LMFDB - Embedded newform 830.6.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [830,6,Mod(1,830)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("830.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(830, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 830 = 2 \cdot 5 \cdot 83 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	830.a (trivial)

Newform invariants

comment:select newform

sage:traces = [19,-76,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$133.118570445$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 3120 x^{17} - 182 x^{16} + 3985173 x^{15} - 520177 x^{14} - 2703387368 x^{13} + \cdots + 14\!\cdots\!80 $ x^19 - 3120x^17 - 182x^16 + 3985173x^15 - 520177x^14 - 2703387368x^13 + 1527939501x^12 + 1057965017865x^11 - 1293897111166x^10 - 242919147398049x^9 + 497019546695269x^8 + 31697757108668755x^7 - 93900689986581909x^6 - 2121345254638742157x^5 + 8373578017102843320x^4 + 52855943066873828244x^3 - 282150136833836903856x^2 + 271275340721620982976*x + 146592101558541530880
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	multiple of $ 2^{6}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-28.0675$ of defining polynomial
Character	$\chi$	$=$	830.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} -28.0675 q^{3} +16.0000 q^{4} -25.0000 q^{5} +112.270 q^{6} -192.152 q^{7} -64.0000 q^{8} +544.786 q^{9} +100.000 q^{10} +97.2153 q^{11} -449.080 q^{12} +909.823 q^{13} +768.609 q^{14} +701.688 q^{15} +256.000 q^{16} +1372.79 q^{17} -2179.14 q^{18} +1037.11 q^{19} -400.000 q^{20} +5393.24 q^{21} -388.861 q^{22} +84.2249 q^{23} +1796.32 q^{24} +625.000 q^{25} -3639.29 q^{26} -8470.38 q^{27} -3074.43 q^{28} +8803.49 q^{29} -2806.75 q^{30} +7589.75 q^{31} -1024.00 q^{32} -2728.59 q^{33} -5491.15 q^{34} +4803.80 q^{35} +8716.57 q^{36} +1496.09 q^{37} -4148.43 q^{38} -25536.5 q^{39} +1600.00 q^{40} -3553.20 q^{41} -21572.9 q^{42} +20829.7 q^{43} +1555.45 q^{44} -13619.6 q^{45} -336.900 q^{46} -2615.07 q^{47} -7185.29 q^{48} +20115.4 q^{49} -2500.00 q^{50} -38530.8 q^{51} +14557.2 q^{52} -10555.9 q^{53} +33881.5 q^{54} -2430.38 q^{55} +12297.7 q^{56} -29109.0 q^{57} -35214.0 q^{58} +563.974 q^{59} +11227.0 q^{60} +54946.5 q^{61} -30359.0 q^{62} -104682. q^{63} +4096.00 q^{64} -22745.6 q^{65} +10914.4 q^{66} +29477.2 q^{67} +21964.6 q^{68} -2363.98 q^{69} -19215.2 q^{70} +58504.5 q^{71} -34866.3 q^{72} -51465.7 q^{73} -5984.35 q^{74} -17542.2 q^{75} +16593.7 q^{76} -18680.1 q^{77} +102146. q^{78} -66833.0 q^{79} -6400.00 q^{80} +105360. q^{81} +14212.8 q^{82} +6889.00 q^{83} +86291.8 q^{84} -34319.7 q^{85} -83318.9 q^{86} -247092. q^{87} -6221.78 q^{88} +40223.5 q^{89} +54478.6 q^{90} -174825. q^{91} +1347.60 q^{92} -213025. q^{93} +10460.3 q^{94} -25927.7 q^{95} +28741.1 q^{96} +148545. q^{97} -80461.8 q^{98} +52961.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 76 q^{2} + 304 q^{4} - 475 q^{5} + 287 q^{7} - 1216 q^{8} + 1623 q^{9} + 1900 q^{10} + 743 q^{11} + 918 q^{13} - 1148 q^{14} + 4864 q^{16} - 189 q^{17} - 6492 q^{18} + 2871 q^{19} - 7600 q^{20} + 5594 q^{21}+ \cdots + 482584 q^{99}+O(q^{100}) $ 19 * q - 76 * q^2 + 304 * q^4 - 475 * q^5 + 287 * q^7 - 1216 * q^8 + 1623 * q^9 + 1900 * q^10 + 743 * q^11 + 918 * q^13 - 1148 * q^14 + 4864 * q^16 - 189 * q^17 - 6492 * q^18 + 2871 * q^19 - 7600 * q^20 + 5594 * q^21 - 2972 * q^22 + 5711 * q^23 + 11875 * q^25 - 3672 * q^26 + 546 * q^27 + 4592 * q^28 + 8661 * q^29 + 21202 * q^31 - 19456 * q^32 - 7082 * q^33 + 756 * q^34 - 7175 * q^35 + 25968 * q^36 + 11396 * q^37 - 11484 * q^38 + 4110 * q^39 + 30400 * q^40 - 9470 * q^41 - 22376 * q^42 - 9024 * q^43 + 11888 * q^44 - 40575 * q^45 - 22844 * q^46 - 57499 * q^47 + 62522 * q^49 - 47500 * q^50 - 49150 * q^51 + 14688 * q^52 - 48794 * q^53 - 2184 * q^54 - 18575 * q^55 - 18368 * q^56 - 59416 * q^57 - 34644 * q^58 - 11195 * q^59 + 19177 * q^61 - 84808 * q^62 - 62149 * q^63 + 77824 * q^64 - 22950 * q^65 + 28328 * q^66 - 6486 * q^67 - 3024 * q^68 - 9252 * q^69 + 28700 * q^70 - 69656 * q^71 - 103872 * q^72 + 24875 * q^73 - 45584 * q^74 + 45936 * q^76 - 102852 * q^77 - 16440 * q^78 + 82194 * q^79 - 121600 * q^80 + 101079 * q^81 + 37880 * q^82 + 130891 * q^83 + 89504 * q^84 + 4725 * q^85 + 36096 * q^86 - 22509 * q^87 - 47552 * q^88 + 56210 * q^89 + 162300 * q^90 + 154764 * q^91 + 91376 * q^92 - 319309 * q^93 + 229996 * q^94 - 71775 * q^95 + 445215 * q^97 - 250088 * q^98 + 482584 * q^99

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−28.0675	−1.80053	−0.900266	−	0.435340i	$-0.856628\pi$
$3$	−28.0675	−1.80053	−0.900266	+	0.435340i	$0.856628\pi$
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	112.270	1.27317
$7$	−192.152	−1.48218	−0.741089	−	0.671407i	$-0.765690\pi$
$7$	−192.152	−1.48218	−0.741089	+	0.671407i	$0.765690\pi$
$8$	−64.0000	−0.353553
$9$	544.786	2.24192
$10$	100.000	0.316228
$11$	97.2153	0.242244	0.121122	−	0.992638i	$-0.461351\pi$
$11$	97.2153	0.242244	0.121122	+	0.992638i	$0.461351\pi$
$12$	−449.080	−0.900266
$13$	909.823	1.49313	0.746567	−	0.665310i	$-0.231701\pi$
$13$	909.823	1.49313	0.746567	+	0.665310i	$0.231701\pi$
$14$	768.609	1.04806
$15$	701.688	0.805223
$16$	256.000	0.250000
$17$	1372.79	1.15208	0.576038	−	0.817423i	$-0.304598\pi$
$17$	1372.79	1.15208	0.576038	+	0.817423i	$0.304598\pi$
$18$	−2179.14	−1.58527
$19$	1037.11	0.659082	0.329541	−	0.944141i	$-0.393106\pi$
$19$	1037.11	0.659082	0.329541	+	0.944141i	$0.393106\pi$
$20$	−400.000	−0.223607
$21$	5393.24	2.66871
$22$	−388.861	−0.171292
$23$	84.2249	0.0331987	0.0165993	−	0.999862i	$-0.494716\pi$
$23$	84.2249	0.0331987	0.0165993	+	0.999862i	$0.494716\pi$
$24$	1796.32	0.636584
$25$	625.000	0.200000
$26$	−3639.29	−1.05581
$27$	−8470.38	−2.23611
$28$	−3074.43	−0.741089
$29$	8803.49	1.94384	0.971919	−	0.235317i	$-0.0756130\pi$
$29$	8803.49	1.94384	0.971919	+	0.235317i	$0.0756130\pi$
$30$	−2806.75	−0.569378
$31$	7589.75	1.41848	0.709240	−	0.704968i	$-0.249038\pi$
$31$	7589.75	1.41848	0.709240	+	0.704968i	$0.249038\pi$
$32$	−1024.00	−0.176777
$33$	−2728.59	−0.436168
$34$	−5491.15	−0.814641
$35$	4803.80	0.662850
$36$	8716.57	1.12096
$37$	1496.09	0.179661	0.0898303	−	0.995957i	$-0.471368\pi$
$37$	1496.09	0.179661	0.0898303	+	0.995957i	$0.471368\pi$
$38$	−4148.43	−0.466041
$39$	−25536.5	−2.68844
$40$	1600.00	0.158114
$41$	−3553.20	−0.330111	−0.165056	−	0.986284i	$-0.552780\pi$
$41$	−3553.20	−0.330111	−0.165056	+	0.986284i	$0.552780\pi$
$42$	−21572.9	−1.88706
$43$	20829.7	1.71796	0.858979	−	0.512011i	$-0.171099\pi$
$43$	20829.7	1.71796	0.858979	+	0.512011i	$0.171099\pi$
$44$	1555.45	0.121122
$45$	−13619.6	−1.00262
$46$	−336.900	−0.0234750
$47$	−2615.07	−0.172678	−0.0863392	−	0.996266i	$-0.527517\pi$
$47$	−2615.07	−0.172678	−0.0863392	+	0.996266i	$0.527517\pi$
$48$	−7185.29	−0.450133
$49$	20115.4	1.19685
$50$	−2500.00	−0.141421
$51$	−38530.8	−2.07435
$52$	14557.2	0.746567
$53$	−10555.9	−0.516185	−0.258092	−	0.966120i	$-0.583094\pi$
$53$	−10555.9	−0.516185	−0.258092	+	0.966120i	$0.583094\pi$
$54$	33881.5	1.58117
$55$	−2430.38	−0.108335
$56$	12297.7	0.524029
$57$	−29109.0	−1.18670
$58$	−35214.0	−1.37450
$59$	563.974	0.0210926	0.0105463	−	0.999944i	$-0.496643\pi$
$59$	563.974	0.0210926	0.0105463	+	0.999944i	$0.496643\pi$
$60$	11227.0	0.402611
$61$	54946.5	1.89067	0.945334	−	0.326105i	$-0.105736\pi$
$61$	54946.5	1.89067	0.945334	+	0.326105i	$0.105736\pi$
$62$	−30359.0	−1.00302
$63$	−104682.	−3.32292
$64$	4096.00	0.125000
$65$	−22745.6	−0.667750
$66$	10914.4	0.308417
$67$	29477.2	0.802232	0.401116	−	0.916027i	$-0.368622\pi$
$67$	29477.2	0.802232	0.401116	+	0.916027i	$0.368622\pi$
$68$	21964.6	0.576038
$69$	−2363.98	−0.0597753
$70$	−19215.2	−0.468706
$71$	58504.5	1.37735	0.688673	−	0.725072i	$-0.258193\pi$
$71$	58504.5	1.37735	0.688673	+	0.725072i	$0.258193\pi$
$72$	−34866.3	−0.792637
$73$	−51465.7	−1.13035	−0.565173	−	0.824973i	$-0.691190\pi$
$73$	−51465.7	−1.13035	−0.565173	+	0.824973i	$0.691190\pi$
$74$	−5984.35	−0.127039
$75$	−17542.2	−0.360106
$76$	16593.7	0.329541
$77$	−18680.1	−0.359049
$78$	102146.	1.90101
$79$	−66833.0	−1.20482	−0.602411	−	0.798186i	$-0.705793\pi$
$79$	−66833.0	−1.20482	−0.602411	+	0.798186i	$0.705793\pi$
$80$	−6400.00	−0.111803
$81$	105360.	1.78428
$82$	14212.8	0.233424
$83$	6889.00	0.109764
$83$	6889.00	0.109764
$84$	86291.8	1.33435
$85$	−34319.7	−0.515224
$86$	−83318.9	−1.21478
$87$	−247092.	−3.49994
$88$	−6221.78	−0.0856462
$89$	40223.5	0.538275	0.269138	−	0.963102i	$-0.413261\pi$
$89$	40223.5	0.538275	0.269138	+	0.963102i	$0.413261\pi$
$90$	54478.6	0.708956
$91$	−174825.	−2.21309
$92$	1347.60	0.0165993
$93$	−213025.	−2.55402
$94$	10460.3	0.122102
$95$	−25927.7	−0.294750
$96$	28741.1	0.318292
$97$	148545.	1.60299	0.801494	−	0.598003i	$-0.204039\pi$
$97$	148545.	1.60299	0.801494	+	0.598003i	$0.204039\pi$
$98$	−80461.8	−0.846300
$99$	52961.5	0.543091
$100$	10000.0	0.100000
$101$	145195.	1.41628	0.708141	−	0.706071i	$-0.249534\pi$
$101$	145195.	1.41628	0.708141	+	0.706071i	$0.249534\pi$
$102$	154123.	1.46679
$103$	−176732.	−1.64142	−0.820712	−	0.571342i	$-0.806423\pi$
$103$	−176732.	−1.64142	−0.820712	+	0.571342i	$0.806423\pi$
$104$	−58228.7	−0.527903
$105$	−134831.	−1.19348
$106$	42223.5	0.364998
$107$	−28065.2	−0.236978	−0.118489	−	0.992955i	$-0.537805\pi$
$107$	−28065.2	−0.236978	−0.118489	+	0.992955i	$0.537805\pi$
$108$	−135526.	−1.11806
$109$	43860.7	0.353598	0.176799	−	0.984247i	$-0.443426\pi$
$109$	43860.7	0.353598	0.176799	+	0.984247i	$0.443426\pi$
$110$	9721.53	0.0766043
$111$	−41991.5	−0.323485
$112$	−49191.0	−0.370544
$113$	154568.	1.13874	0.569370	−	0.822082i	$-0.307187\pi$
$113$	154568.	1.13874	0.569370	+	0.822082i	$0.307187\pi$
$114$	116436.	0.839123
$115$	−2105.62	−0.0148469
$116$	140856.	0.971919
$117$	495659.	3.34748
$118$	−2255.90	−0.0149147
$119$	−263784.	−1.70758
$120$	−44908.0	−0.284689
$121$	−151600.	−0.941318
$122$	−219786.	−1.33690
$123$	99729.5	0.594376
$124$	121436.	0.709240
$125$	−15625.0	−0.0894427
$126$	418727.	2.34966
$127$	−191140.	−1.05158	−0.525790	−	0.850614i	$-0.676230\pi$
$127$	−191140.	−1.05158	−0.525790	+	0.850614i	$0.676230\pi$
$128$	−16384.0	−0.0883883
$129$	−584639.	−3.09324
$130$	90982.3	0.472170
$131$	70118.2	0.356987	0.178494	−	0.983941i	$-0.442878\pi$
$131$	70118.2	0.356987	0.178494	+	0.983941i	$0.442878\pi$
$132$	−43657.5	−0.218084
$133$	−199282.	−0.976876
$134$	−117909.	−0.567264
$135$	211760.	1.00002
$136$	−87858.4	−0.407320
$137$	121966.	0.555186	0.277593	−	0.960699i	$-0.410463\pi$
$137$	121966.	0.555186	0.277593	+	0.960699i	$0.410463\pi$
$138$	9455.94	0.0422675
$139$	−226115.	−0.992643	−0.496321	−	0.868139i	$-0.665316\pi$
$139$	−226115.	−0.992643	−0.496321	+	0.868139i	$0.665316\pi$
$140$	76860.9	0.331425
$141$	73398.4	0.310913
$142$	−234018.	−0.973931
$143$	88448.8	0.361703
$144$	139465.	0.560479
$145$	−220087.	−0.869310
$146$	205863.	0.799275
$147$	−564591.	−2.15497
$148$	23937.4	0.0898303
$149$	−421575.	−1.55564	−0.777821	−	0.628486i	$-0.783675\pi$
$149$	−421575.	−1.55564	−0.777821	+	0.628486i	$0.783675\pi$
$150$	70168.8	0.254634
$151$	305591.	1.09068	0.545342	−	0.838214i	$-0.316400\pi$
$151$	305591.	1.09068	0.545342	+	0.838214i	$0.316400\pi$
$152$	−66374.8	−0.233021
$153$	747875.	2.58286
$154$	74720.5	0.253886
$155$	−189744.	−0.634363
$156$	−408584.	−1.34422
$157$	−86840.6	−0.281173	−0.140586	−	0.990068i	$-0.544899\pi$
$157$	−86840.6	−0.281173	−0.140586	+	0.990068i	$0.544899\pi$
$158$	267332.	0.851938
$159$	296278.	0.929407
$160$	25600.0	0.0790569
$161$	−16184.0	−0.0492063
$162$	−421439.	−1.26167
$163$	−116653.	−0.343897	−0.171949	−	0.985106i	$-0.555006\pi$
$163$	−116653.	−0.343897	−0.171949	+	0.985106i	$0.555006\pi$
$164$	−56851.2	−0.165056
$165$	68214.8	0.195060
$166$	−27556.0	−0.0776151
$167$	−641686.	−1.78046	−0.890229	−	0.455514i	$-0.849456\pi$
$167$	−641686.	−1.78046	−0.890229	+	0.455514i	$0.849456\pi$
$168$	−345167.	−0.943531
$169$	456486.	1.22945
$170$	137279.	0.364318
$171$	565001.	1.47761
$172$	333276.	0.858979
$173$	250461.	0.636247	0.318123	−	0.948049i	$-0.396947\pi$
$173$	250461.	0.636247	0.318123	+	0.948049i	$0.396947\pi$
$174$	988369.	2.47483
$175$	−120095.	−0.296435
$176$	24887.1	0.0605610
$177$	−15829.4	−0.0379778
$178$	−160894.	−0.380618
$179$	255258.	0.595452	0.297726	−	0.954651i	$-0.403772\pi$
$179$	255258.	0.595452	0.297726	+	0.954651i	$0.403772\pi$
$180$	−217914.	−0.501308
$181$	561844.	1.27473	0.637367	−	0.770561i	$-0.280024\pi$
$181$	561844.	1.27473	0.637367	+	0.770561i	$0.280024\pi$
$182$	699298.	1.56489
$183$	−1.54221e6	−3.40421
$184$	−5390.39	−0.0117375
$185$	−37402.2	−0.0803467
$186$	852101.	1.80596
$187$	133456.	0.279083
$188$	−41841.0	−0.0863392
$189$	1.62760e6	3.31431
$190$	103711.	0.208420
$191$	127748.	0.253379	0.126690	−	0.991942i	$-0.459565\pi$
$191$	127748.	0.253379	0.126690	+	0.991942i	$0.459565\pi$
$192$	−114965.	−0.225067
$193$	833104.	1.60993	0.804963	−	0.593325i	$-0.202185\pi$
$193$	833104.	1.60993	0.804963	+	0.593325i	$0.202185\pi$
$194$	−594182.	−1.13348
$195$	638412.	1.20231
$196$	321847.	0.598425
$197$	343532.	0.630669	0.315334	−	0.948981i	$-0.397883\pi$
$197$	343532.	0.630669	0.315334	+	0.948981i	$0.397883\pi$
$198$	−211846.	−0.384023
$199$	785839.	1.40670	0.703349	−	0.710845i	$-0.251687\pi$
$199$	785839.	1.40670	0.703349	+	0.710845i	$0.251687\pi$
$200$	−40000.0	−0.0707107
$201$	−827353.	−1.44444
$202$	−580782.	−1.00146
$203$	−1.69161e6	−2.88111
$204$	−616492.	−1.03718
$205$	88830.0	0.147630
$206$	706926.	1.16066
$207$	45884.5	0.0744287
$208$	232915.	0.373283
$209$	100823.	0.159659
$210$	539324.	0.843920
$211$	1.07241e6	1.65826	0.829131	−	0.559055i	$-0.188836\pi$
$211$	1.07241e6	1.65826	0.829131	+	0.559055i	$0.188836\pi$
$212$	−168894.	−0.258092
$213$	−1.64208e6	−2.47996
$214$	112261.	0.167569
$215$	−520743.	−0.768294
$216$	542104.	0.790585
$217$	−1.45839e6	−2.10244
$218$	−175443.	−0.250032
$219$	1.44452e6	2.03522
$220$	−38886.1	−0.0541674
$221$	1.24899e6	1.72020
$222$	167966.	0.228738
$223$	562600.	0.757595	0.378798	−	0.925480i	$-0.376338\pi$
$223$	562600.	0.757595	0.378798	+	0.925480i	$0.376338\pi$
$224$	196764.	0.262014
$225$	340491.	0.448383
$226$	−618273.	−0.805210
$227$	1.22507e6	1.57797	0.788984	−	0.614414i	$-0.210608\pi$
$227$	1.22507e6	1.57797	0.788984	+	0.614414i	$0.210608\pi$
$228$	−465744.	−0.593349
$229$	54900.3	0.0691809	0.0345905	−	0.999402i	$-0.488987\pi$
$229$	54900.3	0.0691809	0.0345905	+	0.999402i	$0.488987\pi$
$230$	8422.49	0.0104983
$231$	524305.	0.646479
$232$	−563423.	−0.687250
$233$	105087.	0.126812	0.0634059	−	0.997988i	$-0.479804\pi$
$233$	105087.	0.126812	0.0634059	+	0.997988i	$0.479804\pi$
$234$	−1.98264e6	−2.36703
$235$	65376.6	0.0772241
$236$	9023.59	0.0105463
$237$	1.87584e6	2.16932
$238$	1.05514e6	1.20744
$239$	1.62530e6	1.84051	0.920256	−	0.391317i	$-0.127981\pi$
$239$	1.62530e6	1.84051	0.920256	+	0.391317i	$0.127981\pi$
$240$	179632.	0.201306
$241$	582976.	0.646559	0.323279	−	0.946304i	$-0.395215\pi$
$241$	582976.	0.646559	0.323279	+	0.946304i	$0.395215\pi$
$242$	606401.	0.665612
$243$	−898883.	−0.976534
$244$	879143.	0.945334
$245$	−502886.	−0.535247
$246$	−398918.	−0.420287
$247$	943584.	0.984098
$248$	−485744.	−0.501508
$249$	−193357.	−0.197634
$250$	62500.0	0.0632456
$251$	−1.44430e6	−1.44701	−0.723507	−	0.690317i	$-0.757471\pi$
$251$	−1.44430e6	−1.44701	−0.723507	+	0.690317i	$0.757471\pi$
$252$	−1.67491e6	−1.66146
$253$	8187.95	0.00804218
$254$	764560.	0.743580
$255$	963269.	0.927678
$256$	65536.0	0.0625000
$257$	719521.	0.679534	0.339767	−	0.940510i	$-0.389652\pi$
$257$	719521.	0.679534	0.339767	+	0.940510i	$0.389652\pi$
$258$	2.33856e6	2.18725
$259$	−287477.	−0.266289
$260$	−363929.	−0.333875
$261$	4.79602e6	4.35792
$262$	−280473.	−0.252428
$263$	1.21020e6	1.07887	0.539435	−	0.842027i	$-0.318638\pi$
$263$	1.21020e6	1.07887	0.539435	+	0.842027i	$0.318638\pi$
$264$	174630.	0.154209
$265$	263897.	0.230845
$266$	797129.	0.690756
$267$	−1.12897e6	−0.969182
$268$	471636.	0.401116
$269$	758981.	0.639514	0.319757	−	0.947500i	$-0.396399\pi$
$269$	758981.	0.639514	0.319757	+	0.947500i	$0.396399\pi$
$270$	−847038.	−0.707121
$271$	−337731.	−0.279349	−0.139675	−	0.990197i	$-0.544606\pi$
$271$	−337731.	−0.279349	−0.139675	+	0.990197i	$0.544606\pi$
$272$	351434.	0.288019
$273$	4.90689e6	3.98474
$274$	−487865.	−0.392576
$275$	60759.6	0.0484488
$276$	−37823.7	−0.0298877
$277$	145064.	0.113595	0.0567976	−	0.998386i	$-0.481911\pi$
$277$	145064.	0.113595	0.0567976	+	0.998386i	$0.481911\pi$
$278$	904461.	0.701904
$279$	4.13479e6	3.18011
$280$	−307443.	−0.234353
$281$	−1.09533e6	−0.827518	−0.413759	−	0.910386i	$-0.635784\pi$
$281$	−1.09533e6	−0.827518	−0.413759	+	0.910386i	$0.635784\pi$
$282$	−293594.	−0.219849
$283$	1.05176e6	0.780642	0.390321	−	0.920679i	$-0.372364\pi$
$283$	1.05176e6	0.780642	0.390321	+	0.920679i	$0.372364\pi$
$284$	936072.	0.688673
$285$	727725.	0.530708
$286$	−353795.	−0.255762
$287$	682755.	0.489283
$288$	−557861.	−0.396319
$289$	464690.	0.327279
$290$	880349.	0.614695
$291$	−4.16930e6	−2.88623
$292$	−823452.	−0.565173
$293$	−1.02232e6	−0.695696	−0.347848	−	0.937551i	$-0.613088\pi$
$293$	−1.02232e6	−0.695696	−0.347848	+	0.937551i	$0.613088\pi$
$294$	2.25836e6	1.52379
$295$	−14099.4	−0.00943288
$296$	−95749.7	−0.0635196
$297$	−823451.	−0.541685
$298$	1.68630e6	1.10000
$299$	76629.8	0.0495701
$300$	−280675.	−0.180053
$301$	−4.00248e6	−2.54632
$302$	−1.22237e6	−0.771230
$303$	−4.07528e6	−2.55006
$304$	265499.	0.164770
$305$	−1.37366e6	−0.845532
$306$	−2.99150e6	−1.82636
$307$	−2.56832e6	−1.55526	−0.777631	−	0.628721i	$-0.783579\pi$
$307$	−2.56832e6	−1.55526	−0.777631	+	0.628721i	$0.783579\pi$
$308$	−298882.	−0.179524
$309$	4.96042e6	2.95544
$310$	758975.	0.448563
$311$	−2.75284e6	−1.61391	−0.806955	−	0.590613i	$-0.798886\pi$
$311$	−2.75284e6	−1.61391	−0.806955	+	0.590613i	$0.798886\pi$
$312$	1.63434e6	0.950506
$313$	−2.17863e6	−1.25696	−0.628481	−	0.777825i	$-0.716323\pi$
$313$	−2.17863e6	−1.25696	−0.628481	+	0.777825i	$0.716323\pi$
$314$	347362.	0.198819
$315$	2.61704e6	1.48605
$316$	−1.06933e6	−0.602411
$317$	−1.58029e6	−0.883262	−0.441631	−	0.897197i	$-0.645600\pi$
$317$	−1.58029e6	−0.883262	−0.441631	+	0.897197i	$0.645600\pi$
$318$	−1.18511e6	−0.657190
$319$	855834.	0.470883
$320$	−102400.	−0.0559017
$321$	787720.	0.426687
$322$	64736.0	0.0347941
$323$	1.42373e6	0.759312
$324$	1.68575e6	0.892138
$325$	568640.	0.298627
$326$	466614.	0.243172
$327$	−1.23106e6	−0.636665
$328$	227405.	0.116712
$329$	502490.	0.255940
$330$	−272859.	−0.137928
$331$	−359081.	−0.180145	−0.0900725	−	0.995935i	$-0.528710\pi$
$331$	−359081.	−0.180145	−0.0900725	+	0.995935i	$0.528710\pi$
$332$	110224.	0.0548821
$333$	815048.	0.402784
$334$	2.56675e6	1.25897
$335$	−736931.	−0.358769
$336$	1.38067e6	0.667177
$337$	−744940.	−0.357311	−0.178656	−	0.983912i	$-0.557175\pi$
$337$	−744940.	−0.357311	−0.178656	+	0.983912i	$0.557175\pi$
$338$	−1.82594e6	−0.869351
$339$	−4.33835e6	−2.05034
$340$	−549115.	−0.257612
$341$	737840.	0.343618
$342$	−2.26000e6	−1.04483
$343$	−635726.	−0.291766
$344$	−1.33310e6	−0.607390
$345$	59099.6	0.0267323
$346$	−1.00185e6	−0.449894
$347$	−4.15513e6	−1.85251	−0.926256	−	0.376896i	$-0.876991\pi$
$347$	−4.15513e6	−1.85251	−0.926256	+	0.376896i	$0.876991\pi$
$348$	−3.95347e6	−1.74997
$349$	−144345.	−0.0634362	−0.0317181	−	0.999497i	$-0.510098\pi$
$349$	−144345.	−0.0634362	−0.0317181	+	0.999497i	$0.510098\pi$
$350$	480380.	0.209612
$351$	−7.70655e6	−3.33881
$352$	−99548.5	−0.0428231
$353$	−2.31427e6	−0.988502	−0.494251	−	0.869319i	$-0.664558\pi$
$353$	−2.31427e6	−0.988502	−0.494251	+	0.869319i	$0.664558\pi$
$354$	63317.5	0.0268544
$355$	−1.46261e6	−0.615968
$356$	643576.	0.269138
$357$	7.40377e6	3.07455
$358$	−1.02103e6	−0.421048
$359$	2.06775e6	0.846762	0.423381	−	0.905952i	$-0.360843\pi$
$359$	2.06775e6	0.846762	0.423381	+	0.905952i	$0.360843\pi$
$360$	871657.	0.354478
$361$	−1.40051e6	−0.565611
$362$	−2.24738e6	−0.901373
$363$	4.25504e6	1.69487
$364$	−2.79719e6	−1.10654
$365$	1.28664e6	0.505506
$366$	6.16884e6	2.40714
$367$	−3.03769e6	−1.17727	−0.588637	−	0.808397i	$-0.700335\pi$
$367$	−3.03769e6	−1.17727	−0.588637	+	0.808397i	$0.700335\pi$
$368$	21561.6	0.00829967
$369$	−1.93573e6	−0.740082
$370$	149609.	0.0568137
$371$	2.02834e6	0.765077
$372$	−3.40841e6	−1.27701
$373$	1.49831e6	0.557608	0.278804	−	0.960348i	$-0.410062\pi$
$373$	1.49831e6	0.557608	0.278804	+	0.960348i	$0.410062\pi$
$374$	−533824.	−0.197342
$375$	438555.	0.161045
$376$	167364.	0.0610510
$377$	8.00962e6	2.90241
$378$	−6.51041e6	−2.34357
$379$	−2.12100e6	−0.758476	−0.379238	−	0.925299i	$-0.623814\pi$
$379$	−2.12100e6	−0.758476	−0.379238	+	0.925299i	$0.623814\pi$
$380$	−414843.	−0.147375
$381$	5.36483e6	1.89340
$382$	−510992.	−0.179166
$383$	36620.1	0.0127562	0.00637812	−	0.999980i	$-0.497970\pi$
$383$	36620.1	0.0127562	0.00637812	+	0.999980i	$0.497970\pi$
$384$	459858.	0.159146
$385$	467003.	0.160571
$386$	−3.33241e6	−1.13839
$387$	1.13477e7	3.85152
$388$	2.37673e6	0.801494
$389$	−2.56843e6	−0.860584	−0.430292	−	0.902690i	$-0.641589\pi$
$389$	−2.56843e6	−0.860584	−0.430292	+	0.902690i	$0.641589\pi$
$390$	−2.55365e6	−0.850158
$391$	115623.	0.0382474
$392$	−1.28739e6	−0.423150
$393$	−1.96804e6	−0.642767
$394$	−1.37413e6	−0.445950
$395$	1.67082e6	0.538813
$396$	847385.	0.271545
$397$	5.76867e6	1.83696	0.918479	−	0.395470i	$-0.129418\pi$
$397$	5.76867e6	1.83696	0.918479	+	0.395470i	$0.129418\pi$
$398$	−3.14336e6	−0.994686
$399$	5.59336e6	1.75890
$400$	160000.	0.0500000
$401$	−1.47448e6	−0.457906	−0.228953	−	0.973437i	$-0.573530\pi$
$401$	−1.47448e6	−0.457906	−0.228953	+	0.973437i	$0.573530\pi$
$402$	3.30941e6	1.02138
$403$	6.90533e6	2.11798
$404$	2.32313e6	0.708141
$405$	−2.63399e6	−0.797952
$406$	6.76644e6	2.03725
$407$	145443.	0.0435217
$408$	2.46597e6	0.733394
$409$	4.00211e6	1.18299	0.591495	−	0.806309i	$-0.298538\pi$
$409$	4.00211e6	1.18299	0.591495	+	0.806309i	$0.298538\pi$
$410$	−355320.	−0.104390
$411$	−3.42329e6	−0.999630
$412$	−2.82770e6	−0.820712
$413$	−108369.	−0.0312629
$414$	−183538.	−0.0526290
$415$	−172225.	−0.0490881
$416$	−931659.	−0.263951
$417$	6.34650e6	1.78729
$418$	−403291.	−0.112896
$419$	−2.78717e6	−0.775582	−0.387791	−	0.921747i	$-0.626762\pi$
$419$	−2.78717e6	−0.775582	−0.387791	+	0.921747i	$0.626762\pi$
$420$	−2.15729e6	−0.596741
$421$	−3.94957e6	−1.08604	−0.543018	−	0.839721i	$-0.682719\pi$
$421$	−3.94957e6	−1.08604	−0.543018	+	0.839721i	$0.682719\pi$
$422$	−4.28962e6	−1.17257
$423$	−1.42465e6	−0.387131
$424$	675577.	0.182499
$425$	857992.	0.230415
$426$	6.56830e6	1.75359
$427$	−1.05581e7	−2.80230
$428$	−449043.	−0.118489
$429$	−2.48254e6	−0.651257
$430$	2.08297e6	0.543266
$431$	−735837.	−0.190804	−0.0954022	−	0.995439i	$-0.530414\pi$
$431$	−735837.	−0.190804	−0.0954022	+	0.995439i	$0.530414\pi$
$432$	−2.16842e6	−0.559028
$433$	−1.83098e6	−0.469314	−0.234657	−	0.972078i	$-0.575397\pi$
$433$	−1.83098e6	−0.469314	−0.234657	+	0.972078i	$0.575397\pi$
$434$	5.83354e6	1.48665
$435$	6.17730e6	1.56522
$436$	701772.	0.176799
$437$	87350.2	0.0218807
$438$	−5.77806e6	−1.43912
$439$	1.33433e6	0.330447	0.165223	−	0.986256i	$-0.447166\pi$
$439$	1.33433e6	0.330447	0.165223	+	0.986256i	$0.447166\pi$
$440$	155545.	0.0383021
$441$	1.09586e7	2.68324
$442$	−4.99598e6	−1.21637
$443$	−2.85747e6	−0.691786	−0.345893	−	0.938274i	$-0.612424\pi$
$443$	−2.85747e6	−0.691786	−0.345893	+	0.938274i	$0.612424\pi$
$444$	−671864.	−0.161742
$445$	−1.00559e6	−0.240724
$446$	−2.25040e6	−0.535701
$447$	1.18326e7	2.80098
$448$	−787055.	−0.185272
$449$	−3.57967e6	−0.837966	−0.418983	−	0.907994i	$-0.637613\pi$
$449$	−3.57967e6	−0.837966	−0.418983	+	0.907994i	$0.637613\pi$
$450$	−1.36196e6	−0.317055
$451$	−345426.	−0.0799674
$452$	2.47309e6	0.569370
$453$	−8.57720e6	−1.96381
$454$	−4.90030e6	−1.11579
$455$	4.37061e6	0.989724
$456$	1.86298e6	0.419561
$457$	3.28314e6	0.735358	0.367679	−	0.929953i	$-0.380153\pi$
$457$	3.28314e6	0.735358	0.367679	+	0.929953i	$0.380153\pi$
$458$	−219601.	−0.0489183
$459$	−1.16280e7	−2.57617
$460$	−33690.0	−0.00742345
$461$	−6.15166e6	−1.34816	−0.674078	−	0.738660i	$-0.735459\pi$
$461$	−6.15166e6	−1.34816	−0.674078	+	0.738660i	$0.735459\pi$
$462$	−2.09722e6	−0.457129
$463$	3.84606e6	0.833804	0.416902	−	0.908952i	$-0.363116\pi$
$463$	3.84606e6	0.833804	0.416902	+	0.908952i	$0.363116\pi$
$464$	2.25369e6	0.485959
$465$	5.32563e6	1.14219
$466$	−420348.	−0.0896695
$467$	4.65720e6	0.988171	0.494086	−	0.869413i	$-0.335503\pi$
$467$	4.65720e6	0.988171	0.494086	+	0.869413i	$0.335503\pi$
$468$	7.93054e6	1.67374
$469$	−5.66412e6	−1.18905
$470$	−261507.	−0.0546057
$471$	2.43740e6	0.506261
$472$	−36094.4	−0.00745735
$473$	2.02497e6	0.416165
$474$	−7.50334e6	−1.53394
$475$	648192.	0.131816
$476$	−4.22055e6	−0.853790
$477$	−5.75070e6	−1.15724
$478$	−6.50120e6	−1.30144
$479$	2.91876e6	0.581246	0.290623	−	0.956838i	$-0.406138\pi$
$479$	2.91876e6	0.581246	0.290623	+	0.956838i	$0.406138\pi$
$480$	−718529.	−0.142345
$481$	1.36118e6	0.268257
$482$	−2.33190e6	−0.457186
$483$	454245.	0.0885976
$484$	−2.42560e6	−0.470659
$485$	−3.71364e6	−0.716878
$486$	3.59553e6	0.690514
$487$	−4.16988e6	−0.796711	−0.398356	−	0.917231i	$-0.630419\pi$
$487$	−4.16988e6	−0.796711	−0.398356	+	0.917231i	$0.630419\pi$
$488$	−3.51657e6	−0.668452
$489$	3.27417e6	0.619198
$490$	2.01154e6	0.378477
$491$	−8.00133e6	−1.49782	−0.748908	−	0.662674i	$-0.769422\pi$
$491$	−8.00133e6	−1.49782	−0.748908	+	0.662674i	$0.769422\pi$
$492$	1.59567e6	0.297188
$493$	1.20853e7	2.23945
$494$	−3.77434e6	−0.695862
$495$	−1.32404e6	−0.242878
$496$	1.94297e6	0.354620
$497$	−1.12418e7	−2.04147
$498$	773429.	0.139748
$499$	1.03370e7	1.85841	0.929207	−	0.369560i	$-0.120492\pi$
$499$	1.03370e7	1.85841	0.929207	+	0.369560i	$0.120492\pi$
$500$	−250000.	−0.0447214
$501$	1.80105e7	3.20577
$502$	5.77720e6	1.02319
$503$	−871360.	−0.153560	−0.0767799	−	0.997048i	$-0.524464\pi$
$503$	−871360.	−0.153560	−0.0767799	+	0.997048i	$0.524464\pi$
$504$	6.69963e6	1.17483
$505$	−3.62989e6	−0.633380
$506$	−32751.8	−0.00568668
$507$	−1.28124e7	−2.21366
$508$	−3.05824e6	−0.525790
$509$	1.13960e6	0.194966	0.0974828	−	0.995237i	$-0.468921\pi$
$509$	1.13960e6	0.194966	0.0974828	+	0.995237i	$0.468921\pi$
$510$	−3.85308e6	−0.655967
$511$	9.88925e6	1.67537
$512$	−262144.	−0.0441942
$513$	−8.78469e6	−1.47378
$514$	−2.87809e6	−0.480503
$515$	4.41829e6	0.734067
$516$	−9.35422e6	−1.54662
$517$	−254224.	−0.0418303
$518$	1.14991e6	0.188295
$519$	−7.02983e6	−1.14558
$520$	1.45572e6	0.236085
$521$	−4.73664e6	−0.764497	−0.382249	−	0.924059i	$-0.624850\pi$
$521$	−4.73664e6	−0.764497	−0.382249	+	0.924059i	$0.624850\pi$
$522$	−1.91841e7	−3.08152
$523$	3.67448e6	0.587411	0.293705	−	0.955896i	$-0.405112\pi$
$523$	3.67448e6	0.587411	0.293705	+	0.955896i	$0.405112\pi$
$524$	1.12189e6	0.178494
$525$	3.37077e6	0.533742
$526$	−4.84081e6	−0.762876
$527$	1.04191e7	1.63420
$528$	−698520.	−0.109042
$529$	−6.42925e6	−0.998898
$530$	−1.05559e6	−0.163232
$531$	307245.	0.0472878
$532$	−3.18852e6	−0.488438
$533$	−3.23278e6	−0.492900
$534$	4.51589e6	0.685315
$535$	701629.	0.105980
$536$	−1.88654e6	−0.283632
$537$	−7.16446e6	−1.07213
$538$	−3.03593e6	−0.452205
$539$	1.95553e6	0.289930
$540$	3.38815e6	0.500010
$541$	−1.07009e7	−1.57190	−0.785951	−	0.618288i	$-0.787826\pi$
$541$	−1.07009e7	−1.57190	−0.785951	+	0.618288i	$0.787826\pi$
$542$	1.35092e6	0.197530
$543$	−1.57696e7	−2.29520
$544$	−1.40573e6	−0.203660
$545$	−1.09652e6	−0.158134
$546$	−1.96276e7	−2.81764
$547$	−1.47493e6	−0.210767	−0.105383	−	0.994432i	$-0.533607\pi$
$547$	−1.47493e6	−0.210767	−0.105383	+	0.994432i	$0.533607\pi$
$548$	1.95146e6	0.277593
$549$	2.99341e7	4.23872
$550$	−243038.	−0.0342585
$551$	9.13016e6	1.28115
$552$	151295.	0.0211338
$553$	1.28421e7	1.78576
$554$	−580255.	−0.0803239
$555$	1.04979e6	0.144667
$556$	−3.61784e6	−0.496321
$557$	1.41789e6	0.193644	0.0968220	−	0.995302i	$-0.469132\pi$
$557$	1.41789e6	0.193644	0.0968220	+	0.995302i	$0.469132\pi$
$558$	−1.65391e7	−2.24868
$559$	1.89514e7	2.56514
$560$	1.22977e6	0.165712
$561$	−3.74578e6	−0.502499
$562$	4.38130e6	0.585144
$563$	9.58931e6	1.27502	0.637509	−	0.770443i	$-0.279965\pi$
$563$	9.58931e6	1.27502	0.637509	+	0.770443i	$0.279965\pi$
$564$	1.17437e6	0.155456
$565$	−3.86421e6	−0.509260
$566$	−4.20705e6	−0.551997
$567$	−2.02451e7	−2.64461
$568$	−3.74429e6	−0.486966
$569$	−1.02747e7	−1.33042	−0.665210	−	0.746657i	$-0.731658\pi$
$569$	−1.02747e7	−1.33042	−0.665210	+	0.746657i	$0.731658\pi$
$570$	−2.91090e6	−0.375267
$571$	−1.17121e7	−1.50330	−0.751649	−	0.659564i	$-0.770741\pi$
$571$	−1.17121e7	−1.50330	−0.751649	+	0.659564i	$0.770741\pi$
$572$	1.41518e6	0.180851
$573$	−3.58557e6	−0.456217
$574$	−2.73102e6	−0.345975
$575$	52640.6	0.00663974
$576$	2.23144e6	0.280240
$577$	−1.52146e6	−0.190248	−0.0951239	−	0.995465i	$-0.530325\pi$
$577$	−1.52146e6	−0.190248	−0.0951239	+	0.995465i	$0.530325\pi$
$578$	−1.85876e6	−0.231421
$579$	−2.33832e7	−2.89872
$580$	−3.52140e6	−0.434655
$581$	−1.32374e6	−0.162690
$582$	1.66772e7	2.04087
$583$	−1.02619e6	−0.125043
$584$	3.29381e6	0.399637
$585$	−1.23915e7	−1.49704
$586$	4.08930e6	0.491932
$587$	3.87599e6	0.464288	0.232144	−	0.972681i	$-0.425426\pi$
$587$	3.87599e6	0.464288	0.232144	+	0.972681i	$0.425426\pi$
$588$	−9.03345e6	−1.07748
$589$	7.87138e6	0.934894
$590$	56397.4	0.00667005
$591$	−9.64209e6	−1.13554
$592$	382999.	0.0449152
$593$	1.18859e6	0.138802	0.0694008	−	0.997589i	$-0.477891\pi$
$593$	1.18859e6	0.138802	0.0694008	+	0.997589i	$0.477891\pi$
$594$	3.29380e6	0.383029
$595$	6.59460e6	0.763653
$596$	−6.74521e6	−0.777821
$597$	−2.20566e7	−2.53281
$598$	−306519.	−0.0350513
$599$	−4.32787e6	−0.492842	−0.246421	−	0.969163i	$-0.579255\pi$
$599$	−4.32787e6	−0.492842	−0.246421	+	0.969163i	$0.579255\pi$
$600$	1.12270e6	0.127317
$601$	8.80337e6	0.994175	0.497087	−	0.867700i	$-0.334403\pi$
$601$	8.80337e6	0.994175	0.497087	+	0.867700i	$0.334403\pi$
$602$	1.60099e7	1.80052
$603$	1.60588e7	1.79854
$604$	4.88946e6	0.545342
$605$	3.79000e6	0.420970
$606$	1.63011e7	1.80317
$607$	−578879.	−0.0637699	−0.0318849	−	0.999492i	$-0.510151\pi$
$607$	−578879.	−0.0637699	−0.0318849	+	0.999492i	$0.510151\pi$
$608$	−1.06200e6	−0.116510
$609$	4.74793e7	5.18753
$610$	5.49465e6	0.597881
$611$	−2.37925e6	−0.257832
$612$	1.19660e7	1.29143
$613$	8.51400e6	0.915129	0.457564	−	0.889176i	$-0.348722\pi$
$613$	8.51400e6	0.915129	0.457564	+	0.889176i	$0.348722\pi$
$614$	1.02733e7	1.09974
$615$	−2.49324e6	−0.265813
$616$	1.19553e6	0.126943
$617$	6.37505e6	0.674172	0.337086	−	0.941474i	$-0.390559\pi$
$617$	6.37505e6	0.674172	0.337086	+	0.941474i	$0.390559\pi$
$618$	−1.98417e7	−2.08981
$619$	−9.81982e6	−1.03009	−0.515047	−	0.857162i	$-0.672226\pi$
$619$	−9.81982e6	−1.03009	−0.515047	+	0.857162i	$0.672226\pi$
$620$	−3.03590e6	−0.317182
$621$	−713417.	−0.0742360
$622$	1.10113e7	1.14121
$623$	−7.72903e6	−0.797820
$624$	−6.53734e6	−0.672109
$625$	390625.	0.0400000
$626$	8.71451e6	0.888806
$627$	−2.82984e6	−0.287471
$628$	−1.38945e6	−0.140586
$629$	2.05381e6	0.206983
$630$	−1.04682e7	−1.05080
$631$	−666122.	−0.0666009	−0.0333005	−	0.999445i	$-0.510602\pi$
$631$	−666122.	−0.0666009	−0.0333005	+	0.999445i	$0.510602\pi$
$632$	4.27731e6	0.425969
$633$	−3.00998e7	−2.98575
$634$	6.32117e6	0.624561
$635$	4.77850e6	0.470281
$636$	4.74044e6	0.464704
$637$	1.83015e7	1.78706
$638$	−3.42334e6	−0.332964
$639$	3.18724e7	3.08790
$640$	409600.	0.0395285
$641$	8.26589e6	0.794593	0.397296	−	0.917690i	$-0.369949\pi$
$641$	8.26589e6	0.794593	0.397296	+	0.917690i	$0.369949\pi$
$642$	−3.15088e6	−0.301713
$643$	1.64953e7	1.57337	0.786687	−	0.617352i	$-0.211794\pi$
$643$	1.64953e7	1.57337	0.786687	+	0.617352i	$0.211794\pi$
$644$	−258944.	−0.0246032
$645$	1.46160e7	1.38334
$646$	−5.69491e6	−0.536915
$647$	−2.03437e6	−0.191060	−0.0955300	−	0.995427i	$-0.530455\pi$
$647$	−2.03437e6	−0.191060	−0.0955300	+	0.995427i	$0.530455\pi$
$648$	−6.74302e6	−0.630837
$649$	54827.0	0.00510955
$650$	−2.27456e6	−0.211161
$651$	4.09333e7	3.78551
$652$	−1.86645e6	−0.171949
$653$	−1.76780e7	−1.62237	−0.811187	−	0.584787i	$-0.801178\pi$
$653$	−1.76780e7	−1.62237	−0.811187	+	0.584787i	$0.801178\pi$
$654$	4.92425e6	0.450190
$655$	−1.75295e6	−0.159649
$656$	−909619.	−0.0825278
$657$	−2.80378e7	−2.53414
$658$	−2.00996e6	−0.180977
$659$	−8.78497e6	−0.788001	−0.394000	−	0.919110i	$-0.628909\pi$
$659$	−8.78497e6	−0.788001	−0.394000	+	0.919110i	$0.628909\pi$
$660$	1.09144e6	0.0975302
$661$	−1.27679e7	−1.13663	−0.568313	−	0.822813i	$-0.692404\pi$
$661$	−1.27679e7	−1.13663	−0.568313	+	0.822813i	$0.692404\pi$
$662$	1.43632e6	0.127382
$663$	−3.50562e7	−3.09728
$664$	−440896.	−0.0388075
$665$	4.98206e6	0.436872
$666$	−3.26019e6	−0.284811
$667$	741473.	0.0645328
$668$	−1.02670e7	−0.890229
$669$	−1.57908e7	−1.36407
$670$	2.94772e6	0.253688
$671$	5.34164e6	0.458003
$672$	−5.52267e6	−0.471765
$673$	9.43942e6	0.803355	0.401678	−	0.915781i	$-0.368427\pi$
$673$	9.43942e6	0.803355	0.401678	+	0.915781i	$0.368427\pi$
$674$	2.97976e6	0.252657
$675$	−5.29399e6	−0.447222
$676$	7.30377e6	0.614724
$677$	−7.39420e6	−0.620040	−0.310020	−	0.950730i	$-0.600336\pi$
$677$	−7.39420e6	−0.620040	−0.310020	+	0.950730i	$0.600336\pi$
$678$	1.73534e7	1.44981
$679$	−2.85433e7	−2.37591
$680$	2.19646e6	0.182159
$681$	−3.43848e7	−2.84118
$682$	−2.95136e6	−0.242975
$683$	1.37064e7	1.12427	0.562135	−	0.827045i	$-0.309980\pi$
$683$	1.37064e7	1.12427	0.562135	+	0.827045i	$0.309980\pi$
$684$	9.04002e6	0.738804
$685$	−3.04916e6	−0.248287
$686$	2.54290e6	0.206310
$687$	−1.54092e6	−0.124562
$688$	5.33241e6	0.429489
$689$	−9.60399e6	−0.770733
$690$	−236398.	−0.0189026
$691$	1.62811e7	1.29714	0.648572	−	0.761153i	$-0.275366\pi$
$691$	1.62811e7	1.29714	0.648572	+	0.761153i	$0.275366\pi$
$692$	4.00738e6	0.318123
$693$	−1.01767e7	−0.804957
$694$	1.66205e7	1.30992
$695$	5.65288e6	0.443923
$696$	1.58139e7	1.23742
$697$	−4.87779e6	−0.380313
$698$	577378.	0.0448561
$699$	−2.94953e6	−0.228329
$700$	−1.92152e6	−0.148218
$701$	1.79708e7	1.38125	0.690625	−	0.723213i	$-0.257336\pi$
$701$	1.79708e7	1.38125	0.690625	+	0.723213i	$0.257336\pi$
$702$	3.08262e7	2.36090
$703$	1.55160e6	0.118411
$704$	398194.	0.0302805
$705$	−1.83496e6	−0.139044
$706$	9.25709e6	0.698976
$707$	−2.78996e7	−2.09918
$708$	−253270.	−0.0189889
$709$	2.13403e7	1.59436	0.797178	−	0.603745i	$-0.206325\pi$
$709$	2.13403e7	1.59436	0.797178	+	0.603745i	$0.206325\pi$
$710$	5.85045e6	0.435555
$711$	−3.64096e7	−2.70111
$712$	−2.57430e6	−0.190309
$713$	639245.	0.0470916
$714$	−2.96151e7	−2.17404
$715$	−2.21122e6	−0.161758
$716$	4.08413e6	0.297726
$717$	−4.56181e7	−3.31390
$718$	−8.27099e6	−0.598751
$719$	−2.52369e6	−0.182059	−0.0910297	−	0.995848i	$-0.529016\pi$
$719$	−2.52369e6	−0.182059	−0.0910297	+	0.995848i	$0.529016\pi$
$720$	−3.48663e6	−0.250654
$721$	3.39593e7	2.43288
$722$	5.60204e6	0.399947
$723$	−1.63627e7	−1.16415
$724$	8.98951e6	0.637367
$725$	5.50218e6	0.388767
$726$	−1.70202e7	−1.19846
$727$	−1.27558e7	−0.895099	−0.447550	−	0.894259i	$-0.647703\pi$
$727$	−1.27558e7	−0.895099	−0.447550	+	0.894259i	$0.647703\pi$
$728$	1.11888e7	0.782445
$729$	−372997.	−0.0259948
$730$	−5.14657e6	−0.357447
$731$	2.85948e7	1.97922
$732$	−2.46754e7	−1.70210
$733$	6.00004e6	0.412472	0.206236	−	0.978502i	$-0.433879\pi$
$733$	6.00004e6	0.412472	0.206236	+	0.978502i	$0.433879\pi$
$734$	1.21507e7	0.832459
$735$	1.41148e7	0.963730
$736$	−86246.3	−0.00586875
$737$	2.86564e6	0.194336
$738$	7.74293e6	0.523317
$739$	−1.08479e7	−0.730693	−0.365347	−	0.930872i	$-0.619050\pi$
$739$	−1.08479e7	−0.730693	−0.365347	+	0.930872i	$0.619050\pi$
$740$	−598435.	−0.0401733
$741$	−2.64841e7	−1.77190
$742$	−8.11335e6	−0.540991
$743$	9.06462e6	0.602390	0.301195	−	0.953563i	$-0.402615\pi$
$743$	9.06462e6	0.602390	0.301195	+	0.953563i	$0.402615\pi$
$744$	1.36336e7	0.902982
$745$	1.05394e7	0.695704
$746$	−5.99323e6	−0.394288
$747$	3.75303e6	0.246082
$748$	2.13530e6	0.139542
$749$	5.39278e6	0.351244
$750$	−1.75422e6	−0.113876
$751$	−1.67200e7	−1.08177	−0.540887	−	0.841095i	$-0.681911\pi$
$751$	−1.67200e7	−1.08177	−0.540887	+	0.841095i	$0.681911\pi$
$752$	−669457.	−0.0431696
$753$	4.05379e7	2.60540
$754$	−3.20385e7	−2.05231
$755$	−7.63979e6	−0.487769
$756$	2.60416e7	1.65716
$757$	−3.44739e6	−0.218651	−0.109325	−	0.994006i	$-0.534869\pi$
$757$	−3.44739e6	−0.218651	−0.109325	+	0.994006i	$0.534869\pi$
$758$	8.48398e6	0.536323
$759$	−229815.	−0.0144802
$760$	1.65937e6	0.104210
$761$	−1.34495e6	−0.0841871	−0.0420935	−	0.999114i	$-0.513403\pi$
$761$	−1.34495e6	−0.0841871	−0.0420935	+	0.999114i	$0.513403\pi$
$762$	−2.14593e7	−1.33884
$763$	−8.42794e6	−0.524095
$764$	2.04397e6	0.126690
$765$	−1.86969e7	−1.15509
$766$	−146480.	−0.00902002
$767$	513117.	0.0314940
$768$	−1.83943e6	−0.112533
$769$	5.06053e6	0.308589	0.154294	−	0.988025i	$-0.450690\pi$
$769$	5.06053e6	0.308589	0.154294	+	0.988025i	$0.450690\pi$
$770$	−1.86801e6	−0.113541
$771$	−2.01952e7	−1.22352
$772$	1.33297e7	0.804963
$773$	−1.85264e7	−1.11518	−0.557588	−	0.830118i	$-0.688273\pi$
$773$	−1.85264e7	−1.11518	−0.557588	+	0.830118i	$0.688273\pi$
$774$	−4.53910e7	−2.72343
$775$	4.74359e6	0.283696
$776$	−9.50691e6	−0.566742
$777$	8.06876e6	0.479462
$778$	1.02737e7	0.608525
$779$	−3.68505e6	−0.217570
$780$	1.02146e7	0.601153
$781$	5.68753e6	0.333654
$782$	−462492.	−0.0270450
$783$	−7.45689e7	−4.34664
$784$	5.14955e6	0.299212
$785$	2.17101e6	0.125744
$786$	7.87218e6	0.454505
$787$	4.57973e6	0.263574	0.131787	−	0.991278i	$-0.457928\pi$
$787$	4.57973e6	0.263574	0.131787	+	0.991278i	$0.457928\pi$
$788$	5.49651e6	0.315334
$789$	−3.39674e7	−1.94254
$790$	−6.68330e6	−0.380998
$791$	−2.97006e7	−1.68781
$792$	−3.38954e6	−0.192012
$793$	4.99916e7	2.82302
$794$	−2.30747e7	−1.29893
$795$	−7.40694e6	−0.415644
$796$	1.25734e7	0.703349
$797$	3.01782e6	0.168286	0.0841428	−	0.996454i	$-0.473185\pi$
$797$	3.01782e6	0.168286	0.0841428	+	0.996454i	$0.473185\pi$
$798$	−2.23734e7	−1.24373
$799$	−3.58993e6	−0.198939
$800$	−640000.	−0.0353553
$801$	2.19132e7	1.20677
$802$	5.89790e6	0.323789
$803$	−5.00326e6	−0.273819
$804$	−1.32377e7	−0.722222
$805$	404600.	0.0220057
$806$	−2.76213e7	−1.49764
$807$	−2.13027e7	−1.15147
$808$	−9.29251e6	−0.500731
$809$	3.11660e7	1.67421	0.837106	−	0.547041i	$-0.184246\pi$
$809$	3.11660e7	1.67421	0.837106	+	0.547041i	$0.184246\pi$
$810$	1.05360e7	0.564237
$811$	−8.05046e6	−0.429802	−0.214901	−	0.976636i	$-0.568943\pi$
$811$	−8.05046e6	−0.429802	−0.214901	+	0.976636i	$0.568943\pi$
$812$	−2.70658e7	−1.44056
$813$	9.47928e6	0.502978
$814$	−581771.	−0.0307745
$815$	2.91634e6	0.153795
$816$	−9.86387e6	−0.518588
$817$	2.16026e7	1.13227
$818$	−1.60084e7	−0.836500
$819$	−9.52419e7	−4.96156
$820$	1.42128e6	0.0738151
$821$	−3.72050e7	−1.92638	−0.963192	−	0.268814i	$-0.913368\pi$
$821$	−3.72050e7	−1.92638	−0.963192	+	0.268814i	$0.913368\pi$
$822$	1.36932e7	0.706845
$823$	1.34273e7	0.691018	0.345509	−	0.938415i	$-0.387706\pi$
$823$	1.34273e7	0.691018	0.345509	+	0.938415i	$0.387706\pi$
$824$	1.13108e7	0.580331
$825$	−1.70537e6	−0.0872336
$826$	433476.	0.0221062
$827$	−3.65999e7	−1.86087	−0.930436	−	0.366453i	$-0.880572\pi$
$827$	−3.65999e7	−1.86087	−0.930436	+	0.366453i	$0.880572\pi$
$828$	734152.	0.0372143
$829$	−1.00043e7	−0.505591	−0.252796	−	0.967520i	$-0.581350\pi$
$829$	−1.00043e7	−0.505591	−0.252796	+	0.967520i	$0.581350\pi$
$830$	688900.	0.0347105
$831$	−4.07158e6	−0.204532
$832$	3.72664e6	0.186642
$833$	2.76142e7	1.37886
$834$	−2.53860e7	−1.26380
$835$	1.60422e7	0.796245
$836$	1.61316e6	0.0798293
$837$	−6.42880e7	−3.17188
$838$	1.11487e7	0.548420
$839$	−3.13377e7	−1.53696	−0.768479	−	0.639874i	$-0.778986\pi$
$839$	−3.13377e7	−1.53696	−0.768479	+	0.639874i	$0.778986\pi$
$840$	8.62918e6	0.421960
$841$	5.69903e7	2.77850
$842$	1.57983e7	0.767944
$843$	3.07431e7	1.48997
$844$	1.71585e7	0.829131
$845$	−1.14121e7	−0.549826
$846$	5.69860e6	0.273743
$847$	2.91303e7	1.39520
$848$	−2.70231e6	−0.129046
$849$	−2.95204e7	−1.40557
$850$	−3.43197e6	−0.162928
$851$	126008.	0.00596450
$852$	−2.62732e7	−1.23998
$853$	−970203.	−0.0456552	−0.0228276	−	0.999739i	$-0.507267\pi$
$853$	−970203.	−0.0456552	−0.0228276	+	0.999739i	$0.507267\pi$
$854$	4.22323e7	1.98153
$855$	−1.41250e7	−0.660806
$856$	1.79617e6	0.0837844
$857$	−3.23201e7	−1.50321	−0.751607	−	0.659611i	$-0.770721\pi$
$857$	−3.23201e7	−1.50321	−0.751607	+	0.659611i	$0.770721\pi$
$858$	9.93015e6	0.460509
$859$	1.30710e7	0.604400	0.302200	−	0.953244i	$-0.402279\pi$
$859$	1.30710e7	0.604400	0.302200	+	0.953244i	$0.402279\pi$
$860$	−8.33189e6	−0.384147
$861$	−1.91632e7	−0.880970
$862$	2.94335e6	0.134919
$863$	−1.99476e7	−0.911724	−0.455862	−	0.890050i	$-0.650669\pi$
$863$	−1.99476e7	−0.911724	−0.455862	+	0.890050i	$0.650669\pi$
$864$	8.67367e6	0.395293
$865$	−6.26154e6	−0.284538
$866$	7.32392e6	0.331855
$867$	−1.30427e7	−0.589277
$868$	−2.33342e7	−1.05122
$869$	−6.49719e6	−0.291861
$870$	−2.47092e7	−1.10678
$871$	2.68191e7	1.19784
$872$	−2.80709e6	−0.125016
$873$	8.09255e7	3.59376
$874$	−349401.	−0.0154720
$875$	3.00238e6	0.132570
$876$	2.31123e7	1.01761
$877$	7.17824e6	0.315151	0.157576	−	0.987507i	$-0.449632\pi$
$877$	7.17824e6	0.315151	0.157576	+	0.987507i	$0.449632\pi$
$878$	−5.33731e6	−0.233661
$879$	2.86941e7	1.25262
$880$	−622178.	−0.0270837
$881$	−3.96208e6	−0.171982	−0.0859910	−	0.996296i	$-0.527406\pi$
$881$	−3.96208e6	−0.171982	−0.0859910	+	0.996296i	$0.527406\pi$
$882$	−4.38345e7	−1.89734
$883$	1.46882e7	0.633967	0.316984	−	0.948431i	$-0.397330\pi$
$883$	1.46882e7	0.633967	0.316984	+	0.948431i	$0.397330\pi$
$884$	1.99839e7	0.860102
$885$	395734.	0.0169842
$886$	1.14299e7	0.489167
$887$	2.24629e7	0.958642	0.479321	−	0.877640i	$-0.340883\pi$
$887$	2.24629e7	0.958642	0.479321	+	0.877640i	$0.340883\pi$
$888$	2.68746e6	0.114369
$889$	3.67280e7	1.55863
$890$	4.02235e6	0.170218
$891$	1.02426e7	0.432230
$892$	9.00159e6	0.378798
$893$	−2.71210e6	−0.113809
$894$	−4.73303e7	−1.98059
$895$	−6.38145e6	−0.266294
$896$	3.14822e6	0.131007
$897$	−2.15081e6	−0.0892525
$898$	1.43187e7	0.592532
$899$	6.68162e7	2.75729
$900$	5.44786e6	0.224192
$901$	−1.44910e7	−0.594684
$902$	1.38170e6	0.0565455
$903$	1.12340e8	4.58473
$904$	−9.89237e6	−0.402605
$905$	−1.40461e7	−0.570078
$906$	3.43088e7	1.38862
$907$	2.33834e6	0.0943820	0.0471910	−	0.998886i	$-0.484973\pi$
$907$	2.33834e6	0.0943820	0.0471910	+	0.998886i	$0.484973\pi$
$908$	1.96012e7	0.788984
$909$	7.91004e7	3.17519
$910$	−1.74825e7	−0.699840
$911$	−3.46688e7	−1.38402	−0.692011	−	0.721887i	$-0.743275\pi$
$911$	−3.46688e7	−1.38402	−0.692011	+	0.721887i	$0.743275\pi$
$912$	−7.45191e6	−0.296675
$913$	669716.	0.0265897
$914$	−1.31326e7	−0.519977
$915$	3.85553e7	1.52241
$916$	878405.	0.0345905
$917$	−1.34734e7	−0.529118
$918$	4.65122e7	1.82163
$919$	−2.98432e7	−1.16562	−0.582809	−	0.812609i	$-0.698046\pi$
$919$	−2.98432e7	−1.16562	−0.582809	+	0.812609i	$0.698046\pi$
$920$	134760.	0.00524917
$921$	7.20865e7	2.80030
$922$	2.46066e7	0.953290
$923$	5.32287e7	2.05656
$924$	8.38888e6	0.323239
$925$	935055.	0.0359321
$926$	−1.53842e7	−0.589588
$927$	−9.62808e7	−3.67994
$928$	−9.01477e6	−0.343625
$929$	4.31964e7	1.64213	0.821066	−	0.570834i	$-0.193380\pi$
$929$	4.31964e7	1.64213	0.821066	+	0.570834i	$0.193380\pi$
$930$	−2.13025e7	−0.807651
$931$	2.08619e7	0.788822
$932$	1.68139e6	0.0634059
$933$	7.72653e7	2.90590
$934$	−1.86288e7	−0.698743
$935$	−3.33640e6	−0.124810
$936$	−3.17222e7	−1.18351
$937$	−3.00512e7	−1.11818	−0.559092	−	0.829106i	$-0.688850\pi$
$937$	−3.00512e7	−1.11818	−0.559092	+	0.829106i	$0.688850\pi$
$938$	2.26565e7	0.840785
$939$	6.11487e7	2.26320
$940$	1.04603e6	0.0386120
$941$	1.65882e6	0.0610698	0.0305349	−	0.999534i	$-0.490279\pi$
$941$	1.65882e6	0.0610698	0.0305349	+	0.999534i	$0.490279\pi$
$942$	−9.74960e6	−0.357981
$943$	−299268.	−0.0109593
$944$	144377.	0.00527314
$945$	−4.06901e7	−1.48221
$946$	−8.09987e6	−0.294273
$947$	−5.34435e7	−1.93651	−0.968255	−	0.249963i	$-0.919582\pi$
$947$	−5.34435e7	−1.93651	−0.968255	+	0.249963i	$0.919582\pi$
$948$	3.00134e7	1.08466
$949$	−4.68247e7	−1.68776
$950$	−2.59277e6	−0.0932083
$951$	4.43549e7	1.59034
$952$	1.68822e7	0.603721
$953$	−946955.	−0.0337751	−0.0168876	−	0.999857i	$-0.505376\pi$
$953$	−946955.	−0.0337751	−0.0168876	+	0.999857i	$0.505376\pi$
$954$	2.30028e7	0.818295
$955$	−3.19370e6	−0.113315
$956$	2.60048e7	0.920256
$957$	−2.40211e7	−0.847840
$958$	−1.16750e7	−0.411003
$959$	−2.34361e7	−0.822884
$960$	2.87411e6	0.100653
$961$	2.89751e7	1.01208
$962$	−5.44470e6	−0.189687
$963$	−1.52895e7	−0.531285
$964$	9.32761e6	0.323279
$965$	−2.08276e7	−0.719981
$966$	−1.81698e6	−0.0626480
$967$	−1.84144e7	−0.633275	−0.316638	−	0.948547i	$-0.602554\pi$
$967$	−1.84144e7	−0.633275	−0.316638	+	0.948547i	$0.602554\pi$
$968$	9.70241e6	0.332806
$969$	−3.99605e7	−1.36717
$970$	1.48545e7	0.506909
$971$	−4.57586e6	−0.155749	−0.0778745	−	0.996963i	$-0.524813\pi$
$971$	−4.57586e6	−0.155749	−0.0778745	+	0.996963i	$0.524813\pi$
$972$	−1.43821e7	−0.488267
$973$	4.34485e7	1.47127
$974$	1.66795e7	0.563360
$975$	−1.59603e7	−0.537687
$976$	1.40663e7	0.472667
$977$	5.67152e7	1.90092	0.950459	−	0.310851i	$-0.100614\pi$
$977$	5.67152e7	1.90092	0.950459	+	0.310851i	$0.100614\pi$
$978$	−1.30967e7	−0.437839
$979$	3.91034e6	0.130394
$980$	−8.04618e6	−0.267624
$981$	2.38947e7	0.792737
$982$	3.20053e7	1.05912
$983$	5.73245e6	0.189215	0.0946077	−	0.995515i	$-0.469840\pi$
$983$	5.73245e6	0.189215	0.0946077	+	0.995515i	$0.469840\pi$
$984$	−6.38269e6	−0.210144
$985$	−8.58830e6	−0.282044
$986$	−4.83413e7	−1.58353
$987$	−1.41037e7	−0.460828
$988$	1.50973e7	0.492049
$989$	1.75438e6	0.0570339
$990$	5.29615e6	0.171740
$991$	4.00459e7	1.29531	0.647655	−	0.761934i	$-0.275750\pi$
$991$	4.00459e7	1.29531	0.647655	+	0.761934i	$0.275750\pi$
$992$	−7.77190e6	−0.250754
$993$	1.00785e7	0.324357
$994$	4.49670e7	1.44354
$995$	−1.96460e7	−0.629094
$996$	−3.09371e6	−0.0988171
$997$	1.27310e7	0.405624	0.202812	−	0.979218i	$-0.434992\pi$
$997$	1.27310e7	0.405624	0.202812	+	0.979218i	$0.434992\pi$
$998$	−4.13479e7	−1.31410
$999$	−1.26724e7	−0.401741

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	830.6.a.h.1.1	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
830.6.a.h.1.1	✓	19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 830 = 2 \cdot 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	830.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -28.0675 q^{3} +16.0000 q^{4} -25.0000 q^{5} +112.270 q^{6} -192.152 q^{7} -64.0000 q^{8} +544.786 q^{9} +100.000 q^{10} +97.2153 q^{11} -449.080 q^{12} +909.823 q^{13} +768.609 q^{14} +701.688 q^{15} +256.000 q^{16} +1372.79 q^{17} -2179.14 q^{18} +1037.11 q^{19} -400.000 q^{20} +5393.24 q^{21} -388.861 q^{22} +84.2249 q^{23} +1796.32 q^{24} +625.000 q^{25} -3639.29 q^{26} -8470.38 q^{27} -3074.43 q^{28} +8803.49 q^{29} -2806.75 q^{30} +7589.75 q^{31} -1024.00 q^{32} -2728.59 q^{33} -5491.15 q^{34} +4803.80 q^{35} +8716.57 q^{36} +1496.09 q^{37} -4148.43 q^{38} -25536.5 q^{39} +1600.00 q^{40} -3553.20 q^{41} -21572.9 q^{42} +20829.7 q^{43} +1555.45 q^{44} -13619.6 q^{45} -336.900 q^{46} -2615.07 q^{47} -7185.29 q^{48} +20115.4 q^{49} -2500.00 q^{50} -38530.8 q^{51} +14557.2 q^{52} -10555.9 q^{53} +33881.5 q^{54} -2430.38 q^{55} +12297.7 q^{56} -29109.0 q^{57} -35214.0 q^{58} +563.974 q^{59} +11227.0 q^{60} +54946.5 q^{61} -30359.0 q^{62} -104682. q^{63} +4096.00 q^{64} -22745.6 q^{65} +10914.4 q^{66} +29477.2 q^{67} +21964.6 q^{68} -2363.98 q^{69} -19215.2 q^{70} +58504.5 q^{71} -34866.3 q^{72} -51465.7 q^{73} -5984.35 q^{74} -17542.2 q^{75} +16593.7 q^{76} -18680.1 q^{77} +102146. q^{78} -66833.0 q^{79} -6400.00 q^{80} +105360. q^{81} +14212.8 q^{82} +6889.00 q^{83} +86291.8 q^{84} -34319.7 q^{85} -83318.9 q^{86} -247092. q^{87} -6221.78 q^{88} +40223.5 q^{89} +54478.6 q^{90} -174825. q^{91} +1347.60 q^{92} -213025. q^{93} +10460.3 q^{94} -25927.7 q^{95} +28741.1 q^{96} +148545. q^{97} -80461.8 q^{98} +52961.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 76 q^{2} + 304 q^{4} - 475 q^{5} + 287 q^{7} - 1216 q^{8} + 1623 q^{9} + 1900 q^{10} + 743 q^{11} + 918 q^{13} - 1148 q^{14} + 4864 q^{16} - 189 q^{17} - 6492 q^{18} + 2871 q^{19} - 7600 q^{20} + 5594 q^{21}+ \cdots + 482584 q^{99}+O(q^{100}) \) 19 * q - 76 * q^2 + 304 * q^4 - 475 * q^5 + 287 * q^7 - 1216 * q^8 + 1623 * q^9 + 1900 * q^10 + 743 * q^11 + 918 * q^13 - 1148 * q^14 + 4864 * q^16 - 189 * q^17 - 6492 * q^18 + 2871 * q^19 - 7600 * q^20 + 5594 * q^21 - 2972 * q^22 + 5711 * q^23 + 11875 * q^25 - 3672 * q^26 + 546 * q^27 + 4592 * q^28 + 8661 * q^29 + 21202 * q^31 - 19456 * q^32 - 7082 * q^33 + 756 * q^34 - 7175 * q^35 + 25968 * q^36 + 11396 * q^37 - 11484 * q^38 + 4110 * q^39 + 30400 * q^40 - 9470 * q^41 - 22376 * q^42 - 9024 * q^43 + 11888 * q^44 - 40575 * q^45 - 22844 * q^46 - 57499 * q^47 + 62522 * q^49 - 47500 * q^50 - 49150 * q^51 + 14688 * q^52 - 48794 * q^53 - 2184 * q^54 - 18575 * q^55 - 18368 * q^56 - 59416 * q^57 - 34644 * q^58 - 11195 * q^59 + 19177 * q^61 - 84808 * q^62 - 62149 * q^63 + 77824 * q^64 - 22950 * q^65 + 28328 * q^66 - 6486 * q^67 - 3024 * q^68 - 9252 * q^69 + 28700 * q^70 - 69656 * q^71 - 103872 * q^72 + 24875 * q^73 - 45584 * q^74 + 45936 * q^76 - 102852 * q^77 - 16440 * q^78 + 82194 * q^79 - 121600 * q^80 + 101079 * q^81 + 37880 * q^82 + 130891 * q^83 + 89504 * q^84 + 4725 * q^85 + 36096 * q^86 - 22509 * q^87 - 47552 * q^88 + 56210 * q^89 + 162300 * q^90 + 154764 * q^91 + 91376 * q^92 - 319309 * q^93 + 229996 * q^94 - 71775 * q^95 + 445215 * q^97 - 250088 * q^98 + 482584 * q^99

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