LMFDB - Embedded newform 315.4.d.a.64.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(64,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("315.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.d (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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.84052224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 6x^{3} + 36x^{2} - 36x + 18 $ x^6 - 2x^5 + 2x^4 + 6x^3 + 36x^2 - 36*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.6
Root			$-1.55322 + 1.55322i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.4.d.a.64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.10645i q^{2} -8.86293 q^{4} +(6.58150 + 9.03792i) q^{5} +7.00000i q^{7} -3.54358i q^{8} +O(q^{10})$	\(q+4.10645i q^{2} -8.86293 q^{4} +(6.58150 + 9.03792i) q^{5} +7.00000i q^{7} -3.54358i q^{8} +(-37.1137 + 27.0266i) q^{10} +8.17432 q^{11} +19.2242i q^{13} -28.7451 q^{14} -56.3519 q^{16} +18.8982i q^{17} +76.5016 q^{19} +(-58.3313 - 80.1024i) q^{20} +33.5675i q^{22} +142.692i q^{23} +(-38.3678 + 118.966i) q^{25} -78.9433 q^{26} -62.0405i q^{28} -96.1582 q^{29} -270.708 q^{31} -259.755i q^{32} -77.6047 q^{34} +(-63.2654 + 46.0705i) q^{35} -335.614i q^{37} +314.150i q^{38} +(32.0266 - 23.3221i) q^{40} +122.965 q^{41} -492.574i q^{43} -72.4485 q^{44} -585.957 q^{46} +96.9049i q^{47} -49.0000 q^{49} +(-488.528 - 157.556i) q^{50} -170.383i q^{52} +388.602i q^{53} +(53.7993 + 73.8788i) q^{55} +24.8051 q^{56} -394.869i q^{58} +112.896 q^{59} +347.160 q^{61} -1111.65i q^{62} +615.855 q^{64} +(-173.747 + 126.524i) q^{65} +101.653i q^{67} -167.494i q^{68} +(-189.186 - 259.796i) q^{70} -304.685 q^{71} +753.987i q^{73} +1378.18 q^{74} -678.029 q^{76} +57.2203i q^{77} +1164.38 q^{79} +(-370.880 - 509.304i) q^{80} +504.949i q^{82} -889.188i q^{83} +(-170.801 + 124.379i) q^{85} +2022.73 q^{86} -28.9664i q^{88} -938.829 q^{89} -134.570 q^{91} -1264.67i q^{92} -397.935 q^{94} +(503.495 + 691.415i) q^{95} -1206.06i q^{97} -201.216i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} + 14 q^{5}+O(q^{10}) $ 6 * q - 6 * q^4 + 14 * q^5	$ 6 q - 6 q^{4} + 14 q^{5} - 84 q^{10} + 132 q^{11} - 14 q^{14} - 138 q^{16} + 276 q^{19} - 334 q^{20} + 366 q^{25} + 196 q^{26} + 340 q^{29} - 732 q^{31} + 72 q^{34} - 56 q^{35} + 12 q^{40} + 412 q^{41} - 612 q^{44} - 1344 q^{46} - 294 q^{49} - 1216 q^{50} + 1860 q^{55} + 294 q^{56} - 1760 q^{59} - 1740 q^{61} + 1626 q^{64} + 16 q^{65} + 126 q^{70} + 2036 q^{71} + 1960 q^{74} - 900 q^{76} + 3240 q^{79} - 3794 q^{80} + 432 q^{85} + 5864 q^{86} - 3876 q^{89} - 1428 q^{91} - 4224 q^{94} + 828 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 + 14 * q^5 - 84 * q^10 + 132 * q^11 - 14 * q^14 - 138 * q^16 + 276 * q^19 - 334 * q^20 + 366 * q^25 + 196 * q^26 + 340 * q^29 - 732 * q^31 + 72 * q^34 - 56 * q^35 + 12 * q^40 + 412 * q^41 - 612 * q^44 - 1344 * q^46 - 294 * q^49 - 1216 * q^50 + 1860 * q^55 + 294 * q^56 - 1760 * q^59 - 1740 * q^61 + 1626 * q^64 + 16 * q^65 + 126 * q^70 + 2036 * q^71 + 1960 * q^74 - 900 * q^76 + 3240 * q^79 - 3794 * q^80 + 432 * q^85 + 5864 * q^86 - 3876 * q^89 - 1428 * q^91 - 4224 * q^94 + 828 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			4.10645i			1.45185i	0.687774	+	0.725925i	$0.258588\pi$
$2$			4.10645i			1.45185i	−0.687774	+	0.725925i	$0.741412\pi$
$3$		0			0
$3$		0			0
$4$	−8.86293			−1.10787
$5$	6.58150	+	9.03792i	0.588667	+	0.808376i
$5$	6.58150	+	9.03792i	0.588667	+	0.808376i
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		−	3.54358i		−	0.156606i
$9$		0			0
$10$	−37.1137	+	27.0266i	−1.17364	+	0.854656i
$11$	8.17432			0.224059			0.112030	−	0.993705i	$-0.464265\pi$
$11$	8.17432			0.224059			0.112030	+	0.993705i	$0.464265\pi$
$12$		0			0
$13$			19.2242i			0.410142i	0.978747	+	0.205071i	$0.0657425\pi$
$13$			19.2242i			0.410142i	−0.978747	+	0.205071i	$0.934258\pi$
$14$	−28.7451			−0.548747
$15$		0			0
$16$	−56.3519			−0.880499
$17$			18.8982i			0.269618i	0.990872	+	0.134809i	$0.0430420\pi$
$17$			18.8982i			0.269618i	−0.990872	+	0.134809i	$0.956958\pi$
$18$		0			0
$19$	76.5016			0.923720			0.461860	−	0.886953i	$-0.347182\pi$
$19$	76.5016			0.923720			0.461860	+	0.886953i	$0.347182\pi$
$20$	−58.3313	−	80.1024i	−0.652164	−	0.895572i
$21$		0			0
$22$			33.5675i			0.325300i
$23$			142.692i			1.29362i	0.762650	+	0.646811i	$0.223898\pi$
$23$			142.692i			1.29362i	−0.762650	+	0.646811i	$0.776102\pi$
$24$		0			0
$25$	−38.3678	+	118.966i	−0.306943	+	0.951728i
$26$	−78.9433			−0.595464
$27$		0			0
$28$		−	62.0405i		−	0.418734i
$29$	−96.1582			−0.615728			−0.307864	−	0.951430i	$-0.599614\pi$
$29$	−96.1582			−0.615728			−0.307864	+	0.951430i	$0.599614\pi$
$30$		0			0
$31$	−270.708			−1.56840			−0.784202	−	0.620505i	$-0.786928\pi$
$31$	−270.708			−1.56840			−0.784202	+	0.620505i	$0.786928\pi$
$32$		−	259.755i		−	1.43496i
$33$		0			0
$34$	−77.6047			−0.391444
$35$	−63.2654	+	46.0705i	−0.305537	+	0.222495i
$36$		0			0
$37$		−	335.614i		−	1.49120i	−0.666391	−	0.745602i	$-0.732162\pi$
$37$		−	335.614i		−	1.49120i	0.666391	−	0.745602i	$-0.267838\pi$
$38$			314.150i			1.34110i
$39$		0			0
$40$	32.0266	−	23.3221i	0.126596	−	0.0921886i
$41$	122.965			0.468387			0.234194	−	0.972190i	$-0.424755\pi$
$41$	122.965			0.468387			0.234194	+	0.972190i	$0.424755\pi$
$42$		0			0
$43$		−	492.574i		−	1.74690i	−0.486910	−	0.873452i	$-0.661876\pi$
$43$		−	492.574i		−	1.74690i	0.486910	−	0.873452i	$-0.338124\pi$
$44$	−72.4485			−0.248228
$45$		0			0
$46$	−585.957			−1.87814
$47$			96.9049i			0.300745i	0.988629	+	0.150373i	$0.0480473\pi$
$47$			96.9049i			0.300745i	−0.988629	+	0.150373i	$0.951953\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	−488.528	−	157.556i	−1.38177	−	0.445634i
$51$		0			0
$52$		−	170.383i		−	0.454382i
$53$			388.602i			1.00714i	0.863953	+	0.503572i	$0.167981\pi$
$53$			388.602i			1.00714i	−0.863953	+	0.503572i	$0.832019\pi$
$54$		0			0
$55$	53.7993	+	73.8788i	0.131896	+	0.181124i
$56$	24.8051			0.0591914
$57$		0			0
$58$		−	394.869i		−	0.893945i
$59$	112.896			0.249116			0.124558	−	0.992212i	$-0.460249\pi$
$59$	112.896			0.249116			0.124558	+	0.992212i	$0.460249\pi$
$60$		0			0
$61$	347.160			0.728676			0.364338	−	0.931267i	$-0.381295\pi$
$61$	347.160			0.728676			0.364338	+	0.931267i	$0.381295\pi$
$62$		−	1111.65i		−	2.27709i
$63$		0			0
$64$	615.855			1.20284
$65$	−173.747	+	126.524i	−0.331549	+	0.241437i
$66$		0			0
$67$			101.653i			0.185356i	0.995696	+	0.0926781i	$0.0295427\pi$
$67$			101.653i			0.185356i	−0.995696	+	0.0926781i	$0.970457\pi$
$68$		−	167.494i		−	0.298700i
$69$		0			0
$70$	−189.186	−	259.796i	−0.323029	−	0.443594i
$71$	−304.685			−0.509287			−0.254644	−	0.967035i	$-0.581958\pi$
$71$	−304.685			−0.509287			−0.254644	+	0.967035i	$0.581958\pi$
$72$		0			0
$73$			753.987i			1.20887i	0.796655	+	0.604435i	$0.206601\pi$
$73$			753.987i			1.20887i	−0.796655	+	0.604435i	$0.793399\pi$
$74$	1378.18			2.16500
$75$		0			0
$76$	−678.029			−1.02336
$77$			57.2203i			0.0846864i
$78$		0			0
$79$	1164.38			1.65827			0.829135	−	0.559049i	$-0.188834\pi$
$79$	1164.38			1.65827			0.829135	+	0.559049i	$0.188834\pi$
$80$	−370.880	−	509.304i	−0.518320	−	0.711774i
$81$		0			0
$82$			504.949i			0.680027i
$83$		−	889.188i		−	1.17592i	−0.808891	−	0.587958i	$-0.799932\pi$
$83$		−	889.188i		−	1.17592i	0.808891	−	0.587958i	$-0.200068\pi$
$84$		0			0
$85$	−170.801	+	124.379i	−0.217952	+	0.158715i
$86$	2022.73			2.53624
$87$		0			0
$88$		−	28.9664i		−	0.0350889i
$89$	−938.829			−1.11815			−0.559077	−	0.829116i	$-0.688844\pi$
$89$	−938.829			−1.11815			−0.559077	+	0.829116i	$0.688844\pi$
$90$		0			0
$91$	−134.570			−0.155019
$92$		−	1264.67i		−	1.43316i
$93$		0			0
$94$	−397.935			−0.436637
$95$	503.495	+	691.415i	0.543763	+	0.746713i
$96$		0			0
$97$		−	1206.06i		−	1.26244i	−0.775605	−	0.631218i	$-0.782555\pi$
$97$		−	1206.06i		−	1.26244i	0.775605	−	0.631218i	$-0.217445\pi$
$98$		−	201.216i		−	0.207407i
$99$		0			0
$100$	340.051	−	1054.39i	0.340051	−	1.05439i
$101$	−436.407			−0.429942			−0.214971	−	0.976620i	$-0.568966\pi$
$101$	−436.407			−0.429942			−0.214971	+	0.976620i	$0.568966\pi$
$102$		0			0
$103$			1765.48i			1.68891i	0.535629	+	0.844454i	$0.320075\pi$
$103$			1765.48i			1.68891i	−0.535629	+	0.844454i	$0.679925\pi$
$104$	68.1226			0.0642305
$105$		0			0
$106$	−1595.78			−1.46222
$107$			304.599i			0.275203i	0.990488	+	0.137601i	$0.0439393\pi$
$107$			304.599i			0.275203i	−0.990488	+	0.137601i	$0.956061\pi$
$108$		0			0
$109$	1037.76			0.911918			0.455959	−	0.890001i	$-0.349296\pi$
$109$	1037.76			0.911918			0.455959	+	0.890001i	$0.349296\pi$
$110$	−303.380	+	220.924i	−0.262965	+	0.191493i
$111$		0			0
$112$		−	394.463i		−	0.332797i
$113$			1479.63i			1.23178i	0.787830	+	0.615892i	$0.211204\pi$
$113$			1479.63i			1.23178i	−0.787830	+	0.615892i	$0.788796\pi$
$114$		0			0
$115$	−1289.64	+	939.126i	−1.04573	+	0.761513i
$116$	852.243			0.682145
$117$		0			0
$118$			463.604i			0.361679i
$119$	−132.288			−0.101906
$120$		0			0
$121$	−1264.18			−0.949797
$122$			1425.59i			1.05793i
$123$		0			0
$124$	2399.26			1.73758
$125$	−1327.72	+	436.209i	−0.950041	+	0.312126i
$126$		0			0
$127$			1633.52i			1.14135i	0.821175	+	0.570676i	$0.193319\pi$
$127$			1633.52i			1.14135i	−0.821175	+	0.570676i	$0.806681\pi$
$128$			450.940i			0.311389i
$129$		0			0
$130$	−519.565	−	713.483i	−0.350530	−	0.481359i
$131$	359.287			0.239627			0.119813	−	0.992796i	$-0.461770\pi$
$131$	359.287			0.239627			0.119813	+	0.992796i	$0.461770\pi$
$132$		0			0
$133$			535.511i			0.349133i
$134$	−417.432			−0.269109
$135$		0			0
$136$	66.9675			0.0422236
$137$			2573.71i			1.60501i	0.596643	+	0.802507i	$0.296501\pi$
$137$			2573.71i			1.60501i	−0.596643	+	0.802507i	$0.703499\pi$
$138$		0			0
$139$	2267.12			1.38341			0.691706	−	0.722179i	$-0.256859\pi$
$139$	2267.12			1.38341			0.691706	+	0.722179i	$0.256859\pi$
$140$	560.717	−	408.319i	0.338494	−	0.246495i
$141$		0			0
$142$		−	1251.17i		−	0.739409i
$143$			157.145i			0.0918960i
$144$		0			0
$145$	−632.865	−	869.069i	−0.362459	−	0.497740i
$146$	−3096.21			−1.75510
$147$		0			0
$148$			2974.52i			1.65206i
$149$	457.166			0.251359			0.125680	−	0.992071i	$-0.459889\pi$
$149$	457.166			0.251359			0.125680	+	0.992071i	$0.459889\pi$
$150$		0			0
$151$	2260.38			1.21819			0.609096	−	0.793097i	$-0.291532\pi$
$151$	2260.38			1.21819			0.609096	+	0.793097i	$0.291532\pi$
$152$		−	271.090i		−	0.144660i
$153$		0			0
$154$	−234.972			−0.122952
$155$	−1781.66	−	2446.63i	−0.923268	−	1.26786i
$156$		0			0
$157$		−	643.877i		−	0.327305i	−0.986518	−	0.163653i	$-0.947672\pi$
$157$		−	643.877i		−	0.327305i	0.986518	−	0.163653i	$-0.0523276\pi$
$158$			4781.48i			2.40756i
$159$		0			0
$160$	2347.64	−	1709.58i	1.15998	−	0.844712i
$161$	−998.843			−0.488943
$162$		0			0
$163$			3089.66i			1.48467i	0.670030	+	0.742334i	$0.266281\pi$
$163$			3089.66i			1.48467i	−0.670030	+	0.742334i	$0.733719\pi$
$164$	−1089.83			−0.518910
$165$		0			0
$166$	3651.41			1.70725
$167$			2726.84i			1.26353i	0.775161	+	0.631764i	$0.217669\pi$
$167$			2726.84i			1.26353i	−0.775161	+	0.631764i	$0.782331\pi$
$168$		0			0
$169$	1827.43			0.831784
$170$	−510.755	−	701.385i	−0.230430	−	0.316434i
$171$		0			0
$172$			4365.65i			1.93534i
$173$		−	901.915i		−	0.396366i	−0.980165	−	0.198183i	$-0.936496\pi$
$173$		−	901.915i		−	0.396366i	0.980165	−	0.198183i	$-0.0635040\pi$
$174$		0			0
$175$	−832.762	−	268.575i	−0.359719	−	0.116013i
$176$	−460.639			−0.197284
$177$		0			0
$178$		−	3855.25i		−	1.62339i
$179$	4547.62			1.89891			0.949455	−	0.313903i	$-0.101637\pi$
$179$	4547.62			1.89891			0.949455	+	0.313903i	$0.101637\pi$
$180$		0			0
$181$	−402.003			−0.165086			−0.0825432	−	0.996587i	$-0.526304\pi$
$181$	−402.003			−0.165086			−0.0825432	+	0.996587i	$0.526304\pi$
$182$		−	552.603i		−	0.225064i
$183$		0			0
$184$	505.640			0.202589
$185$	3033.25	−	2208.84i	1.20545	−	0.877823i
$186$		0			0
$187$			154.480i			0.0604103i
$188$		−	858.861i		−	0.333186i
$189$		0			0
$190$	−2839.26	+	2067.58i	−1.08411	+	0.789462i
$191$	4399.62			1.66673			0.833365	−	0.552723i	$-0.186411\pi$
$191$	4399.62			1.66673			0.833365	+	0.552723i	$0.186411\pi$
$192$		0			0
$193$		−	301.958i		−	0.112619i	−0.998413	−	0.0563094i	$-0.982067\pi$
$193$		−	301.958i		−	0.112619i	0.998413	−	0.0563094i	$-0.0179333\pi$
$194$	4952.61			1.83287
$195$		0			0
$196$	434.284			0.158267
$197$			952.950i			0.344644i	0.985041	+	0.172322i	$0.0551270\pi$
$197$			952.950i			0.344644i	−0.985041	+	0.172322i	$0.944873\pi$
$198$		0			0
$199$	−4124.90			−1.46938			−0.734690	−	0.678403i	$-0.762672\pi$
$199$	−4124.90			−1.46938			−0.734690	+	0.678403i	$0.762672\pi$
$200$	421.566	+	135.959i	0.149046	+	0.0480689i
$201$		0			0
$202$		−	1792.08i		−	0.624210i
$203$		−	673.107i		−	0.232723i
$204$		0			0
$205$	809.292	+	1111.35i	0.275724	+	0.378633i
$206$	−7249.83			−2.45204
$207$		0			0
$208$		−	1083.32i		−	0.361129i
$209$	625.349			0.206968
$210$		0			0
$211$	−1014.74			−0.331079			−0.165540	−	0.986203i	$-0.552937\pi$
$211$	−1014.74			−0.331079			−0.165540	+	0.986203i	$0.552937\pi$
$212$		−	3444.15i		−	1.11578i
$213$		0			0
$214$	−1250.82			−0.399553
$215$	4451.85	−	3241.88i	1.41215	−	1.02834i
$216$		0			0
$217$		−	1894.95i		−	0.592801i
$218$			4261.50i			1.32397i
$219$		0			0
$220$	−476.819	−	654.783i	−0.146123	−	0.200661i
$221$	−363.304			−0.110581
$222$		0			0
$223$		−	5042.95i		−	1.51435i	−0.653210	−	0.757177i	$-0.726578\pi$
$223$		−	5042.95i		−	1.51435i	0.653210	−	0.757177i	$-0.273422\pi$
$224$	1818.28			0.542363
$225$		0			0
$226$	−6076.02			−1.78837
$227$		−	5831.62i		−	1.70510i	−0.522643	−	0.852552i	$-0.675054\pi$
$227$		−	5831.62i		−	1.70510i	0.522643	−	0.852552i	$-0.324946\pi$
$228$		0			0
$229$	−18.5657			−0.00535744			−0.00267872	−	0.999996i	$-0.500853\pi$
$229$	−18.5657			−0.00535744			−0.00267872	+	0.999996i	$0.500853\pi$
$230$	−3856.47	−	5295.83i	−1.10560	−	1.51825i
$231$		0			0
$232$			340.744i			0.0964265i
$233$			1732.62i			0.487158i	0.969881	+	0.243579i	$0.0783216\pi$
$233$			1732.62i			0.487158i	−0.969881	+	0.243579i	$0.921678\pi$
$234$		0			0
$235$	−875.818	+	637.779i	−0.243115	+	0.177039i
$236$	−1000.59			−0.275988
$237$		0			0
$238$		−	543.233i		−	0.147952i
$239$	−1937.54			−0.524389			−0.262195	−	0.965015i	$-0.584446\pi$
$239$	−1937.54			−0.524389			−0.262195	+	0.965015i	$0.584446\pi$
$240$		0			0
$241$	5613.02			1.50027			0.750137	−	0.661282i	$-0.229987\pi$
$241$	5613.02			1.50027			0.750137	+	0.661282i	$0.229987\pi$
$242$		−	5191.29i		−	1.37896i
$243$		0			0
$244$	−3076.85			−0.807275
$245$	−322.493	−	442.858i	−0.0840953	−	0.115482i
$246$		0			0
$247$			1470.69i			0.378856i
$248$			959.275i			0.245621i
$249$		0			0
$250$	−1791.27	−	5452.23i	−0.453160	−	1.37932i
$251$	5132.49			1.29068			0.645338	−	0.763897i	$-0.276716\pi$
$251$	5132.49			1.29068			0.645338	+	0.763897i	$0.276716\pi$
$252$		0			0
$253$			1166.41i			0.289848i
$254$	−6707.98			−1.65707
$255$		0			0
$256$	3075.08			0.750752
$257$		−	4689.21i		−	1.13815i	−0.822285	−	0.569075i	$-0.807301\pi$
$257$		−	4689.21i		−	1.13815i	0.822285	−	0.569075i	$-0.192699\pi$
$258$		0			0
$259$	2349.30			0.563622
$260$	1539.91	−	1121.38i	0.367312	−	0.267480i
$261$		0			0
$262$			1475.40i			0.347902i
$263$			1767.86i			0.414491i	0.978289	+	0.207246i	$0.0664499\pi$
$263$			1767.86i			0.414491i	−0.978289	+	0.207246i	$0.933550\pi$
$264$		0			0
$265$	−3512.15	+	2557.58i	−0.814151	+	0.592872i
$266$	−2199.05			−0.506889
$267$		0			0
$268$		−	900.941i		−	0.205350i
$269$	−7869.14			−1.78361			−0.891803	−	0.452425i	$-0.850559\pi$
$269$	−7869.14			−1.78361			−0.891803	+	0.452425i	$0.850559\pi$
$270$		0			0
$271$	4820.52			1.08054			0.540268	−	0.841493i	$-0.318323\pi$
$271$	4820.52			1.08054			0.540268	+	0.841493i	$0.318323\pi$
$272$		−	1064.95i		−	0.237398i
$273$		0			0
$274$	−10568.8			−2.33024
$275$	−313.631	+	972.467i	−0.0687733	+	0.213243i
$276$		0			0
$277$		−	1783.66i		−	0.386895i	−0.981111	−	0.193448i	$-0.938033\pi$
$277$		−	1783.66i		−	0.386895i	0.981111	−	0.193448i	$-0.0619670\pi$
$278$			9309.80i			2.00851i
$279$		0			0
$280$	163.254	+	224.186i	0.0348440	+	0.0478489i
$281$	−1779.27			−0.377730			−0.188865	−	0.982003i	$-0.560481\pi$
$281$	−1779.27			−0.377730			−0.188865	+	0.982003i	$0.560481\pi$
$282$		0			0
$283$			7374.55i			1.54902i	0.632564	+	0.774508i	$0.282002\pi$
$283$			7374.55i			1.54902i	−0.632564	+	0.774508i	$0.717998\pi$
$284$	2700.40			0.564222
$285$		0			0
$286$	−645.308			−0.133419
$287$			860.753i			0.177034i
$288$		0			0
$289$	4555.86			0.927306
$290$	3568.79	−	2598.83i	0.722643	−	0.526236i
$291$		0			0
$292$		−	6682.53i		−	1.33927i
$293$			934.020i			0.186232i	0.995655	+	0.0931161i	$0.0296828\pi$
$293$			934.020i			0.186232i	−0.995655	+	0.0931161i	$0.970317\pi$
$294$		0			0
$295$	743.028	+	1020.35i	0.146647	+	0.201380i
$296$	−1189.27			−0.233531
$297$		0			0
$298$			1877.33i			0.364936i
$299$	−2743.14			−0.530568
$300$		0			0
$301$	3448.02			0.660268
$302$			9282.13i			1.76863i
$303$		0			0
$304$	−4311.01			−0.813334
$305$	2284.83	+	3137.60i	0.428947	+	0.589044i
$306$		0			0
$307$			1929.10i			0.358630i	0.983792	+	0.179315i	$0.0573882\pi$
$307$			1929.10i			0.358630i	−0.983792	+	0.179315i	$0.942612\pi$
$308$		−	507.139i		−	0.0938212i
$309$		0			0
$310$	10047.0	−	7316.31i	1.84074	−	1.34045i
$311$	4465.18			0.814138			0.407069	−	0.913397i	$-0.366551\pi$
$311$	4465.18			0.814138			0.407069	+	0.913397i	$0.366551\pi$
$312$		0			0
$313$		−	7457.31i		−	1.34668i	−0.739332	−	0.673342i	$-0.764858\pi$
$313$		−	7457.31i		−	1.34668i	0.739332	−	0.673342i	$-0.235142\pi$
$314$	2644.05			0.475198
$315$		0			0
$316$	−10319.8			−1.83714
$317$		−	2024.84i		−	0.358758i	−0.983780	−	0.179379i	$-0.942591\pi$
$317$		−	2024.84i		−	0.358758i	0.983780	−	0.179379i	$-0.0574089\pi$
$318$		0			0
$319$	−786.028			−0.137960
$320$	4053.25	+	5566.05i	0.708074	+	0.972349i
$321$		0			0
$322$		−	4101.70i		−	0.709872i
$323$			1445.75i			0.249051i
$324$		0			0
$325$	−2287.03	−	737.592i	−0.390343	−	0.125890i
$326$	−12687.5			−2.15551
$327$		0			0
$328$		−	435.736i		−	0.0733521i
$329$	−678.334			−0.113671
$330$		0			0
$331$	3153.56			0.523671			0.261836	−	0.965112i	$-0.415672\pi$
$331$	3153.56			0.523671			0.261836	+	0.965112i	$0.415672\pi$
$332$			7880.81i			1.30276i
$333$		0			0
$334$	−11197.6			−1.83445
$335$	−918.729	+	669.027i	−0.149837	+	0.109113i
$336$		0			0
$337$		−	5058.35i		−	0.817644i	−0.912614	−	0.408822i	$-0.865940\pi$
$337$		−	5058.35i		−	0.817644i	0.912614	−	0.408822i	$-0.134060\pi$
$338$			7504.25i			1.20762i
$339$		0			0
$340$	1513.80	−	1102.36i	0.241462	−	0.175835i
$341$	−2212.85			−0.351416
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$	−1745.48			−0.273575
$345$		0			0
$346$	3703.67			0.575464
$347$		−	7950.25i		−	1.22995i	−0.788548	−	0.614973i	$-0.789167\pi$
$347$		−	7950.25i		−	1.22995i	0.788548	−	0.614973i	$-0.210833\pi$
$348$		0			0
$349$	−4711.92			−0.722702			−0.361351	−	0.932430i	$-0.617684\pi$
$349$	−4711.92			−0.722702			−0.361351	+	0.932430i	$0.617684\pi$
$350$	1102.89	−	3419.70i	0.168434	−	0.522258i
$351$		0			0
$352$		−	2123.32i		−	0.321515i
$353$		−	308.497i		−	0.0465146i	−0.999730	−	0.0232573i	$-0.992596\pi$
$353$		−	308.497i		−	0.0465146i	0.999730	−	0.0232573i	$-0.00740370\pi$
$354$		0			0
$355$	−2005.28	−	2753.71i	−0.299801	−	0.411696i
$356$	8320.77			1.23876
$357$		0			0
$358$			18674.6i			2.75693i
$359$	−7598.91			−1.11715			−0.558573	−	0.829456i	$-0.688651\pi$
$359$	−7598.91			−1.11715			−0.558573	+	0.829456i	$0.688651\pi$
$360$		0			0
$361$	−1006.50			−0.146742
$362$		−	1650.80i		−	0.239681i
$363$		0			0
$364$	1192.68			0.171740
$365$	−6814.47	+	4962.36i	−0.977221	+	0.711621i
$366$		0			0
$367$		−	2057.48i		−	0.292642i	−0.989237	−	0.146321i	$-0.953257\pi$
$367$		−	2057.48i		−	0.292642i	0.989237	−	0.146321i	$-0.0467432\pi$
$368$		−	8040.96i		−	1.13903i
$369$		0			0
$370$	9070.49	+	12455.9i	1.27447	+	1.75014i
$371$	−2720.22			−0.380665
$372$		0			0
$373$			5661.44i			0.785893i	0.919561	+	0.392947i	$0.128544\pi$
$373$			5661.44i			0.785893i	−0.919561	+	0.392947i	$0.871456\pi$
$374$	−634.366			−0.0877066
$375$		0			0
$376$	343.390			0.0470984
$377$		−	1848.57i		−	0.252536i
$378$		0			0
$379$	1822.92			0.247063			0.123532	−	0.992341i	$-0.460578\pi$
$379$	1822.92			0.247063			0.123532	+	0.992341i	$0.460578\pi$
$380$	−4462.44	−	6127.97i	−0.602417	−	0.827258i
$381$		0			0
$382$			18066.8i			2.41984i
$383$			1886.34i			0.251664i	0.992052	+	0.125832i	$0.0401600\pi$
$383$			1886.34i			0.251664i	−0.992052	+	0.125832i	$0.959840\pi$
$384$		0			0
$385$	−517.152	+	376.595i	−0.0684584	+	0.0498521i
$386$	1239.98			0.163505
$387$		0			0
$388$			10689.2i			1.39861i
$389$	−9079.55			−1.18342			−0.591711	−	0.806150i	$-0.701547\pi$
$389$	−9079.55			−1.18342			−0.591711	+	0.806150i	$0.701547\pi$
$390$		0			0
$391$	−2696.63			−0.348783
$392$			173.635i			0.0223722i
$393$		0			0
$394$	−3913.24			−0.500371
$395$	7663.38	+	10523.6i	0.976168	+	1.34050i
$396$		0			0
$397$		−	2620.95i		−	0.331339i	−0.986181	−	0.165669i	$-0.947022\pi$
$397$		−	2620.95i		−	0.331339i	0.986181	−	0.165669i	$-0.0529785\pi$
$398$		−	16938.7i		−	2.13332i
$399$		0			0
$400$	2162.10	−	6703.96i	0.270262	−	0.837995i
$401$	10752.1			1.33899			0.669493	−	0.742818i	$-0.266511\pi$
$401$	10752.1			1.33899			0.669493	+	0.742818i	$0.266511\pi$
$402$		0			0
$403$		−	5204.15i		−	0.643268i
$404$	3867.84			0.476318
$405$		0			0
$406$	2764.08			0.337879
$407$		−	2743.42i		−	0.334118i
$408$		0			0
$409$	−13382.0			−1.61785			−0.808923	−	0.587914i	$-0.799949\pi$
$409$	−13382.0			−1.61785			−0.808923	+	0.587914i	$0.799949\pi$
$410$	−4563.68	+	3323.32i	−0.549718	+	0.400310i
$411$		0			0
$412$		−	15647.3i		−	1.87108i
$413$			790.275i			0.0941571i
$414$		0			0
$415$	8036.41	−	5852.19i	0.950583	−	0.692223i
$416$	4993.59			0.588536
$417$		0			0
$418$			2567.96i			0.300486i
$419$	2335.91			0.272354			0.136177	−	0.990684i	$-0.456518\pi$
$419$	2335.91			0.272354			0.136177	+	0.990684i	$0.456518\pi$
$420$		0			0
$421$	4262.71			0.493472			0.246736	−	0.969083i	$-0.420642\pi$
$421$	4262.71			0.493472			0.246736	+	0.969083i	$0.420642\pi$
$422$		−	4166.99i		−	0.480677i
$423$		0			0
$424$	1377.04			0.157724
$425$	−2248.25	−	725.085i	−0.256603	−	0.0827571i
$426$		0			0
$427$			2430.12i			0.275414i
$428$		−	2699.64i		−	0.304888i
$429$		0			0
$430$	13312.6	+	18281.3i	1.49300	+	2.05024i
$431$	−5233.78			−0.584924			−0.292462	−	0.956277i	$-0.594474\pi$
$431$	−5233.78			−0.584924			−0.292462	+	0.956277i	$0.594474\pi$
$432$		0			0
$433$		−	14627.5i		−	1.62345i	−0.584039	−	0.811726i	$-0.698528\pi$
$433$		−	14627.5i		−	1.62345i	0.584039	−	0.811726i	$-0.301472\pi$
$434$	7781.54			0.860658
$435$		0			0
$436$	−9197.57			−1.01028
$437$			10916.2i			1.19494i
$438$		0			0
$439$	−35.2964			−0.00383737			−0.00191868	−	0.999998i	$-0.500611\pi$
$439$	−35.2964			−0.00383737			−0.00191868	+	0.999998i	$0.500611\pi$
$440$	261.796	−	190.642i	0.0283650	−	0.0206557i
$441$		0			0
$442$		−	1491.89i		−	0.160548i
$443$			2805.40i			0.300877i	0.988619	+	0.150439i	$0.0480686\pi$
$443$			2805.40i			0.300877i	−0.988619	+	0.150439i	$0.951931\pi$
$444$		0			0
$445$	−6178.90	−	8485.05i	−0.658220	−	0.903888i
$446$	20708.6			2.19861
$447$		0			0
$448$			4310.99i			0.454632i
$449$	14903.0			1.56641			0.783203	−	0.621766i	$-0.213585\pi$
$449$	14903.0			1.56641			0.783203	+	0.621766i	$0.213585\pi$
$450$		0			0
$451$	1005.15			0.104946
$452$		−	13113.8i		−	1.36465i
$453$		0			0
$454$	23947.3			2.47555
$455$	−885.669	−	1216.23i	−0.0912546	−	0.125314i
$456$		0			0
$457$			2633.85i			0.269598i	0.990873	+	0.134799i	$0.0430388\pi$
$457$			2633.85i			0.269598i	−0.990873	+	0.134799i	$0.956961\pi$
$458$		−	76.2390i		−	0.00777820i
$459$		0			0
$460$	11430.0	−	8323.41i	1.15853	−	0.843654i
$461$	18415.2			1.86048			0.930239	−	0.366955i	$-0.119600\pi$
$461$	18415.2			1.86048			0.930239	+	0.366955i	$0.119600\pi$
$462$		0			0
$463$			14811.0i			1.48666i	0.668923	+	0.743332i	$0.266756\pi$
$463$			14811.0i			1.48666i	−0.668923	+	0.743332i	$0.733244\pi$
$464$	5418.70			0.542148
$465$		0			0
$466$	−7114.93			−0.707280
$467$		−	14407.1i		−	1.42758i	−0.700360	−	0.713790i	$-0.746977\pi$
$467$		−	14407.1i		−	1.42758i	0.700360	−	0.713790i	$-0.253023\pi$
$468$		0			0
$469$	−711.569			−0.0700580
$470$	−2619.01	−	3596.50i	−0.257034	−	0.352967i
$471$		0			0
$472$		−	400.058i		−	0.0390130i
$473$		−	4026.46i		−	0.391410i
$474$		0			0
$475$	−2935.20	+	9101.09i	−0.283529	+	0.879130i
$476$	1172.46			0.112898
$477$		0			0
$478$		−	7956.41i		−	0.761334i
$479$	−6629.16			−0.632347			−0.316174	−	0.948701i	$-0.602398\pi$
$479$	−6629.16			−0.632347			−0.316174	+	0.948701i	$0.602398\pi$
$480$		0			0
$481$	6451.92			0.611605
$482$			23049.6i			2.17817i
$483$		0			0
$484$	11204.3			1.05225
$485$	10900.2	−	7937.65i	1.02052	−	0.743155i
$486$		0			0
$487$			3890.46i			0.361999i	0.983483	+	0.180999i	$0.0579332\pi$
$487$			3890.46i			0.361999i	−0.983483	+	0.180999i	$0.942067\pi$
$488$		−	1230.19i		−	0.114115i
$489$		0			0
$490$	1818.57	−	1324.30i	0.167663	−	0.122094i
$491$	12804.1			1.17687			0.588433	−	0.808546i	$-0.299745\pi$
$491$	12804.1			1.17687			0.588433	+	0.808546i	$0.299745\pi$
$492$		0			0
$493$		−	1817.22i		−	0.166011i
$494$	−6039.29			−0.550042
$495$		0			0
$496$	15254.9			1.38098
$497$		−	2132.79i		−	0.192493i
$498$		0			0
$499$	5842.56			0.524146			0.262073	−	0.965048i	$-0.415594\pi$
$499$	5842.56			0.524146			0.262073	+	0.965048i	$0.415594\pi$
$500$	11767.5	−	3866.09i	1.05252	−	0.345794i
$501$		0			0
$502$			21076.3i			1.87387i
$503$			18045.9i			1.59966i	0.600227	+	0.799830i	$0.295077\pi$
$503$			18045.9i			1.59966i	−0.600227	+	0.799830i	$0.704923\pi$
$504$		0			0
$505$	−2872.21	−	3944.21i	−0.253092	−	0.347554i
$506$	−4789.80			−0.420816
$507$		0			0
$508$		−	14477.8i		−	1.26447i
$509$	−21723.0			−1.89166			−0.945831	−	0.324659i	$-0.894750\pi$
$509$	−21723.0			−1.89166			−0.945831	+	0.324659i	$0.894750\pi$
$510$		0			0
$511$	−5277.91			−0.456910
$512$			16235.2i			1.40137i
$513$		0			0
$514$	19256.0			1.65242
$515$	−15956.2	+	11619.5i	−1.36527	+	0.994204i
$516$		0			0
$517$			792.132i			0.0673847i
$518$			9647.27i			0.818295i
$519$		0			0
$520$	448.349	+	615.686i	0.0378104	+	0.0519224i
$521$	−17101.2			−1.43804			−0.719018	−	0.694992i	$-0.755408\pi$
$521$	−17101.2			−1.43804			−0.719018	+	0.694992i	$0.755408\pi$
$522$		0			0
$523$		−	4938.15i		−	0.412869i	−0.978460	−	0.206434i	$-0.933814\pi$
$523$		−	4938.15i		−	0.412869i	0.978460	−	0.206434i	$-0.0661860\pi$
$524$	−3184.34			−0.265474
$525$		0			0
$526$	−7259.64			−0.601779
$527$		−	5115.90i		−	0.422869i
$528$		0			0
$529$	−8193.97			−0.673458
$530$	−10502.6	−	14422.5i	−0.860761	−	1.18202i
$531$		0			0
$532$		−	4746.20i		−	0.386793i
$533$			2363.90i			0.192105i
$534$		0			0
$535$	−2752.94	+	2004.72i	−0.222467	+	0.162003i
$536$	360.215			0.0290278
$537$		0			0
$538$		−	32314.2i		−	2.58953i
$539$	−400.542			−0.0320085
$540$		0			0
$541$	11481.7			0.912451			0.456226	−	0.889864i	$-0.349201\pi$
$541$	11481.7			0.912451			0.456226	+	0.889864i	$0.349201\pi$
$542$			19795.2i			1.56878i
$543$		0			0
$544$	4908.91			0.386890
$545$	6829.99	+	9379.16i	0.536816	+	0.737173i
$546$		0			0
$547$			18561.2i			1.45086i	0.688298	+	0.725428i	$0.258358\pi$
$547$			18561.2i			1.45086i	−0.688298	+	0.725428i	$0.741642\pi$
$548$		−	22810.6i		−	1.77814i
$549$		0			0
$550$	−3993.39	−	1287.91i	−0.309597	−	0.0998485i
$551$	−7356.26			−0.568761
$552$		0			0
$553$			8150.68i			0.626767i
$554$	7324.53			0.561714
$555$		0			0
$556$	−20093.3			−1.53264
$557$		−	17604.9i		−	1.33922i	−0.742714	−	0.669608i	$-0.766462\pi$
$557$		−	17604.9i		−	1.33922i	0.742714	−	0.669608i	$-0.233538\pi$
$558$		0			0
$559$	9469.36			0.716478
$560$	3565.13	−	2596.16i	0.269025	−	0.195907i
$561$		0			0
$562$		−	7306.47i		−	0.548407i
$563$		−	3586.87i		−	0.268505i	−0.990947	−	0.134253i	$-0.957137\pi$
$563$		−	3586.87i		−	0.268505i	0.990947	−	0.134253i	$-0.0428634\pi$
$564$		0			0
$565$	−13372.8	+	9738.17i	−0.995745	+	0.725111i
$566$	−30283.2			−2.24894
$567$		0			0
$568$			1079.67i			0.0797573i
$569$	−3213.58			−0.236767			−0.118383	−	0.992968i	$-0.537771\pi$
$569$	−3213.58			−0.236767			−0.118383	+	0.992968i	$0.537771\pi$
$570$		0			0
$571$	−14706.3			−1.07783			−0.538913	−	0.842361i	$-0.681165\pi$
$571$	−14706.3			−1.07783			−0.538913	+	0.842361i	$0.681165\pi$
$572$		−	1392.77i		−	0.101809i
$573$		0			0
$574$	−3534.64			−0.257026
$575$	−16975.5	−	5474.78i	−1.23118	−	0.397068i
$576$		0			0
$577$		−	3632.05i		−	0.262052i	−0.991379	−	0.131026i	$-0.958173\pi$
$577$		−	3632.05i		−	0.262052i	0.991379	−	0.131026i	$-0.0418272\pi$
$578$			18708.4i			1.34631i
$579$		0			0
$580$	5609.04	+	7702.50i	0.401556	+	0.551429i
$581$	6224.32			0.444455
$582$		0			0
$583$			3176.56i			0.225660i
$584$	2671.81			0.189316
$585$		0			0
$586$	−3835.51			−0.270381
$587$			15671.9i			1.10196i	0.834520	+	0.550978i	$0.185745\pi$
$587$			15671.9i			1.10196i	−0.834520	+	0.550978i	$0.814255\pi$
$588$		0			0
$589$	−20709.6			−1.44877
$590$	−4190.01	+	3051.21i	−0.292373	+	0.212909i
$591$		0			0
$592$			18912.5i			1.31300i
$593$		−	6007.59i		−	0.416024i	−0.978126	−	0.208012i	$-0.933301\pi$
$593$		−	6007.59i		−	0.416024i	0.978126	−	0.208012i	$-0.0666992\pi$
$594$		0			0
$595$	−870.651	−	1195.61i	−0.0599886	−	0.0823782i
$596$	−4051.83			−0.278472
$597$		0			0
$598$		−	11264.6i		−	0.770305i
$599$	10787.4			0.735832			0.367916	−	0.929859i	$-0.380071\pi$
$599$	10787.4			0.735832			0.367916	+	0.929859i	$0.380071\pi$
$600$		0			0
$601$	−9839.14			−0.667798			−0.333899	−	0.942609i	$-0.608364\pi$
$601$	−9839.14			−0.667798			−0.333899	+	0.942609i	$0.608364\pi$
$602$			14159.1i			0.958609i
$603$		0			0
$604$	−20033.6			−1.34959
$605$	−8320.20	−	11425.6i	−0.559114	−	0.767793i
$606$		0			0
$607$		−	5572.23i		−	0.372603i	−0.982493	−	0.186301i	$-0.940350\pi$
$607$		−	5572.23i		−	0.372603i	0.982493	−	0.186301i	$-0.0596501\pi$
$608$		−	19871.7i		−	1.32550i
$609$		0			0
$610$	−12884.4	+	9382.54i	−0.855203	+	0.622767i
$611$	−1862.92			−0.123348
$612$		0			0
$613$		−	27431.5i		−	1.80742i	−0.428147	−	0.903709i	$-0.640833\pi$
$613$		−	27431.5i		−	1.80742i	0.428147	−	0.903709i	$-0.359167\pi$
$614$	−7921.75			−0.520677
$615$		0			0
$616$	202.765			0.0132624
$617$		−	6390.91i		−	0.416999i	−0.978022	−	0.208499i	$-0.933142\pi$
$617$		−	6390.91i		−	0.416999i	0.978022	−	0.208499i	$-0.0668580\pi$
$618$		0			0
$619$	−5725.64			−0.371782			−0.185891	−	0.982570i	$-0.559517\pi$
$619$	−5725.64			−0.371782			−0.185891	+	0.982570i	$0.559517\pi$
$620$	15790.7	+	21684.3i	1.02286	+	1.40462i
$621$		0			0
$622$			18336.0i			1.18201i
$623$		−	6571.80i		−	0.422622i
$624$		0			0
$625$	−12680.8	−	9128.93i	−0.811573	−	0.584252i
$626$	30623.1			1.95518
$627$		0			0
$628$			5706.63i			0.362611i
$629$	6342.51			0.402055
$630$		0			0
$631$	16835.6			1.06215			0.531074	−	0.847326i	$-0.321789\pi$
$631$	16835.6			1.06215			0.531074	+	0.847326i	$0.321789\pi$
$632$		−	4126.08i		−	0.259694i
$633$		0			0
$634$	8314.91			0.520863
$635$	−14763.6	+	10751.0i	−0.922641	+	0.671876i
$636$		0			0
$637$		−	941.987i		−	0.0585917i
$638$		−	3227.78i		−	0.200297i
$639$		0			0
$640$	−4075.55	+	2967.86i	−0.251719	+	0.183304i
$641$	−12957.8			−0.798446			−0.399223	−	0.916854i	$-0.630720\pi$
$641$	−12957.8			−0.798446			−0.399223	+	0.916854i	$0.630720\pi$
$642$		0			0
$643$			5665.07i			0.347447i	0.984794	+	0.173724i	$0.0555800\pi$
$643$			5665.07i			0.347447i	−0.984794	+	0.173724i	$0.944420\pi$
$644$	8852.68			0.541684
$645$		0			0
$646$	−5936.89			−0.361585
$647$			11556.3i			0.702201i	0.936338	+	0.351101i	$0.114192\pi$
$647$			11556.3i			0.702201i	−0.936338	+	0.351101i	$0.885808\pi$
$648$		0			0
$649$	922.852			0.0558168
$650$	3028.88	−	9391.57i	0.182773	−	0.566720i
$651$		0			0
$652$		−	27383.4i		−	1.64481i
$653$			21891.6i			1.31192i	0.754795	+	0.655961i	$0.227736\pi$
$653$			21891.6i			1.31192i	−0.754795	+	0.655961i	$0.772264\pi$
$654$		0			0
$655$	2364.65	+	3247.21i	0.141060	+	0.193708i
$656$	−6929.30			−0.412414
$657$		0			0
$658$		−	2785.54i		−	0.165033i
$659$	12118.3			0.716332			0.358166	−	0.933658i	$-0.383402\pi$
$659$	12118.3			0.716332			0.358166	+	0.933658i	$0.383402\pi$
$660$		0			0
$661$	739.495			0.0435144			0.0217572	−	0.999763i	$-0.493074\pi$
$661$	739.495			0.0435144			0.0217572	+	0.999763i	$0.493074\pi$
$662$			12949.9i			0.760292i
$663$		0			0
$664$	−3150.91			−0.184155
$665$	−4839.91	+	3524.47i	−0.282231	+	0.205523i
$666$		0			0
$667$		−	13721.0i		−	0.796520i
$668$		−	24167.8i		−	1.39982i
$669$		0			0
$670$	−2747.33	−	3772.71i	−0.158416	−	0.217541i
$671$	2837.79			0.163267
$672$		0			0
$673$			21080.0i			1.20739i	0.797215	+	0.603695i	$0.206306\pi$
$673$			21080.0i			1.20739i	−0.797215	+	0.603695i	$0.793694\pi$
$674$	20771.9			1.18710
$675$		0			0
$676$	−16196.4			−0.921505
$677$		−	9588.70i		−	0.544348i	−0.962248	−	0.272174i	$-0.912257\pi$
$677$		−	9588.70i		−	0.544348i	0.962248	−	0.272174i	$-0.0877427\pi$
$678$		0			0
$679$	8442.39			0.477156
$680$	440.746	+	605.246i	0.0248557	+	0.0341326i
$681$		0			0
$682$		−	9086.97i		−	0.510202i
$683$		−	33084.2i		−	1.85348i	−0.375698	−	0.926742i	$-0.622597\pi$
$683$		−	33084.2i		−	1.85348i	0.375698	−	0.926742i	$-0.377403\pi$
$684$		0			0
$685$	−23261.0	+	16938.9i	−1.29745	+	0.944818i
$686$	1408.51			0.0783925
$687$		0			0
$688$			27757.5i			1.53815i
$689$	−7470.58			−0.413072
$690$		0			0
$691$	−6292.90			−0.346445			−0.173222	−	0.984883i	$-0.555418\pi$
$691$	−6292.90			−0.346445			−0.173222	+	0.984883i	$0.555418\pi$
$692$			7993.61i			0.439120i
$693$		0			0
$694$	32647.3			1.78570
$695$	14921.0	+	20490.0i	0.814369	+	1.11832i
$696$		0			0
$697$			2323.82i			0.126285i
$698$		−	19349.2i		−	1.04925i
$699$		0			0
$700$	7380.71	+	2380.36i	0.398521	+	0.128527i
$701$	−23822.6			−1.28355			−0.641775	−	0.766893i	$-0.721802\pi$
$701$	−23822.6			−1.28355			−0.641775	+	0.766893i	$0.721802\pi$
$702$		0			0
$703$		−	25675.0i		−	1.37746i
$704$	5034.20			0.269508
$705$		0			0
$706$	1266.83			0.0675322
$707$		−	3054.85i		−	0.162503i
$708$		0			0
$709$	−10562.0			−0.559471			−0.279736	−	0.960077i	$-0.590247\pi$
$709$	−10562.0			−0.559471			−0.279736	+	0.960077i	$0.590247\pi$
$710$	11308.0	−	8234.58i	0.597720	−	0.435265i
$711$		0			0
$712$			3326.82i			0.175109i
$713$		−	38627.8i		−	2.02892i
$714$		0			0
$715$	−1420.26	+	1034.25i	−0.0742865	+	0.0540962i
$716$	−40305.2			−2.10374
$717$		0			0
$718$		−	31204.5i		−	1.62193i
$719$	−3538.42			−0.183534			−0.0917670	−	0.995781i	$-0.529251\pi$
$719$	−3538.42			−0.183534			−0.0917670	+	0.995781i	$0.529251\pi$
$720$		0			0
$721$	−12358.3			−0.638347
$722$		−	4133.14i		−	0.213047i
$723$		0			0
$724$	3562.92			0.182894
$725$	3689.38	−	11439.6i	0.188993	−	0.586006i
$726$		0			0
$727$			14066.6i			0.717608i	0.933413	+	0.358804i	$0.116815\pi$
$727$			14066.6i			0.717608i	−0.933413	+	0.358804i	$0.883185\pi$
$728$			476.858i			0.0242768i
$729$		0			0
$730$	−20377.7	−	27983.3i	−1.03317	−	1.41878i
$731$	9308.79			0.470996
$732$		0			0
$733$			34204.1i			1.72354i	0.507298	+	0.861771i	$0.330644\pi$
$733$			34204.1i			1.72354i	−0.507298	+	0.861771i	$0.669356\pi$
$734$	8448.94			0.424872
$735$		0			0
$736$	37064.9			1.85629
$737$			830.943i			0.0415308i
$738$		0			0
$739$	24949.9			1.24195			0.620973	−	0.783832i	$-0.286738\pi$
$739$	24949.9			1.24195			0.620973	+	0.783832i	$0.286738\pi$
$740$	−26883.5	+	19576.8i	−1.33548	+	0.972510i
$741$		0			0
$742$		−	11170.4i		−	0.552668i
$743$			16171.5i			0.798487i	0.916845	+	0.399244i	$0.130727\pi$
$743$			16171.5i			0.798487i	−0.916845	+	0.399244i	$0.869273\pi$
$744$		0			0
$745$	3008.84	+	4131.83i	0.147967	+	0.203193i
$746$	−23248.4			−1.14100
$747$		0			0
$748$		−	1369.15i		−	0.0669265i
$749$	−2132.19			−0.104017
$750$		0			0
$751$	18661.4			0.906744			0.453372	−	0.891321i	$-0.350221\pi$
$751$	18661.4			0.906744			0.453372	+	0.891321i	$0.350221\pi$
$752$		−	5460.77i		−	0.264806i
$753$		0			0
$754$	7591.05			0.366644
$755$	14876.7	+	20429.1i	0.717109	+	0.984757i
$756$		0			0
$757$		−	23051.3i		−	1.10675i	−0.832931	−	0.553377i	$-0.813339\pi$
$757$		−	23051.3i		−	1.10675i	0.832931	−	0.553377i	$-0.186661\pi$
$758$			7485.71i			0.358698i
$759$		0			0
$760$	2450.09	−	1784.18i	0.116939	−	0.0851564i
$761$	18693.9			0.890478			0.445239	−	0.895412i	$-0.353119\pi$
$761$	18693.9			0.890478			0.445239	+	0.895412i	$0.353119\pi$
$762$		0			0
$763$			7264.30i			0.344673i
$764$	−38993.5			−1.84651
$765$		0			0
$766$	−7746.14			−0.365378
$767$			2170.35i			0.102173i
$768$		0			0
$769$	−6553.93			−0.307335			−0.153668	−	0.988123i	$-0.549109\pi$
$769$	−6553.93			−0.307335			−0.153668	+	0.988123i	$0.549109\pi$
$770$	−1546.47	−	2123.66i	−0.0723777	−	0.0993913i
$771$		0			0
$772$			2676.23i			0.124767i
$773$		−	10818.2i		−	0.503368i	−0.967809	−	0.251684i	$-0.919016\pi$
$773$		−	10818.2i		−	0.503368i	0.967809	−	0.251684i	$-0.0809844\pi$
$774$		0			0
$775$	10386.5	−	32205.0i	0.481410	−	1.49269i
$776$	−4273.75			−0.197705
$777$		0			0
$778$		−	37284.7i		−	1.71815i
$779$	9407.01			0.432658
$780$		0			0
$781$	−2490.59			−0.114111
$782$		−	11073.6i		−	0.506381i
$783$		0			0
$784$	2761.24			0.125786
$785$	5819.30	−	4237.67i	0.264586	−	0.192674i
$786$		0			0
$787$		−	28571.9i		−	1.29413i	−0.762436	−	0.647064i	$-0.775997\pi$
$787$		−	28571.9i		−	1.29413i	0.762436	−	0.647064i	$-0.224003\pi$
$788$		−	8445.93i		−	0.381819i
$789$		0			0
$790$	−43214.6	+	31469.3i	−1.94621	+	1.41725i
$791$	−10357.4			−0.465571
$792$		0			0
$793$			6673.88i			0.298860i
$794$	10762.8			0.481054
$795$		0			0
$796$	36558.7			1.62788
$797$			531.831i			0.0236367i	0.999930	+	0.0118183i	$0.00376198\pi$
$797$			531.831i			0.0236367i	−0.999930	+	0.0118183i	$0.996238\pi$
$798$		0			0
$799$	−1831.33			−0.0810862
$800$	30902.0	+	9966.23i	1.36569	+	0.440449i
$801$		0			0
$802$			44152.9i			1.94401i
$803$			6163.33i			0.270858i
$804$		0			0
$805$	−6573.88	−	9027.46i	−0.287825	−	0.395250i
$806$	21370.6			0.933929
$807$		0			0
$808$			1546.44i			0.0673313i
$809$	44326.9			1.92639			0.963196	−	0.268801i	$-0.0866274\pi$
$809$	44326.9			1.92639			0.963196	+	0.268801i	$0.0866274\pi$
$810$		0			0
$811$	9509.56			0.411746			0.205873	−	0.978579i	$-0.433997\pi$
$811$	9509.56			0.411746			0.205873	+	0.978579i	$0.433997\pi$
$812$			5965.70i			0.257827i
$813$		0			0
$814$	11265.7			0.485089
$815$	−27924.1	+	20334.6i	−1.20017	+	0.873974i
$816$		0			0
$817$		−	37682.7i		−	1.61365i
$818$		−	54952.7i		−	2.34887i
$819$		0			0
$820$	−7172.70	−	9849.78i	−0.305465	−	0.419474i
$821$	−45600.8			−1.93846			−0.969232	−	0.246148i	$-0.920835\pi$
$821$	−45600.8			−1.93846			−0.969232	+	0.246148i	$0.920835\pi$
$822$		0			0
$823$		−	14111.4i		−	0.597681i	−0.954303	−	0.298841i	$-0.903400\pi$
$823$		−	14111.4i		−	0.597681i	0.954303	−	0.298841i	$-0.0965999\pi$
$824$	6256.10			0.264492
$825$		0			0
$826$	−3245.23			−0.136702
$827$		−	10907.3i		−	0.458628i	−0.973352	−	0.229314i	$-0.926352\pi$
$827$		−	10907.3i		−	0.458628i	0.973352	−	0.229314i	$-0.0736483\pi$
$828$		0			0
$829$	6653.10			0.278736			0.139368	−	0.990241i	$-0.455493\pi$
$829$	6653.10			0.278736			0.139368	+	0.990241i	$0.455493\pi$
$830$	24031.7	+	33001.1i	1.00500	+	1.38010i
$831$		0			0
$832$			11839.3i			0.493336i
$833$		−	926.014i		−	0.0385168i
$834$		0			0
$835$	−24644.9	+	17946.7i	−1.02140	+	0.743797i
$836$	−5542.43			−0.229293
$837$		0			0
$838$			9592.28i			0.395418i
$839$	−4960.81			−0.204131			−0.102066	−	0.994778i	$-0.532545\pi$
$839$	−4960.81			−0.204131			−0.102066	+	0.994778i	$0.532545\pi$
$840$		0			0
$841$	−15142.6			−0.620878
$842$			17504.6i			0.716447i
$843$		0			0
$844$	8993.59			0.366791
$845$	12027.2	+	16516.1i	0.489644	+	0.672394i
$846$		0			0
$847$		−	8849.26i		−	0.358990i
$848$		−	21898.5i		−	0.886789i
$849$		0			0
$850$	2977.52	−	9232.32i	0.120151	−	0.372548i
$851$	47889.4			1.92906
$852$		0			0
$853$		−	26053.2i		−	1.04577i	−0.852402	−	0.522887i	$-0.824855\pi$
$853$		−	26053.2i		−	1.04577i	0.852402	−	0.522887i	$-0.175145\pi$
$854$	−9979.15			−0.399859
$855$		0			0
$856$	1079.37			0.0430983
$857$		−	42983.0i		−	1.71327i	−0.515924	−	0.856635i	$-0.672551\pi$
$857$		−	42983.0i		−	1.71327i	0.515924	−	0.856635i	$-0.327449\pi$
$858$		0			0
$859$	31824.0			1.26405			0.632027	−	0.774946i	$-0.282223\pi$
$859$	31824.0			1.26405			0.632027	+	0.774946i	$0.282223\pi$
$860$	−39456.4	+	28732.5i	−1.56448	+	1.13927i
$861$		0			0
$862$		−	21492.2i		−	0.849221i
$863$			20388.4i			0.804207i	0.915594	+	0.402103i	$0.131721\pi$
$863$			20388.4i			0.804207i	−0.915594	+	0.402103i	$0.868279\pi$
$864$		0			0
$865$	8151.43	−	5935.95i	0.320413	−	0.233327i
$866$	60067.2			2.35701
$867$		0			0
$868$			16794.8i			0.656745i
$869$	9518.04			0.371551
$870$		0			0
$871$	−1954.20			−0.0760223
$872$		−	3677.38i		−	0.142812i
$873$		0			0
$874$	−44826.7			−1.73488
$875$	−3053.46	−	9294.06i	−0.117972	−	0.359082i
$876$		0			0
$877$			15298.2i			0.589034i	0.955646	+	0.294517i	$0.0951588\pi$
$877$			15298.2i			0.589034i	−0.955646	+	0.294517i	$0.904841\pi$
$878$		−	144.943i		−	0.00557128i
$879$		0			0
$880$	−3031.69	−	4163.21i	−0.116134	−	0.159479i
$881$	−22994.9			−0.879361			−0.439681	−	0.898154i	$-0.644908\pi$
$881$	−22994.9			−0.879361			−0.439681	+	0.898154i	$0.644908\pi$
$882$		0			0
$883$			799.661i			0.0304765i	0.999884	+	0.0152382i	$0.00485067\pi$
$883$			799.661i			0.0304765i	−0.999884	+	0.0152382i	$0.995149\pi$
$884$	3219.94			0.122509
$885$		0			0
$886$	−11520.2			−0.436828
$887$			6611.21i			0.250262i	0.992140	+	0.125131i	$0.0399352\pi$
$887$			6611.21i			0.250262i	−0.992140	+	0.125131i	$0.960065\pi$
$888$		0			0
$889$	−11434.7			−0.431391
$890$	34843.4	−	25373.3i	1.31231	−	0.955636i
$891$		0			0
$892$			44695.3i			1.67770i
$893$			7413.38i			0.277804i
$894$		0			0
$895$	29930.1	+	41101.0i	1.11783	+	1.53503i
$896$	−3156.58			−0.117694
$897$		0			0
$898$			61198.4i			2.27418i
$899$	26030.8			0.965712
$900$		0			0
$901$	−7343.90			−0.271544
$902$			4127.61i			0.152366i
$903$		0			0
$904$	5243.18			0.192904
$905$	−2645.78	−	3633.27i	−0.0971809	−	0.133452i
$906$		0			0
$907$		−	11367.9i		−	0.416169i	−0.978111	−	0.208084i	$-0.933277\pi$
$907$		−	11367.9i		−	0.416169i	0.978111	−	0.208084i	$-0.0667228\pi$
$908$			51685.3i			1.88903i
$909$		0			0
$910$	4994.38	−	3636.96i	0.181936	−	0.132488i
$911$	18431.4			0.670318			0.335159	−	0.942162i	$-0.391210\pi$
$911$	18431.4			0.670318			0.335159	+	0.942162i	$0.391210\pi$
$912$		0			0
$913$		−	7268.51i		−	0.263475i
$914$	−10815.8			−0.391415
$915$		0			0
$916$	164.546			0.00593533
$917$			2515.01i			0.0905703i
$918$		0			0
$919$	16263.1			0.583755			0.291877	−	0.956456i	$-0.405720\pi$
$919$	16263.1			0.583755			0.291877	+	0.956456i	$0.405720\pi$
$920$	3327.87	+	4569.93i	0.119257	+	0.163768i
$921$		0			0
$922$			75621.0i			2.70113i
$923$		−	5857.33i		−	0.208880i
$924$		0			0
$925$	39926.6	+	12876.8i	1.41922	+	0.457714i
$926$	−60820.6			−2.15841
$927$		0			0
$928$			24977.6i			0.883544i
$929$	−3972.84			−0.140306			−0.0701532	−	0.997536i	$-0.522349\pi$
$929$	−3972.84			−0.140306			−0.0701532	+	0.997536i	$0.522349\pi$
$930$		0			0
$931$	−3748.58			−0.131960
$932$		−	15356.1i		−	0.539706i
$933$		0			0
$934$	59161.9			2.07263
$935$	−1396.18	+	1016.71i	−0.0488342	+	0.0355615i
$936$		0			0
$937$		−	144.199i		−	0.00502750i	−0.999997	−	0.00251375i	$-0.999200\pi$
$937$		−	144.199i		−	0.00502750i	0.999997	−	0.00251375i	$-0.000800152\pi$
$938$		−	2922.02i		−	0.101714i
$939$		0			0
$940$	7762.31	−	5652.59i	0.269339	−	0.196135i
$941$	−48589.1			−1.68327			−0.841637	−	0.540044i	$-0.818407\pi$
$941$	−48589.1			−1.68327			−0.841637	+	0.540044i	$0.818407\pi$
$942$		0			0
$943$			17546.1i			0.605916i
$944$	−6361.93			−0.219347
$945$		0			0
$946$	16534.5			0.568268
$947$		−	11977.2i		−	0.410989i	−0.978658	−	0.205494i	$-0.934120\pi$
$947$		−	11977.2i		−	0.410989i	0.978658	−	0.205494i	$-0.0658802\pi$
$948$		0			0
$949$	−14494.8			−0.495808
$950$	−37373.2	−	12053.3i	−1.27636	−	0.411641i
$951$		0			0
$952$			468.772i			0.0159590i
$953$			45863.7i			1.55894i	0.626439	+	0.779471i	$0.284512\pi$
$953$			45863.7i			1.55894i	−0.626439	+	0.779471i	$0.715488\pi$
$954$		0			0
$955$	28956.1	+	39763.4i	0.981149	+	1.34734i
$956$	17172.3			0.580953
$957$		0			0
$958$		−	27222.3i		−	0.918073i
$959$	−18016.0			−0.606638
$960$		0			0
$961$	43491.7			1.45989
$962$			26494.5i			0.887959i
$963$		0			0
$964$	−49747.8			−1.66210
$965$	2729.07	−	1987.34i	0.0910383	−	0.0662949i
$966$		0			0
$967$		−	19843.3i		−	0.659894i	−0.944000	−	0.329947i	$-0.892969\pi$
$967$		−	19843.3i		−	0.659894i	0.944000	−	0.329947i	$-0.107031\pi$
$968$			4479.73i			0.148744i
$969$		0			0
$970$	32595.6	+	44761.2i	1.07895	+	1.48165i
$971$	45035.7			1.48843			0.744214	−	0.667941i	$-0.232824\pi$
$971$	45035.7			1.48843			0.744214	+	0.667941i	$0.232824\pi$
$972$		0			0
$973$			15869.8i			0.522881i
$974$	−15976.0			−0.525568
$975$		0			0
$976$	−19563.1			−0.641598
$977$		−	25098.2i		−	0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$977$		−	25098.2i		−	0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$978$		0			0
$979$	−7674.29			−0.250533
$980$	2858.24	+	3925.02i	0.0931663	+	0.127939i
$981$		0			0
$982$			52579.4i			1.70863i
$983$			388.045i			0.0125908i	0.999980	+	0.00629538i	$0.00200389\pi$
$983$			388.045i			0.0125908i	−0.999980	+	0.00629538i	$0.997996\pi$
$984$		0			0
$985$	−8612.68	+	6271.84i	−0.278602	+	0.202881i
$986$	7462.33			0.241023
$987$		0			0
$988$		−	13034.6i		−	0.419722i
$989$	70286.4			2.25983
$990$		0			0
$991$	53520.8			1.71558			0.857792	−	0.513997i	$-0.171836\pi$
$991$	53520.8			1.71558			0.857792	+	0.513997i	$0.171836\pi$
$992$			70317.7i			2.25059i
$993$		0			0
$994$	8758.20			0.279470
$995$	−27148.0	−	37280.5i	−0.864975	−	1.18781i
$996$		0			0
$997$			5968.90i			0.189606i	0.995496	+	0.0948028i	$0.0302221\pi$
$997$			5968.90i			0.189606i	−0.995496	+	0.0948028i	$0.969778\pi$
$998$			23992.2i			0.760981i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.d.a.64.6		6
3.2	odd	2		105.4.d.a.64.1	✓	6
5.2	odd	4		1575.4.a.bc.1.1		3
5.3	odd	4		1575.4.a.bd.1.3		3
5.4	even	2	inner	315.4.d.a.64.1		6
15.2	even	4		525.4.a.r.1.3		3
15.8	even	4		525.4.a.q.1.1		3
15.14	odd	2		105.4.d.a.64.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.a.64.1	✓	6	3.2	odd	2
105.4.d.a.64.6	yes	6	15.14	odd	2
315.4.d.a.64.1		6	5.4	even	2	inner
315.4.d.a.64.6		6	1.1	even	1	trivial
525.4.a.q.1.1		3	15.8	even	4
525.4.a.r.1.3		3	15.2	even	4
1575.4.a.bc.1.1		3	5.2	odd	4
1575.4.a.bd.1.3		3	5.3	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.10645i q^{2} -8.86293 q^{4} +(6.58150 + 9.03792i) q^{5} +7.00000i q^{7} -3.54358i q^{8} +O(q^{10})\)	\(q+4.10645i q^{2} -8.86293 q^{4} +(6.58150 + 9.03792i) q^{5} +7.00000i q^{7} -3.54358i q^{8} +(-37.1137 + 27.0266i) q^{10} +8.17432 q^{11} +19.2242i q^{13} -28.7451 q^{14} -56.3519 q^{16} +18.8982i q^{17} +76.5016 q^{19} +(-58.3313 - 80.1024i) q^{20} +33.5675i q^{22} +142.692i q^{23} +(-38.3678 + 118.966i) q^{25} -78.9433 q^{26} -62.0405i q^{28} -96.1582 q^{29} -270.708 q^{31} -259.755i q^{32} -77.6047 q^{34} +(-63.2654 + 46.0705i) q^{35} -335.614i q^{37} +314.150i q^{38} +(32.0266 - 23.3221i) q^{40} +122.965 q^{41} -492.574i q^{43} -72.4485 q^{44} -585.957 q^{46} +96.9049i q^{47} -49.0000 q^{49} +(-488.528 - 157.556i) q^{50} -170.383i q^{52} +388.602i q^{53} +(53.7993 + 73.8788i) q^{55} +24.8051 q^{56} -394.869i q^{58} +112.896 q^{59} +347.160 q^{61} -1111.65i q^{62} +615.855 q^{64} +(-173.747 + 126.524i) q^{65} +101.653i q^{67} -167.494i q^{68} +(-189.186 - 259.796i) q^{70} -304.685 q^{71} +753.987i q^{73} +1378.18 q^{74} -678.029 q^{76} +57.2203i q^{77} +1164.38 q^{79} +(-370.880 - 509.304i) q^{80} +504.949i q^{82} -889.188i q^{83} +(-170.801 + 124.379i) q^{85} +2022.73 q^{86} -28.9664i q^{88} -938.829 q^{89} -134.570 q^{91} -1264.67i q^{92} -397.935 q^{94} +(503.495 + 691.415i) q^{95} -1206.06i q^{97} -201.216i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} + 14 q^{5}+O(q^{10}) \) 6 * q - 6 * q^4 + 14 * q^5	\( 6 q - 6 q^{4} + 14 q^{5} - 84 q^{10} + 132 q^{11} - 14 q^{14} - 138 q^{16} + 276 q^{19} - 334 q^{20} + 366 q^{25} + 196 q^{26} + 340 q^{29} - 732 q^{31} + 72 q^{34} - 56 q^{35} + 12 q^{40} + 412 q^{41} - 612 q^{44} - 1344 q^{46} - 294 q^{49} - 1216 q^{50} + 1860 q^{55} + 294 q^{56} - 1760 q^{59} - 1740 q^{61} + 1626 q^{64} + 16 q^{65} + 126 q^{70} + 2036 q^{71} + 1960 q^{74} - 900 q^{76} + 3240 q^{79} - 3794 q^{80} + 432 q^{85} + 5864 q^{86} - 3876 q^{89} - 1428 q^{91} - 4224 q^{94} + 828 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 + 14 * q^5 - 84 * q^10 + 132 * q^11 - 14 * q^14 - 138 * q^16 + 276 * q^19 - 334 * q^20 + 366 * q^25 + 196 * q^26 + 340 * q^29 - 732 * q^31 + 72 * q^34 - 56 * q^35 + 12 * q^40 + 412 * q^41 - 612 * q^44 - 1344 * q^46 - 294 * q^49 - 1216 * q^50 + 1860 * q^55 + 294 * q^56 - 1760 * q^59 - 1740 * q^61 + 1626 * q^64 + 16 * q^65 + 126 * q^70 + 2036 * q^71 + 1960 * q^74 - 900 * q^76 + 3240 * q^79 - 3794 * q^80 + 432 * q^85 + 5864 * q^86 - 3876 * q^89 - 1428 * q^91 - 4224 * q^94 + 828 * q^95

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