452	−	0.475987i	$-0.157909\pi$
$744$		−	22.5464i		−	0.826589i
$745$	−15.4225			−0.565035
$746$	−12.3821			−0.453341
$747$	0.336875			0.0123256
$748$	3.71380	−	17.3789i	0.135790	−	0.635435i
$749$		0			0
$750$		−	9.67945i		−	0.353444i
$751$	12.3170			0.449454			0.224727	−	0.974422i	$-0.427851\pi$
$751$	12.3170			0.449454			0.224727	+	0.974422i	$0.427851\pi$
$752$		−	10.1593i		−	0.370470i
$753$	5.97859			0.217872
$754$	49.2277			1.79277
$755$	9.76642			0.355437
$756$		0			0
$757$	25.7311			0.935212			0.467606	−	0.883937i	$-0.345117\pi$
$757$	25.7311			0.935212			0.467606	+	0.883937i	$0.345117\pi$
$758$			37.0064i			1.34413i
$759$	23.9347	+	5.11475i	0.868775	+	0.185654i
$760$	−30.2292			−1.09653
$761$	−6.17012			−0.223667			−0.111833	−	0.993727i	$-0.535672\pi$
$761$	−6.17012			−0.223667			−0.111833	+	0.993727i	$0.535672\pi$
$762$		−	9.96244i		−	0.360901i
$763$		0			0
$764$	−10.8272			−0.391715
$765$		−	20.3800i		−	0.736840i
$766$	5.74543			0.207591
$767$			48.6630i			1.75712i
$768$		−	16.7697i		−	0.605123i
$769$	5.02245			0.181114			0.0905571	−	0.995891i	$-0.471135\pi$
$769$	5.02245			0.181114			0.0905571	+	0.995891i	$0.471135\pi$
$770$		0			0
$771$	6.41238			0.230936
$772$		−	1.25215i		−	0.0450658i
$773$		−	4.47728i		−	0.161037i	−0.996753	−	0.0805183i	$-0.974342\pi$
$773$		−	4.47728i		−	0.161037i	0.996753	−	0.0805183i	$-0.0256575\pi$
$774$	6.64559			0.238871
$775$			56.8529i			2.04222i
$776$	33.0549			1.18660
$777$		0			0
$778$		−	6.34505i		−	0.227481i
$779$	23.4526			0.840278
$780$	−16.3634			−0.585904
$781$	9.19941	−	43.0490i	0.329181	−	1.54041i
$782$		−	43.6653i		−	1.56147i
$783$	9.68475			0.346105
$784$		0			0
$785$	66.9165			2.38835
$786$	1.10144			0.0392871
$787$	22.6798			0.808447			0.404224	−	0.914660i	$-0.367542\pi$
$787$	22.6798			0.808447			0.404224	+	0.914660i	$0.367542\pi$
$788$		−	12.3482i		−	0.439887i
$789$	−12.3087			−0.438203
$790$			22.4304i			0.798039i
$791$		0			0
$792$	9.82429	+	2.09941i	0.349091	+	0.0745994i
$793$	−32.2261			−1.14438
$794$	−3.23252			−0.114718
$795$	12.6444			0.448450
$796$		−	5.85143i		−	0.207398i
$797$			24.0188i			0.850790i	0.905008	+	0.425395i	$0.139865\pi$
$797$			24.0188i			0.850790i	−0.905008	+	0.425395i	$0.860135\pi$
$798$		0			0
$799$			46.3309i			1.63907i
$800$			36.3610i			1.28555i
$801$		−	3.49186i		−	0.123379i
$802$		−	31.9387i		−	1.12779i
$803$	3.02138	−	14.1387i	0.106622	−	0.498943i
$804$			5.49322i			0.193731i
$805$		0			0
$806$	−37.8352			−1.33269
$807$	−16.1799			−0.569558
$808$		−	16.4538i		−	0.578843i
$809$			23.0431i			0.810153i	0.914283	+	0.405076i	$0.132755\pi$
$809$			23.0431i			0.810153i	−0.914283	+	0.405076i	$0.867245\pi$
$810$	3.66928			0.128925
$811$	−13.7970			−0.484477			−0.242238	−	0.970217i	$-0.577882\pi$
$811$	−13.7970			−0.484477			−0.242238	+	0.970217i	$0.577882\pi$
$812$		0			0
$813$		−	21.1208i		−	0.740738i
$814$	−0.197466	−	0.0421977i	−0.00692118	−	0.00147903i
$815$		−	30.4039i		−	1.06500i
$816$		−	7.20646i		−	0.252277i
$817$			18.0750i			0.632363i
$818$		−	9.74686i		−	0.340791i
$819$		0			0
$820$		−	27.7588i		−	0.969380i
$821$		−	9.94000i		−	0.346908i	−0.984842	−	0.173454i	$-0.944507\pi$
$821$		−	9.94000i		−	0.346908i	0.984842	−	0.173454i	$-0.0554928\pi$
$822$	−6.52178			−0.227473
$823$	20.8213			0.725783			0.362892	−	0.931831i	$-0.381789\pi$
$823$	20.8213			0.725783			0.362892	+	0.931831i	$0.381789\pi$
$824$	−2.18814			−0.0762276
$825$	−24.7729	−	5.29388i	−0.862483	−	0.184309i
$826$		0			0
$827$		−	25.0049i		−	0.869506i	−0.900550	−	0.434753i	$-0.856836\pi$
$827$		−	25.0049i		−	0.869506i	0.900550	−	0.434753i	$-0.143164\pi$
$828$	−6.89740			−0.239701
$829$		−	46.0661i		−	1.59994i	−0.600040	−	0.799970i	$-0.704849\pi$
$829$		−	46.0661i		−	1.59994i	0.600040	−	0.799970i	$-0.295151\pi$
$830$	−1.23609			−0.0429053
$831$	−20.4296			−0.708693
$832$	−36.5792			−1.26816
$833$		0			0
$834$	4.32183			0.149653
$835$			57.2036i			1.97961i
$836$	−1.81861	+	8.51027i	−0.0628980	+	0.294334i
$837$	−7.44346			−0.257284
$838$	−10.4796			−0.362010
$839$		−	54.5999i		−	1.88500i	−0.334209	−	0.942499i	$-0.608469\pi$
$839$		−	54.5999i		−	1.88500i	0.334209	−	0.942499i	$-0.391531\pi$
$840$		0			0
$841$	−64.7944			−2.23429
$842$		−	18.5886i		−	0.640604i
$843$	31.2575			1.07657
$844$		−	3.03874i		−	0.104598i
$845$			40.0025i			1.37613i
$846$	−8.34157			−0.286789
$847$		0			0
$848$	4.47111			0.153539
$849$		−	12.8756i		−	0.441890i
$850$			45.1945i			1.55016i
$851$	0.435289			0.0149215
$852$			12.4057i			0.425011i
$853$	−37.1893			−1.27334			−0.636668	−	0.771138i	$-0.719688\pi$
$853$	−37.1893			−1.27334			−0.636668	+	0.771138i	$0.719688\pi$
$854$		0			0
$855$			9.97987i			0.341304i
$856$	28.8615			0.986465
$857$	−19.1553			−0.654332			−0.327166	−	0.944967i	$-0.606094\pi$
$857$	−19.1553			−0.654332			−0.327166	+	0.944967i	$0.606094\pi$
$858$	3.52304	−	16.4862i	0.120274	−	0.562830i
$859$		−	54.1421i		−	1.84730i	−0.383232	−	0.923652i	$-0.625189\pi$
$859$		−	54.1421i		−	1.84730i	0.383232	−	0.923652i	$-0.374811\pi$
$860$	21.3937			0.729521
$861$		0			0
$862$	−8.37214			−0.285156
$863$	0.667003			0.0227050			0.0113525	−	0.999936i	$-0.496386\pi$
$863$	0.667003			0.0227050			0.0113525	+	0.999936i	$0.496386\pi$
$864$	−4.76056			−0.161957
$865$		−	50.4974i		−	1.71696i
$866$	−19.3857			−0.658752
$867$			15.8648i			0.538796i
$868$		0			0
$869$	19.8270	+	4.23695i	0.672584	+	0.143729i
$870$	−35.5361			−1.20479
$871$	28.9433			0.980707
$872$	−28.6991			−0.971874
$873$		−	10.9128i		−	0.369341i
$874$			21.3824i			0.723272i
$875$		0			0
$876$			4.07443i			0.137662i
$877$			40.1831i			1.35689i	0.734653	+	0.678443i	$0.237345\pi$
$877$			40.1831i			1.35689i	−0.734653	+	0.678443i	$0.762655\pi$
$878$			20.4741i			0.690968i
$879$		−	30.0461i		−	1.01343i
$880$	−14.4942	−	3.09736i	−0.488600	−	0.104412i
$881$			8.49844i			0.286320i	0.989700	+	0.143160i	$0.0457263\pi$
$881$			8.49844i			0.286320i	−0.989700	+	0.143160i	$0.954274\pi$
$882$		0			0
$883$	19.3467			0.651069			0.325534	−	0.945530i	$-0.394456\pi$
$883$	19.3467			0.651069			0.325534	+	0.945530i	$0.394456\pi$
$884$	26.3876			0.887512
$885$		−	35.1285i		−	1.18083i
$886$		−	18.2260i		−	0.612315i
$887$	33.8313			1.13594			0.567972	−	0.823048i	$-0.307728\pi$
$887$	33.8313			1.13594			0.567972	+	0.823048i	$0.307728\pi$
$888$	0.178670			0.00599577
$889$		0			0
$890$			12.8126i			0.429480i
$891$	0.693101	−	3.24339i	0.0232198	−	0.108658i
$892$		−	4.04829i		−	0.135547i
$893$		−	22.6878i		−	0.759217i
$894$			4.47773i			0.149758i
$895$		−	10.4150i		−	0.348137i
$896$		0			0
$897$			36.3418i			1.21342i
$898$		−	26.6983i		−	0.890935i
$899$	72.0881			2.40427
$900$	7.13896			0.237965
$901$	−20.3903			−0.679301
$902$	27.9671	+	5.97647i	0.931204	+	0.198995i
$903$		0			0
$904$			44.2171i			1.47064i
$905$	13.8265			0.459610
$906$		−	2.83556i		−	0.0942053i
$907$	6.84469			0.227274			0.113637	−	0.993522i	$-0.463750\pi$
$907$	6.84469			0.227274			0.113637	+	0.993522i	$0.463750\pi$
$908$	−1.94382			−0.0645078
$909$	−5.43207			−0.180170
$910$		0			0
$911$	3.51008			0.116294			0.0581471	−	0.998308i	$-0.481481\pi$
$911$	3.51008			0.116294			0.0581471	+	0.998308i	$0.481481\pi$
$912$			3.52893i			0.116855i
$913$	−0.233488	+	1.09262i	−0.00772734	+	0.0361604i
$914$	−18.9735			−0.627588
$915$	23.2631			0.769055
$916$		−	2.74593i		−	0.0907281i
$917$		0			0
$918$	−5.91709			−0.195293
$919$		−	42.6745i		−	1.40770i	−0.710348	−	0.703850i	$-0.751463\pi$
$919$		−	42.6745i		−	1.40770i	0.710348	−	0.703850i	$-0.248537\pi$
$920$	79.4636			2.61984
$921$		−	29.2380i		−	0.963426i
$922$			36.8746i			1.21440i
$923$	65.3644			2.15150
$924$		0			0
$925$	−0.450534			−0.0148135
$926$		−	15.5030i		−	0.509461i
$927$			0.722394i			0.0237265i
$928$	46.1048			1.51346
$929$		−	37.4798i		−	1.22967i	−0.788655	−	0.614836i	$-0.789222\pi$
$929$		−	37.4798i		−	1.22967i	0.788655	−	0.614836i	$-0.210778\pi$
$930$	27.3122			0.895601
$931$		0			0
$932$			5.18809i			0.169942i
$933$	2.45932			0.0805144
$934$	−0.704534			−0.0230531
$935$	66.1004	+	14.1254i	2.16171	+	0.461950i
$936$			14.9169i			0.487575i
$937$	−20.7231			−0.676995			−0.338498	−	0.940967i	$-0.609919\pi$
$937$	−20.7231			−0.676995			−0.338498	+	0.940967i	$0.609919\pi$
$938$		0			0
$939$	−29.5027			−0.962784
$940$	−26.8535			−0.875865
$941$	−24.0647			−0.784486			−0.392243	−	0.919862i	$-0.628301\pi$
$941$	−24.0647			−0.784486			−0.392243	+	0.919862i	$0.628301\pi$
$942$		−	19.4284i		−	0.633011i
$943$	−61.6501			−2.00760
$944$		−	12.4216i		−	0.404289i
$945$		0			0
$946$	−4.60607	+	21.5543i	−0.149756	+	0.700790i
$947$	1.23498			0.0401313			0.0200657	−	0.999799i	$-0.493612\pi$
$947$	1.23498			0.0401313			0.0200657	+	0.999799i	$0.493612\pi$
$948$	−5.71365			−0.185571
$949$	21.4678			0.696874
$950$		−	22.1313i		−	0.718034i
$951$			14.7399i			0.477975i
$952$		0			0
$953$		−	48.8669i		−	1.58296i	−0.611198	−	0.791478i	$-0.709312\pi$
$953$		−	48.8669i		−	1.58296i	0.611198	−	0.791478i	$-0.290688\pi$
$954$		−	3.67115i		−	0.118858i
$955$		−	41.1811i		−	1.33259i
$956$		−	21.7271i		−	0.702704i
$957$	−6.71251	+	31.4115i	−0.216985	+	1.01539i
$958$		−	27.3100i		−	0.882345i
$959$		0			0
$960$	26.4055			0.852234
$961$	−24.4051			−0.787261
$962$		−	0.299827i		−	0.00966680i
$963$		−	9.52833i		−	0.307046i
$964$	12.7045			0.409184
$965$	4.76252			0.153311
$966$		0			0
$967$			3.29950i			0.106105i	0.998592	+	0.0530524i	$0.0168950\pi$
$967$			3.29950i			0.106105i	−0.998592	+	0.0530524i	$0.983105\pi$
$968$	−13.6185	+	30.4090i	−0.437714	+	0.977381i
$969$		−	16.0935i		−	0.516999i
$970$			40.0420i			1.28567i
$971$			43.7452i			1.40385i	0.712251	+	0.701925i	$0.247676\pi$
$971$			43.7452i			1.40385i	−0.712251	+	0.701925i	$0.752324\pi$
$972$			0.934667i			0.0299795i
$973$		0			0
$974$			9.52769i			0.305287i
$975$		−	37.6146i		−	1.20463i
$976$	8.22595			0.263306
$977$	−38.9657			−1.24662			−0.623312	−	0.781974i	$-0.714213\pi$
$977$	−38.9657			−1.24662			−0.623312	+	0.781974i	$0.714213\pi$
$978$	−8.82741			−0.282269
$979$	11.3255	+	2.42021i	0.361964	+	0.0773503i
$980$		0			0
$981$			9.47473i			0.302505i
$982$	7.39595			0.236014
$983$			31.2466i			0.996610i	0.867002	+	0.498305i	$0.166044\pi$
$983$			31.2466i			0.996610i	−0.867002	+	0.498305i	$0.833956\pi$
$984$	−25.3050			−0.806695
$985$	46.9663			1.49647
$986$	57.3055			1.82498
$987$		0			0
$988$	−12.9218			−0.411096
$989$		−	47.5138i		−	1.51085i
$990$	−2.54318	+	11.9009i	−0.0808277	+	0.378237i
$991$	−1.36533			−0.0433711			−0.0216856	−	0.999765i	$-0.506903\pi$
$991$	−1.36533			−0.0433711			−0.0216856	+	0.999765i	$0.506903\pi$
$992$	−35.4350			−1.12506
$993$			14.1350i			0.448562i
$994$		0			0
$995$	22.2558			0.705556
$996$		−	0.314866i		−	0.00997692i
$997$	−40.0336			−1.26788			−0.633938	−	0.773384i	$-0.718563\pi$
$997$	−40.0336			−1.26788			−0.633938	+	0.773384i	$0.718563\pi$
$998$		−	16.6713i		−	0.527721i
$999$		−	0.0589861i		−	0.00186624i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.c.b.538.18	yes	48
7.6	odd	2	inner	1617.2.c.b.538.17	✓	48
11.10	odd	2	inner	1617.2.c.b.538.32	yes	48
77.76	even	2	inner	1617.2.c.b.538.31	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.c.b.538.17	✓	48	7.6	odd	2	inner
1617.2.c.b.538.18	yes	48	1.1	even	1	trivial
1617.2.c.b.538.31	yes	48	77.76	even	2	inner
1617.2.c.b.538.32	yes	48	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.03215i q^{2} +1.00000i q^{3} +0.934667 q^{4} +3.55499i q^{5} +1.03215 q^{6} -3.02902i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.03215i q^{2} +1.00000i q^{3} +0.934667 q^{4} +3.55499i q^{5} +1.03215 q^{6} -3.02902i q^{8} -1.00000 q^{9} +3.66928 q^{10} +(0.693101 - 3.24339i) q^{11} +0.934667i q^{12} +4.92468 q^{13} -3.55499 q^{15} -1.25706 q^{16} +5.73278 q^{17} +1.03215i q^{18} -2.80728 q^{19} +3.32274i q^{20} +(-3.34767 - 0.715384i) q^{22} +7.37952 q^{23} +3.02902 q^{24} -7.63797 q^{25} -5.08301i q^{26} -1.00000i q^{27} +9.68475i q^{29} +3.66928i q^{30} -7.44346i q^{31} -4.76056i q^{32} +(3.24339 + 0.693101i) q^{33} -5.91709i q^{34} -0.934667 q^{36} +0.0589861 q^{37} +2.89754i q^{38} +4.92468i q^{39} +10.7681 q^{40} -8.35421 q^{41} -6.43860i q^{43} +(0.647819 - 3.03150i) q^{44} -3.55499i q^{45} -7.61677i q^{46} +8.08175i q^{47} -1.25706i q^{48} +7.88352i q^{50} +5.73278i q^{51} +4.60294 q^{52} -3.55680 q^{53} -1.03215 q^{54} +(11.5302 + 2.46397i) q^{55} -2.80728i q^{57} +9.99611 q^{58} +9.88146i q^{59} -3.32274 q^{60} -6.54379 q^{61} -7.68276 q^{62} -7.42773 q^{64} +17.5072i q^{65} +(0.715384 - 3.34767i) q^{66} +5.87720 q^{67} +5.35824 q^{68} +7.37952i q^{69} +13.2728 q^{71} +3.02902i q^{72} +4.35922 q^{73} -0.0608825i q^{74} -7.63797i q^{75} -2.62388 q^{76} +5.08301 q^{78} +6.11303i q^{79} -4.46885i q^{80} +1.00000 q^{81} +8.62279i q^{82} -0.336875 q^{83} +20.3800i q^{85} -6.64559 q^{86} -9.68475 q^{87} +(-9.82429 - 2.09941i) q^{88} +3.49186i q^{89} -3.66928 q^{90} +6.89740 q^{92} +7.44346 q^{93} +8.34157 q^{94} -9.97987i q^{95} +4.76056 q^{96} +10.9128i q^{97} +(-0.693101 + 3.24339i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) \) 48 * q - 64 * q^4 - 48 * q^9	\( 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) \) 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Introduction

L-functions

Modular forms

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