LMFDB - Embedded newform 325.6.b.b.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.b.274.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.438447i q^{2} +26.3693i q^{3} +31.8078 q^{4} +11.5616 q^{6} -162.309i q^{7} -27.9763i q^{8} -452.341 q^{9} +O(q^{10})$	\(q-0.438447i q^{2} +26.3693i q^{3} +31.8078 q^{4} +11.5616 q^{6} -162.309i q^{7} -27.9763i q^{8} -452.341 q^{9} -361.170 q^{11} +838.749i q^{12} +169.000i q^{13} -71.1638 q^{14} +1005.58 q^{16} -1578.88i q^{17} +198.328i q^{18} +98.2765 q^{19} +4279.97 q^{21} +158.354i q^{22} -1607.16i q^{23} +737.717 q^{24} +74.0976 q^{26} -5520.18i q^{27} -5162.68i q^{28} +307.045 q^{29} +2936.42 q^{31} -1336.14i q^{32} -9523.82i q^{33} -692.255 q^{34} -14388.0 q^{36} -12222.3i q^{37} -43.0891i q^{38} -4456.41 q^{39} -104.151 q^{41} -1876.54i q^{42} -10936.4i q^{43} -11488.0 q^{44} -704.654 q^{46} -14949.2i q^{47} +26516.5i q^{48} -9537.11 q^{49} +41634.0 q^{51} +5375.51i q^{52} +35911.5i q^{53} -2420.31 q^{54} -4540.80 q^{56} +2591.48i q^{57} -134.623i q^{58} +1598.46 q^{59} +20156.2 q^{61} -1287.47i q^{62} +73418.9i q^{63} +31592.8 q^{64} -4175.69 q^{66} -35368.1i q^{67} -50220.6i q^{68} +42379.7 q^{69} +26140.3 q^{71} +12654.8i q^{72} -75468.4i q^{73} -5358.81 q^{74} +3125.96 q^{76} +58621.1i q^{77} +1953.90i q^{78} +7576.72 q^{79} +35644.4 q^{81} +45.6648i q^{82} +912.974i q^{83} +136136. q^{84} -4795.04 q^{86} +8096.58i q^{87} +10104.2i q^{88} -106709. q^{89} +27430.2 q^{91} -51120.1i q^{92} +77431.5i q^{93} -6554.44 q^{94} +35233.0 q^{96} +103676. i q^{97} +4181.52i q^{98} +163372. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9}+O(q^{10}) $ 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9	$ 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9} - 752 q^{11} + 1010 q^{14} + 930 q^{16} + 624 q^{19} + 8148 q^{21} + 2118 q^{24} + 1690 q^{26} + 1624 q^{29} + 15440 q^{31} - 10974 q^{34} - 23396 q^{36} - 9464 q^{39} + 15680 q^{41} - 23308 q^{44} + 37192 q^{46} - 17368 q^{49} + 86696 q^{51} + 2350 q^{54} + 40690 q^{56} + 77872 q^{59} + 3968 q^{61} - 6434 q^{64} - 8572 q^{66} + 70960 q^{69} + 134792 q^{71} - 53010 q^{74} + 11036 q^{76} + 110592 q^{79} + 185524 q^{81} + 267662 q^{84} + 68106 q^{86} - 233016 q^{89} + 12168 q^{91} + 214250 q^{94} + 82986 q^{96} + 319616 q^{99}+O(q^{100}) $ 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9 - 752 * q^11 + 1010 * q^14 + 930 * q^16 + 624 * q^19 + 8148 * q^21 + 2118 * q^24 + 1690 * q^26 + 1624 * q^29 + 15440 * q^31 - 10974 * q^34 - 23396 * q^36 - 9464 * q^39 + 15680 * q^41 - 23308 * q^44 + 37192 * q^46 - 17368 * q^49 + 86696 * q^51 + 2350 * q^54 + 40690 * q^56 + 77872 * q^59 + 3968 * q^61 - 6434 * q^64 - 8572 * q^66 + 70960 * q^69 + 134792 * q^71 - 53010 * q^74 + 11036 * q^76 + 110592 * q^79 + 185524 * q^81 + 267662 * q^84 + 68106 * q^86 - 233016 * q^89 + 12168 * q^91 + 214250 * q^94 + 82986 * q^96 + 319616 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 0.438447i	− 0.0775072i	−0.999249	−	0.0387536i	$-0.987661\pi$
$2$	− 0.438447i	− 0.0775072i	0.999249	−	0.0387536i	$-0.0123388\pi$
$3$	26.3693i	1.69159i	0.533506	+	0.845796i	$0.320874\pi$
$3$	26.3693i	1.69159i	−0.533506	+	0.845796i	$0.679126\pi$
$4$	31.8078	0.993993
$5$	0	0
$5$	0	0
$6$	11.5616	0.131111
$7$	− 162.309i	− 1.25198i	−0.779832	−	0.625989i	$-0.784695\pi$
$7$	− 162.309i	− 1.25198i	0.779832	−	0.625989i	$-0.215305\pi$
$8$	− 27.9763i	− 0.154549i
$9$	−452.341	−1.86149
$10$	0	0
$11$	−361.170	−0.899975	−0.449988	−	0.893035i	$-0.648572\pi$
$11$	−361.170	−0.899975	−0.449988	+	0.893035i	$0.648572\pi$
$12$	838.749i	1.68143i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−71.1638	−0.0970374
$15$	0	0
$16$	1005.58	0.982014
$17$	− 1578.88i	− 1.32503i	−0.749048	−	0.662516i	$-0.769489\pi$
$17$	− 1578.88i	− 1.32503i	0.749048	−	0.662516i	$-0.230511\pi$
$18$	198.328i	0.144279i
$19$	98.2765	0.0624548	0.0312274	−	0.999512i	$-0.490058\pi$
$19$	98.2765	0.0624548	0.0312274	+	0.999512i	$0.490058\pi$
$20$	0	0
$21$	4279.97	2.11784
$22$	158.354i	0.0697546i
$23$	− 1607.16i	− 0.633489i	−0.948511	−	0.316745i	$-0.897410\pi$
$23$	− 1607.16i	− 0.633489i	0.948511	−	0.316745i	$-0.102590\pi$
$24$	737.717	0.261434
$25$	0	0
$26$	74.0976	0.0214966
$27$	− 5520.18i	− 1.45728i
$28$	− 5162.68i	− 1.24446i
$29$	307.045	0.0677966	0.0338983	−	0.999425i	$-0.489208\pi$
$29$	307.045	0.0677966	0.0338983	+	0.999425i	$0.489208\pi$
$30$	0	0
$31$	2936.42	0.548801	0.274400	−	0.961616i	$-0.411521\pi$
$31$	2936.42	0.548801	0.274400	+	0.961616i	$0.411521\pi$
$32$	− 1336.14i	− 0.230662i
$33$	− 9523.82i	− 1.52239i
$34$	−692.255	−0.102700
$35$	0	0
$36$	−14388.0	−1.85030
$37$	− 12222.3i	− 1.46773i	−0.679294	−	0.733867i	$-0.737714\pi$
$37$	− 12222.3i	− 1.46773i	0.679294	−	0.733867i	$-0.262286\pi$
$38$	− 43.0891i	− 0.00484070i
$39$	−4456.41	−0.469163
$40$	0	0
$41$	−104.151	−0.00967619	−0.00483809	−	0.999988i	$-0.501540\pi$
$41$	−104.151	−0.00967619	−0.00483809	+	0.999988i	$0.501540\pi$
$42$	− 1876.54i	− 0.164148i
$43$	− 10936.4i	− 0.901994i	−0.892526	−	0.450997i	$-0.851069\pi$
$43$	− 10936.4i	− 0.901994i	0.892526	−	0.450997i	$-0.148931\pi$
$44$	−11488.0	−0.894569
$45$	0	0
$46$	−704.654	−0.0491000
$47$	− 14949.2i	− 0.987129i	−0.869709	−	0.493564i	$-0.835694\pi$
$47$	− 14949.2i	− 0.987129i	0.869709	−	0.493564i	$-0.164306\pi$
$48$	26516.5i	1.66117i
$49$	−9537.11	−0.567449
$50$	0	0
$51$	41634.0	2.24141
$52$	5375.51i	0.275684i
$53$	35911.5i	1.75608i	0.478587	+	0.878040i	$0.341149\pi$
$53$	35911.5i	1.75608i	−0.478587	+	0.878040i	$0.658851\pi$
$54$	−2420.31	−0.112950
$55$	0	0
$56$	−4540.80	−0.193492
$57$	2591.48i	0.105648i
$58$	− 134.623i	− 0.00525472i
$59$	1598.46	0.0597822	0.0298911	−	0.999553i	$-0.490484\pi$
$59$	1598.46	0.0597822	0.0298911	+	0.999553i	$0.490484\pi$
$60$	0	0
$61$	20156.2	0.693560	0.346780	−	0.937947i	$-0.387275\pi$
$61$	20156.2	0.693560	0.346780	+	0.937947i	$0.387275\pi$
$62$	− 1287.47i	− 0.0425360i
$63$	73418.9i	2.33054i
$64$	31592.8	0.964136
$65$	0	0
$66$	−4175.69	−0.117996
$67$	− 35368.1i	− 0.962552i	−0.876569	−	0.481276i	$-0.840173\pi$
$67$	− 35368.1i	− 0.962552i	0.876569	−	0.481276i	$-0.159827\pi$
$68$	− 50220.6i	− 1.31707i
$69$	42379.7	1.07161
$70$	0	0
$71$	26140.3	0.615411	0.307706	−	0.951482i	$-0.400439\pi$
$71$	26140.3	0.615411	0.307706	+	0.951482i	$0.400439\pi$
$72$	12654.8i	0.287690i
$73$	− 75468.4i	− 1.65752i	−0.559606	−	0.828759i	$-0.689048\pi$
$73$	− 75468.4i	− 1.65752i	0.559606	−	0.828759i	$-0.310952\pi$
$74$	−5358.81	−0.113760
$75$	0	0
$76$	3125.96	0.0620796
$77$	58621.1i	1.12675i
$78$	1953.90i	0.0363636i
$79$	7576.72	0.136588	0.0682942	−	0.997665i	$-0.478244\pi$
$79$	7576.72	0.136588	0.0682942	+	0.997665i	$0.478244\pi$
$80$	0	0
$81$	35644.4	0.603642
$82$	45.6648i	0 0.000749974i
$83$	912.974i	0.0145466i	0.999974	+	0.00727332i	$0.00231519\pi$
$83$	912.974i	0.0145466i	−0.999974	+	0.00727332i	$0.997685\pi$
$84$	136136.	2.10511
$85$	0	0
$86$	−4795.04	−0.0699110
$87$	8096.58i	0.114684i
$88$	10104.2i	0.139090i
$89$	−106709.	−1.42799	−0.713995	−	0.700151i	$-0.753116\pi$
$89$	−106709.	−1.42799	−0.713995	+	0.700151i	$0.753116\pi$
$90$	0	0
$91$	27430.2	0.347236
$92$	− 51120.1i	− 0.629684i
$93$	77431.5i	0.928347i
$94$	−6554.44	−0.0765096
$95$	0	0
$96$	35233.0	0.390186
$97$	103676.i	1.11879i	0.828900	+	0.559397i	$0.188967\pi$
$97$	103676.i	1.11879i	−0.828900	+	0.559397i	$0.811033\pi$
$98$	4181.52i	0.0439814i
$99$	163372.	1.67529
$100$	0	0
$101$	−3079.33	−0.0300367	−0.0150183	−	0.999887i	$-0.504781\pi$
$101$	−3079.33	−0.0300367	−0.0150183	+	0.999887i	$0.504781\pi$
$102$	− 18254.3i	− 0.173726i
$103$	113712.i	1.05612i	0.849206	+	0.528061i	$0.177081\pi$
$103$	113712.i	1.05612i	−0.849206	+	0.528061i	$0.822919\pi$
$104$	4728.00	0.0428641
$105$	0	0
$106$	15745.3	0.136109
$107$	− 189464.i	− 1.59980i	−0.600131	−	0.799901i	$-0.704885\pi$
$107$	− 189464.i	− 1.59980i	0.600131	−	0.799901i	$-0.295115\pi$
$108$	− 175584.i	− 1.44853i
$109$	97525.1	0.786231	0.393116	−	0.919489i	$-0.371397\pi$
$109$	97525.1	0.786231	0.393116	+	0.919489i	$0.371397\pi$
$110$	0	0
$111$	322293.	2.48281
$112$	− 163215.i	− 1.22946i
$113$	− 16930.4i	− 0.124730i	−0.998053	−	0.0623650i	$-0.980136\pi$
$113$	− 16930.4i	− 0.124730i	0.998053	−	0.0623650i	$-0.0198643\pi$
$114$	1136.23	0.00818849
$115$	0	0
$116$	9766.43	0.0673893
$117$	− 76445.6i	− 0.516283i
$118$	− 700.840i	− 0.00463355i
$119$	−256266.	−1.65891
$120$	0	0
$121$	−30606.9	−0.190045
$122$	− 8837.43i	− 0.0537559i
$123$	− 2746.39i	− 0.0163682i
$124$	93401.1	0.545504
$125$	0	0
$126$	32190.3	0.180634
$127$	248343.i	1.36629i	0.730284	+	0.683144i	$0.239388\pi$
$127$	248343.i	1.36629i	−0.730284	+	0.683144i	$0.760612\pi$
$128$	− 56608.2i	− 0.305390i
$129$	288386.	1.52581
$130$	0	0
$131$	−149743.	−0.762372	−0.381186	−	0.924498i	$-0.624484\pi$
$131$	−149743.	−0.762372	−0.381186	+	0.924498i	$0.624484\pi$
$132$	− 302931.i	− 1.51325i
$133$	− 15951.1i	− 0.0781920i
$134$	−15507.0	−0.0746048
$135$	0	0
$136$	−44171.2	−0.204782
$137$	− 268515.i	− 1.22227i	−0.791527	−	0.611134i	$-0.790714\pi$
$137$	− 268515.i	− 1.22227i	0.791527	−	0.611134i	$-0.209286\pi$
$138$	− 18581.3i	− 0.0830572i
$139$	−180749.	−0.793487	−0.396744	−	0.917929i	$-0.629860\pi$
$139$	−180749.	−0.793487	−0.396744	+	0.917929i	$0.629860\pi$
$140$	0	0
$141$	394201.	1.66982
$142$	− 11461.2i	− 0.0476988i
$143$	− 61037.8i	− 0.249608i
$144$	−454866.	−1.82800
$145$	0	0
$146$	−33088.9	−0.128470
$147$	− 251487.i	− 0.959892i
$148$	− 388763.i	− 1.45892i
$149$	471175.	1.73867	0.869334	−	0.494225i	$-0.164548\pi$
$149$	471175.	1.73867	0.869334	+	0.494225i	$0.164548\pi$
$150$	0	0
$151$	342495.	1.22239	0.611197	−	0.791478i	$-0.290688\pi$
$151$	342495.	1.22239	0.611197	+	0.791478i	$0.290688\pi$
$152$	− 2749.42i	− 0.00965232i
$153$	714191.i	2.46653i
$154$	25702.3	0.0873312
$155$	0	0
$156$	−141749.	−0.466345
$157$	− 28287.9i	− 0.0915909i	−0.998951	−	0.0457954i	$-0.985418\pi$
$157$	− 28287.9i	− 0.0915909i	0.998951	−	0.0457954i	$-0.0145822\pi$
$158$	− 3321.99i	− 0.0105866i
$159$	−946963.	−2.97057
$160$	0	0
$161$	−260856.	−0.793114
$162$	− 15628.2i	− 0.0467866i
$163$	219500.i	0.647091i	0.946213	+	0.323545i	$0.104875\pi$
$163$	219500.i	0.647091i	−0.946213	+	0.323545i	$0.895125\pi$
$164$	−3312.81	−0.00961806
$165$	0	0
$166$	400.291	0.00112747
$167$	− 46335.7i	− 0.128566i	−0.997932	−	0.0642828i	$-0.979524\pi$
$167$	− 46335.7i	− 0.128566i	0.997932	−	0.0642828i	$-0.0204760\pi$
$168$	− 119738.i	− 0.327309i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−44454.5	−0.116259
$172$	− 347863.i	− 0.896575i
$173$	− 210808.i	− 0.535514i	−0.963486	−	0.267757i	$-0.913717\pi$
$173$	− 210808.i	− 0.535514i	0.963486	−	0.267757i	$-0.0862825\pi$
$174$	3549.92	0.00888885
$175$	0	0
$176$	−363187.	−0.883788
$177$	42150.3i	0.101127i
$178$	46786.1i	0.110680i
$179$	157511.	0.367432	0.183716	−	0.982979i	$-0.441187\pi$
$179$	157511.	0.367432	0.183716	+	0.982979i	$0.441187\pi$
$180$	0	0
$181$	−236781.	−0.537217	−0.268608	−	0.963249i	$-0.586564\pi$
$181$	−236781.	−0.537217	−0.268608	+	0.963249i	$0.586564\pi$
$182$	− 12026.7i	− 0.0269133i
$183$	531505.i	1.17322i
$184$	−44962.4	−0.0979050
$185$	0	0
$186$	33949.6	0.0719536
$187$	570244.i	1.19250i
$188$	− 475501.i	− 0.981199i
$189$	−895973.	−1.82448
$190$	0	0
$191$	−473016.	−0.938193	−0.469096	−	0.883147i	$-0.655420\pi$
$191$	−473016.	−0.938193	−0.469096	+	0.883147i	$0.655420\pi$
$192$	833081.i	1.63093i
$193$	− 644265.i	− 1.24501i	−0.782618	−	0.622503i	$-0.786116\pi$
$193$	− 644265.i	− 1.24501i	0.782618	−	0.622503i	$-0.213884\pi$
$194$	45456.6	0.0867147
$195$	0	0
$196$	−303354.	−0.564040
$197$	241554.i	0.443453i	0.975109	+	0.221727i	$0.0711693\pi$
$197$	241554.i	0.443453i	−0.975109	+	0.221727i	$0.928831\pi$
$198$	− 71630.1i	− 0.129847i
$199$	558640.	0.999998	0.499999	−	0.866026i	$-0.333334\pi$
$199$	558640.	0.999998	0.499999	+	0.866026i	$0.333334\pi$
$200$	0	0
$201$	932632.	1.62825
$202$	1350.12i	0.00232806i
$203$	− 49836.1i	− 0.0848798i
$204$	1.32428e6	2.22795
$205$	0	0
$206$	49856.8	0.0818571
$207$	726984.i	1.17923i
$208$	169943.i	0.272362i
$209$	−35494.6	−0.0562078
$210$	0	0
$211$	−255863.	−0.395641	−0.197821	−	0.980238i	$-0.563386\pi$
$211$	−255863.	−0.395641	−0.197821	+	0.980238i	$0.563386\pi$
$212$	1.14227e6i	1.74553i
$213$	689303.i	1.04103i
$214$	−83069.8	−0.123996
$215$	0	0
$216$	−154434.	−0.225221
$217$	− 476607.i	− 0.687086i
$218$	− 42759.6i	− 0.0609386i
$219$	1.99005e6	2.80384
$220$	0	0
$221$	266831.	0.367498
$222$	− 141308.i	− 0.192435i
$223$	− 200547.i	− 0.270056i	−0.990842	−	0.135028i	$-0.956888\pi$
$223$	− 200547.i	− 0.270056i	0.990842	−	0.135028i	$-0.0431124\pi$
$224$	−216867.	−0.288784
$225$	0	0
$226$	−7423.09	−0.00966748
$227$	1.06615e6i	1.37327i	0.727005	+	0.686633i	$0.240912\pi$
$227$	1.06615e6i	1.37327i	−0.727005	+	0.686633i	$0.759088\pi$
$228$	82429.3i	0.105013i
$229$	140758.	0.177372	0.0886859	−	0.996060i	$-0.471733\pi$
$229$	140758.	0.177372	0.0886859	+	0.996060i	$0.471733\pi$
$230$	0	0
$231$	−1.54580e6	−1.90600
$232$	− 8590.01i	− 0.0104779i
$233$	616852.i	0.744374i	0.928158	+	0.372187i	$0.121392\pi$
$233$	616852.i	0.744374i	−0.928158	+	0.372187i	$0.878608\pi$
$234$	−33517.4	−0.0400157
$235$	0	0
$236$	50843.4	0.0594231
$237$	199793.i	0.231052i
$238$	112359.i	0.128578i
$239$	−1.23020e6	−1.39310	−0.696548	−	0.717510i	$-0.745282\pi$
$239$	−1.23020e6	−1.39310	−0.696548	+	0.717510i	$0.745282\pi$
$240$	0	0
$241$	73227.9	0.0812146	0.0406073	−	0.999175i	$-0.487071\pi$
$241$	73227.9	0.0812146	0.0406073	+	0.999175i	$0.487071\pi$
$242$	13419.5i	0.0147299i
$243$	− 401483.i	− 0.436166i
$244$	641123.	0.689393
$245$	0	0
$246$	−1204.15	−0.00126865
$247$	16608.7i	0.0173218i
$248$	− 82150.4i	− 0.0848165i
$249$	−24074.5	−0.0246070
$250$	0	0
$251$	−1.91445e6	−1.91805	−0.959024	−	0.283325i	$-0.908562\pi$
$251$	−1.91445e6	−1.91805	−0.959024	+	0.283325i	$0.908562\pi$
$252$	2.33529e6i	2.31654i
$253$	580458.i	0.570124i
$254$	108885.	0.105897
$255$	0	0
$256$	986150.	0.940466
$257$	517982.i	0.489195i	0.969625	+	0.244598i	$0.0786558\pi$
$257$	517982.i	0.489195i	−0.969625	+	0.244598i	$0.921344\pi$
$258$	− 126442.i	− 0.118261i
$259$	−1.98378e6	−1.83757
$260$	0	0
$261$	−138889.	−0.126202
$262$	65654.2i	0.0590894i
$263$	− 46304.4i	− 0.0412793i	−0.999787	−	0.0206397i	$-0.993430\pi$
$263$	− 46304.4i	− 0.0412793i	0.999787	−	0.0206397i	$-0.00657028\pi$
$264$	−266442.	−0.235284
$265$	0	0
$266$	−6993.73	−0.00606045
$267$	− 2.81384e6i	− 2.41558i
$268$	− 1.12498e6i	− 0.956770i
$269$	−473160.	−0.398682	−0.199341	−	0.979930i	$-0.563880\pi$
$269$	−473160.	−0.398682	−0.199341	+	0.979930i	$0.563880\pi$
$270$	0	0
$271$	−282334.	−0.233528	−0.116764	−	0.993160i	$-0.537252\pi$
$271$	−282334.	−0.233528	−0.116764	+	0.993160i	$0.537252\pi$
$272$	− 1.58769e6i	− 1.30120i
$273$	723315.i	0.587382i
$274$	−117729.	−0.0947346
$275$	0	0
$276$	1.34800e6	1.06517
$277$	− 151888.i	− 0.118939i	−0.998230	−	0.0594696i	$-0.981059\pi$
$277$	− 151888.i	− 0.118939i	0.998230	−	0.0594696i	$-0.0189409\pi$
$278$	79249.1i	0.0615010i
$279$	−1.32826e6	−1.02158
$280$	0	0
$281$	−1.82376e6	−1.37785	−0.688925	−	0.724832i	$-0.741917\pi$
$281$	−1.82376e6	−1.37785	−0.688925	+	0.724832i	$0.741917\pi$
$282$	− 172836.i	− 0.129423i
$283$	− 977978.i	− 0.725877i	−0.931813	−	0.362938i	$-0.881774\pi$
$283$	− 977978.i	− 0.725877i	0.931813	−	0.362938i	$-0.118226\pi$
$284$	831466.	0.611714
$285$	0	0
$286$	−26761.9	−0.0193464
$287$	16904.6i	0.0121144i
$288$	604390.i	0.429374i
$289$	−1.07300e6	−0.755711
$290$	0	0
$291$	−2.73388e6	−1.89255
$292$	− 2.40048e6i	− 1.64756i
$293$	− 2.78953e6i	− 1.89829i	−0.314843	−	0.949144i	$-0.601952\pi$
$293$	− 2.78953e6i	− 1.89829i	0.314843	−	0.949144i	$-0.398048\pi$
$294$	−110264.	−0.0743986
$295$	0	0
$296$	−341934.	−0.226837
$297$	1.99372e6i	1.31152i
$298$	− 206585.i	− 0.134759i
$299$	271610.	0.175698
$300$	0	0
$301$	−1.77507e6	−1.12928
$302$	− 150166.i	− 0.0947444i
$303$	− 81199.7i	− 0.0508099i
$304$	98825.1	0.0613315
$305$	0	0
$306$	313135.	0.191174
$307$	− 185868.i	− 0.112553i	−0.998415	−	0.0562766i	$-0.982077\pi$
$307$	− 185868.i	− 0.112553i	0.998415	−	0.0562766i	$-0.0179229\pi$
$308$	1.86461e6i	1.11998i
$309$	−2.99851e6	−1.78653
$310$	0	0
$311$	−1.37067e6	−0.803585	−0.401792	−	0.915731i	$-0.631613\pi$
$311$	−1.37067e6	−0.803585	−0.401792	+	0.915731i	$0.631613\pi$
$312$	124674.i	0.0725087i
$313$	− 2.54459e6i	− 1.46810i	−0.679093	−	0.734052i	$-0.737627\pi$
$313$	− 2.54459e6i	− 1.46810i	0.679093	−	0.734052i	$-0.262373\pi$
$314$	−12402.8	−0.00709896
$315$	0	0
$316$	240999.	0.135768
$317$	1.94759e6i	1.08855i	0.838906	+	0.544276i	$0.183195\pi$
$317$	1.94759e6i	1.08855i	−0.838906	+	0.544276i	$0.816805\pi$
$318$	415193.i	0.230241i
$319$	−110896.	−0.0610152
$320$	0	0
$321$	4.99603e6	2.70621
$322$	114372.i	0.0614721i
$323$	− 155167.i	− 0.0827546i
$324$	1.13377e6	0.600015
$325$	0	0
$326$	96239.1	0.0501542
$327$	2.57167e6i	1.32998i
$328$	2913.77i	0.00149544i
$329$	−2.42639e6	−1.23586
$330$	0	0
$331$	2.80487e6	1.40716	0.703578	−	0.710618i	$-0.251584\pi$
$331$	2.80487e6	1.40716	0.703578	+	0.710618i	$0.251584\pi$
$332$	29039.6i	0.0144593i
$333$	5.52863e6i	2.73216i
$334$	−20315.8	−0.00996476
$335$	0	0
$336$	4.30386e6	2.07975
$337$	− 2.69636e6i	− 1.29331i	−0.762782	−	0.646656i	$-0.776167\pi$
$337$	− 2.69636e6i	− 1.29331i	0.762782	−	0.646656i	$-0.223833\pi$
$338$	12522.5i	0.00596210i
$339$	446443.	0.210992
$340$	0	0
$341$	−1.06055e6	−0.493907
$342$	19490.9i	0.00901089i
$343$	− 1.17997e6i	− 0.541544i
$344$	−305961.	−0.139402
$345$	0	0
$346$	−92428.0	−0.0415063
$347$	− 4.14648e6i	− 1.84865i	−0.381601	−	0.924327i	$-0.624627\pi$
$347$	− 4.14648e6i	− 1.84865i	0.381601	−	0.924327i	$-0.375373\pi$
$348$	257534.i	0.113995i
$349$	372488.	0.163700	0.0818500	−	0.996645i	$-0.473917\pi$
$349$	372488.	0.163700	0.0818500	+	0.996645i	$0.473917\pi$
$350$	0	0
$351$	932910.	0.404177
$352$	482573.i	0.207590i
$353$	1.00030e6i	0.427260i	0.976915	+	0.213630i	$0.0685287\pi$
$353$	1.00030e6i	0.427260i	−0.976915	+	0.213630i	$0.931471\pi$
$354$	18480.7	0.00783808
$355$	0	0
$356$	−3.39417e6	−1.41941
$357$	− 6.75755e6i	− 2.80620i
$358$	− 69060.0i	− 0.0284786i
$359$	2.28110e6	0.934131	0.467066	−	0.884223i	$-0.345311\pi$
$359$	2.28110e6	0.934131	0.467066	+	0.884223i	$0.345311\pi$
$360$	0	0
$361$	−2.46644e6	−0.996099
$362$	103816.i	0.0416382i
$363$	− 807083.i	− 0.321478i
$364$	872492.	0.345150
$365$	0	0
$366$	233037.	0.0909331
$367$	3.68679e6i	1.42884i	0.699718	+	0.714419i	$0.253309\pi$
$367$	3.68679e6i	1.42884i	−0.699718	+	0.714419i	$0.746691\pi$
$368$	− 1.61613e6i	− 0.622095i
$369$	47111.8	0.0180121
$370$	0	0
$371$	5.82876e6	2.19857
$372$	2.46292e6i	0.922770i
$373$	− 2.38226e6i	− 0.886577i	−0.896379	−	0.443289i	$-0.853812\pi$
$373$	− 2.38226e6i	− 0.886577i	0.896379	−	0.443289i	$-0.146188\pi$
$374$	250022.	0.0924271
$375$	0	0
$376$	−418224.	−0.152560
$377$	51890.7i	0.0188034i
$378$	392837.i	0.141411i
$379$	5.36421e6	1.91826	0.959131	−	0.282963i	$-0.0913173\pi$
$379$	5.36421e6	1.91826	0.959131	+	0.282963i	$0.0913173\pi$
$380$	0	0
$381$	−6.54863e6	−2.31120
$382$	207392.i	0.0727167i
$383$	2.11002e6i	0.735004i	0.930023	+	0.367502i	$0.119787\pi$
$383$	2.11002e6i	0.735004i	−0.930023	+	0.367502i	$0.880213\pi$
$384$	1.49272e6	0.516595
$385$	0	0
$386$	−282476.	−0.0964969
$387$	4.94698e6i	1.67905i
$388$	3.29771e6i	1.11207i
$389$	439548.	0.147276	0.0736381	−	0.997285i	$-0.476539\pi$
$389$	439548.	0.147276	0.0736381	+	0.997285i	$0.476539\pi$
$390$	0	0
$391$	−2.53751e6	−0.839394
$392$	266813.i	0.0876986i
$393$	− 3.94861e6i	− 1.28962i
$394$	105909.	0.0343708
$395$	0	0
$396$	5.19650e6	1.66523
$397$	− 4.45376e6i	− 1.41824i	−0.705087	−	0.709121i	$-0.749092\pi$
$397$	− 4.45376e6i	− 1.41824i	0.705087	−	0.709121i	$-0.250908\pi$
$398$	− 244934.i	− 0.0775071i
$399$	420621.	0.132269
$400$	0	0
$401$	−666447.	−0.206969	−0.103484	−	0.994631i	$-0.532999\pi$
$401$	−666447.	−0.206969	−0.103484	+	0.994631i	$0.532999\pi$
$402$	− 408910.i	− 0.126201i
$403$	496256.i	0.152210i
$404$	−97946.5	−0.0298563
$405$	0	0
$406$	−21850.5	−0.00657880
$407$	4.41432e6i	1.32092i
$408$	− 1.16477e6i	− 0.346408i
$409$	2.89544e6	0.855867	0.427934	−	0.903810i	$-0.359242\pi$
$409$	2.89544e6	0.855867	0.427934	+	0.903810i	$0.359242\pi$
$410$	0	0
$411$	7.08055e6	2.06758
$412$	3.61693e6i	1.04978i
$413$	− 259444.i	− 0.0748460i
$414$	318744.	0.0913989
$415$	0	0
$416$	225807.	0.0639741
$417$	− 4.76624e6i	− 1.34226i
$418$	15562.5i	0.00435651i
$419$	4.58473e6	1.27579	0.637895	−	0.770124i	$-0.279806\pi$
$419$	4.58473e6	1.27579	0.637895	+	0.770124i	$0.279806\pi$
$420$	0	0
$421$	−2.73604e6	−0.752346	−0.376173	−	0.926549i	$-0.622760\pi$
$421$	−2.73604e6	−0.752346	−0.376173	+	0.926549i	$0.622760\pi$
$422$	112183.i	0.0306651i
$423$	6.76214e6i	1.83753i
$424$	1.00467e6	0.271400
$425$	0	0
$426$	302223.	0.0806870
$427$	− 3.27153e6i	− 0.868322i
$428$	− 6.02641e6i	− 1.59019i
$429$	1.60953e6	0.422235
$430$	0	0
$431$	5.29736e6	1.37362	0.686810	−	0.726837i	$-0.259010\pi$
$431$	5.29736e6	1.37362	0.686810	+	0.726837i	$0.259010\pi$
$432$	− 5.55099e6i	− 1.43107i
$433$	1.39103e6i	0.356546i	0.983981	+	0.178273i	$0.0570511\pi$
$433$	1.39103e6i	0.356546i	−0.983981	+	0.178273i	$0.942949\pi$
$434$	−208967.	−0.0532542
$435$	0	0
$436$	3.10206e6	0.781508
$437$	− 157946.i	− 0.0395644i
$438$	− 872532.i	− 0.217318i
$439$	2.21416e6	0.548338	0.274169	−	0.961682i	$-0.411597\pi$
$439$	2.21416e6	0.548338	0.274169	+	0.961682i	$0.411597\pi$
$440$	0	0
$441$	4.31403e6	1.05630
$442$	− 116991.i	− 0.0284837i
$443$	2.81977e6i	0.682660i	0.939943	+	0.341330i	$0.110877\pi$
$443$	2.81977e6i	0.682660i	−0.939943	+	0.341330i	$0.889123\pi$
$444$	1.02514e7	2.46789
$445$	0	0
$446$	−87929.2	−0.0209313
$447$	1.24246e7i	2.94112i
$448$	− 5.12779e6i	− 1.20708i
$449$	−7.30191e6	−1.70931	−0.854655	−	0.519197i	$-0.826231\pi$
$449$	−7.30191e6	−1.70931	−0.854655	+	0.519197i	$0.826231\pi$
$450$	0	0
$451$	37616.3	0.00870833
$452$	− 538518.i	− 0.123981i
$453$	9.03135e6i	2.06779i
$454$	467451.	0.106438
$455$	0	0
$456$	72500.2	0.0163278
$457$	− 1.99963e6i	− 0.447877i	−0.974603	−	0.223938i	$-0.928109\pi$
$457$	− 1.99963e6i	− 0.447877i	0.974603	−	0.223938i	$-0.0718914\pi$
$458$	− 61715.0i	− 0.0137476i
$459$	−8.71569e6	−1.93095
$460$	0	0
$461$	4.68772e6	1.02733	0.513664	−	0.857991i	$-0.328288\pi$
$461$	4.68772e6	1.02733	0.513664	+	0.857991i	$0.328288\pi$
$462$	677751.i	0.147729i
$463$	− 743245.i	− 0.161131i	−0.996749	−	0.0805655i	$-0.974327\pi$
$463$	− 743245.i	− 0.161131i	0.996749	−	0.0805655i	$-0.0256726\pi$
$464$	308759.	0.0665772
$465$	0	0
$466$	270457.	0.0576944
$467$	− 498135.i	− 0.105695i	−0.998603	−	0.0528476i	$-0.983170\pi$
$467$	− 498135.i	− 0.105695i	0.998603	−	0.0528476i	$-0.0168297\pi$
$468$	− 2.43156e6i	− 0.513182i
$469$	−5.74054e6	−1.20509
$470$	0	0
$471$	745934.	0.154934
$472$	− 44719.1i	− 0.00923927i
$473$	3.94991e6i	0.811772i
$474$	87598.7	0.0179082
$475$	0	0
$476$	−8.15124e6	−1.64895
$477$	− 1.62443e7i	− 3.26892i
$478$	539378.i	0.107975i
$479$	−2.55822e6	−0.509447	−0.254724	−	0.967014i	$-0.581984\pi$
$479$	−2.55822e6	−0.509447	−0.254724	+	0.967014i	$0.581984\pi$
$480$	0	0
$481$	2.06556e6	0.407076
$482$	− 32106.6i	− 0.00629472i
$483$	− 6.87859e6i	− 1.34163i
$484$	−973538.	−0.188903
$485$	0	0
$486$	−176029.	−0.0338060
$487$	− 4.08401e6i	− 0.780305i	−0.920750	−	0.390153i	$-0.872422\pi$
$487$	− 4.08401e6i	− 0.780305i	0.920750	−	0.390153i	$-0.127578\pi$
$488$	− 563896.i	− 0.107189i
$489$	−5.78806e6	−1.09461
$490$	0	0
$491$	−3.16712e6	−0.592871	−0.296436	−	0.955053i	$-0.595798\pi$
$491$	−3.16712e6	−0.592871	−0.296436	+	0.955053i	$0.595798\pi$
$492$	− 87356.6i	− 0.0162698i
$493$	− 484788.i	− 0.0898326i
$494$	7282.05	0.00134257
$495$	0	0
$496$	2.95282e6	0.538930
$497$	− 4.24281e6i	− 0.770481i
$498$	10555.4i	0.00190722i
$499$	−5.92427e6	−1.06508	−0.532542	−	0.846404i	$-0.678763\pi$
$499$	−5.92427e6	−1.06508	−0.532542	+	0.846404i	$0.678763\pi$
$500$	0	0
$501$	1.22184e6	0.217481
$502$	839385.i	0.148663i
$503$	9.14040e6i	1.61081i	0.592723	+	0.805406i	$0.298053\pi$
$503$	9.14040e6i	1.61081i	−0.592723	+	0.805406i	$0.701947\pi$
$504$	2.05399e6	0.360182
$505$	0	0
$506$	254500.	0.0441888
$507$	− 753134.i	− 0.130122i
$508$	7.89923e6i	1.35808i
$509$	−1.05761e7	−1.80938	−0.904690	−	0.426071i	$-0.859897\pi$
$509$	−1.05761e7	−1.80938	−0.904690	+	0.426071i	$0.859897\pi$
$510$	0	0
$511$	−1.22492e7	−2.07518
$512$	− 2.24384e6i	− 0.378283i
$513$	− 542504.i	− 0.0910142i
$514$	227108.	0.0379162
$515$	0	0
$516$	9.17290e6	1.51664
$517$	5.39921e6i	0.888391i
$518$	869782.i	0.142425i
$519$	5.55886e6	0.905872
$520$	0	0
$521$	2.23591e6	0.360877	0.180439	−	0.983586i	$-0.442248\pi$
$521$	2.23591e6	0.360877	0.180439	+	0.983586i	$0.442248\pi$
$522$	60895.6i	0.00978159i
$523$	2.60360e6i	0.416217i	0.978106	+	0.208108i	$0.0667307\pi$
$523$	2.60360e6i	0.416217i	−0.978106	+	0.208108i	$0.933269\pi$
$524$	−4.76298e6	−0.757792
$525$	0	0
$526$	−20302.0	−0.00319945
$527$	− 4.63626e6i	− 0.727179i
$528$	− 9.57698e6i	− 1.49501i
$529$	3.85338e6	0.598691
$530$	0	0
$531$	−723049.	−0.111284
$532$	− 507370.i	− 0.0777223i
$533$	− 17601.5i	− 0.00268369i
$534$	−1.23372e6	−0.187225
$535$	0	0
$536$	−989469.	−0.148761
$537$	4.15344e6i	0.621545i
$538$	207456.i	0.0309008i
$539$	3.44452e6	0.510690
$540$	0	0
$541$	9.46248e6	1.38999	0.694995	−	0.719014i	$-0.255406\pi$
$541$	9.46248e6	1.38999	0.694995	+	0.719014i	$0.255406\pi$
$542$	123789.i	0.0181001i
$543$	− 6.24374e6i	− 0.908752i
$544$	−2.10960e6	−0.305635
$545$	0	0
$546$	317135.	0.0455264
$547$	1.06969e7i	1.52858i	0.644871	+	0.764292i	$0.276911\pi$
$547$	1.06969e7i	1.52858i	−0.644871	+	0.764292i	$0.723089\pi$
$548$	− 8.54085e6i	− 1.21493i
$549$	−9.11747e6	−1.29105
$550$	0	0
$551$	30175.4	0.00423422
$552$	− 1.18563e6i	− 0.165615i
$553$	− 1.22977e6i	− 0.171006i
$554$	−66595.0	−0.00921864
$555$	0	0
$556$	−5.74923e6	−0.788720
$557$	− 1.23614e7i	− 1.68822i	−0.536170	−	0.844110i	$-0.680129\pi$
$557$	− 1.23614e7i	− 1.68822i	0.536170	−	0.844110i	$-0.319871\pi$
$558$	582374.i	0.0791802i
$559$	1.84825e6	0.250168
$560$	0	0
$561$	−1.50370e7	−2.01722
$562$	799623.i	0.106793i
$563$	879561.i	0.116949i	0.998289	+	0.0584743i	$0.0186236\pi$
$563$	879561.i	0.116949i	−0.998289	+	0.0584743i	$0.981376\pi$
$564$	1.25386e7	1.65979
$565$	0	0
$566$	−428792.	−0.0562607
$567$	− 5.78540e6i	− 0.755746i
$568$	− 731311.i	− 0.0951111i
$569$	3.69345e6	0.478247	0.239123	−	0.970989i	$-0.423140\pi$
$569$	3.69345e6	0.478247	0.239123	+	0.970989i	$0.423140\pi$
$570$	0	0
$571$	1.02916e7	1.32097	0.660483	−	0.750841i	$-0.270352\pi$
$571$	1.02916e7	1.32097	0.660483	+	0.750841i	$0.270352\pi$
$572$	− 1.94148e6i	− 0.248109i
$573$	− 1.24731e7i	− 1.58704i
$574$	7411.79	0.000938952 0
$575$	0	0
$576$	−1.42907e7	−1.79472
$577$	7.71100e6i	0.964209i	0.876114	+	0.482105i	$0.160127\pi$
$577$	7.71100e6i	0.964209i	−0.876114	+	0.482105i	$0.839873\pi$
$578$	470454.i	0.0585731i
$579$	1.69888e7	2.10604
$580$	0	0
$581$	148184.	0.0182121
$582$	1.19866e6i	0.146686i
$583$	− 1.29702e7i	− 1.58043i
$584$	−2.11133e6	−0.256167
$585$	0	0
$586$	−1.22306e6	−0.147131
$587$	− 1.12713e7i	− 1.35014i	−0.737753	−	0.675071i	$-0.764113\pi$
$587$	− 1.12713e7i	− 1.35014i	0.737753	−	0.675071i	$-0.235887\pi$
$588$	− 7.99924e6i	− 0.954126i
$589$	288582.	0.0342752
$590$	0	0
$591$	−6.36960e6	−0.750142
$592$	− 1.22905e7i	− 1.44133i
$593$	1.13296e7i	1.32305i	0.749923	+	0.661525i	$0.230090\pi$
$593$	1.13296e7i	1.32305i	−0.749923	+	0.661525i	$0.769910\pi$
$594$	874143.	0.101652
$595$	0	0
$596$	1.49870e7	1.72822
$597$	1.47309e7i	1.69159i
$598$	− 119087.i	− 0.0136179i
$599$	2.16607e6	0.246664	0.123332	−	0.992365i	$-0.460642\pi$
$599$	2.16607e6	0.246664	0.123332	+	0.992365i	$0.460642\pi$
$600$	0	0
$601$	−1.25868e7	−1.42144	−0.710720	−	0.703475i	$-0.751631\pi$
$601$	−1.25868e7	−1.42144	−0.710720	+	0.703475i	$0.751631\pi$
$602$	778276.i	0.0875271i
$603$	1.59984e7i	1.79178i
$604$	1.08940e7	1.21505
$605$	0	0
$606$	−35601.8	−0.00393813
$607$	1.42492e7i	1.56970i	0.619684	+	0.784852i	$0.287261\pi$
$607$	1.42492e7i	1.56970i	−0.619684	+	0.784852i	$0.712739\pi$
$608$	− 131311.i	− 0.0144060i
$609$	1.31415e6	0.143582
$610$	0	0
$611$	2.52642e6	0.273780
$612$	2.27168e7i	2.45171i
$613$	− 5.77209e6i	− 0.620415i	−0.950669	−	0.310207i	$-0.899602\pi$
$613$	− 5.77209e6i	− 0.620415i	0.950669	−	0.310207i	$-0.100398\pi$
$614$	−81493.2	−0.00872369
$615$	0	0
$616$	1.64000e6	0.174138
$617$	1.22764e7i	1.29825i	0.760681	+	0.649126i	$0.224865\pi$
$617$	1.22764e7i	1.29825i	−0.760681	+	0.649126i	$0.775135\pi$
$618$	1.31469e6i	0.138469i
$619$	−542347.	−0.0568919	−0.0284460	−	0.999595i	$-0.509056\pi$
$619$	−542347.	−0.0568919	−0.0284460	+	0.999595i	$0.509056\pi$
$620$	0	0
$621$	−8.87180e6	−0.923172
$622$	600966.i	0.0622836i
$623$	1.73198e7i	1.78781i
$624$	−4.48129e6	−0.460725
$625$	0	0
$626$	−1.11567e6	−0.113789
$627$	− 935968.i	− 0.0950806i
$628$	− 899776.i	− 0.0910407i
$629$	−1.92975e7	−1.94479
$630$	0	0
$631$	3.52305e6	0.352246	0.176123	−	0.984368i	$-0.443644\pi$
$631$	3.52305e6	0.352246	0.176123	+	0.984368i	$0.443644\pi$
$632$	− 211969.i	− 0.0211096i
$633$	− 6.74694e6i	− 0.669264i
$634$	853915.	0.0843706
$635$	0	0
$636$	−3.01208e7	−2.95273
$637$	− 1.61177e6i	− 0.157382i
$638$	48621.9i	0.00472912i
$639$	−1.18243e7	−1.14558
$640$	0	0
$641$	1.51725e7	1.45852	0.729258	−	0.684239i	$-0.239866\pi$
$641$	1.51725e7	1.45852	0.729258	+	0.684239i	$0.239866\pi$
$642$	− 2.19049e6i	− 0.209751i
$643$	1.62226e7i	1.54737i	0.633573	+	0.773683i	$0.281588\pi$
$643$	1.62226e7i	1.54737i	−0.633573	+	0.773683i	$0.718412\pi$
$644$	−8.29724e6	−0.788350
$645$	0	0
$646$	−68032.4	−0.00641408
$647$	1.91064e7i	1.79440i	0.441625	+	0.897200i	$0.354402\pi$
$647$	1.91064e7i	1.79440i	−0.441625	+	0.897200i	$0.645598\pi$
$648$	− 997201.i	− 0.0932921i
$649$	−577317.	−0.0538025
$650$	0	0
$651$	1.25678e7	1.16227
$652$	6.98180e6i	0.643204i
$653$	3.26760e6i	0.299879i	0.988695	+	0.149939i	$0.0479079\pi$
$653$	3.26760e6i	0.299879i	−0.988695	+	0.149939i	$0.952092\pi$
$654$	1.12754e6	0.103083
$655$	0	0
$656$	−104732.	−0.00950215
$657$	3.41375e7i	3.08544i
$658$	1.06384e6i	0.0957884i
$659$	6.57191e6	0.589492	0.294746	−	0.955576i	$-0.404765\pi$
$659$	6.57191e6	0.589492	0.294746	+	0.955576i	$0.404765\pi$
$660$	0	0
$661$	−7.69757e6	−0.685252	−0.342626	−	0.939472i	$-0.611316\pi$
$661$	−7.69757e6	−0.685252	−0.342626	+	0.939472i	$0.611316\pi$
$662$	− 1.22979e6i	− 0.109065i
$663$	7.03614e6i	0.621657i
$664$	25541.7	0.00224817
$665$	0	0
$666$	2.42401e6	0.211762
$667$	− 493471.i	− 0.0429484i
$668$	− 1.47384e6i	− 0.127793i
$669$	5.28828e6	0.456825
$670$	0	0
$671$	−7.27982e6	−0.624187
$672$	− 5.71863e6i	− 0.488505i
$673$	9.22875e6i	0.785426i	0.919661	+	0.392713i	$0.128463\pi$
$673$	9.22875e6i	0.785426i	−0.919661	+	0.392713i	$0.871537\pi$
$674$	−1.18221e6	−0.100241
$675$	0	0
$676$	−908462.	−0.0764610
$677$	− 3.79269e6i	− 0.318035i	−0.987276	−	0.159018i	$-0.949167\pi$
$677$	− 3.79269e6i	− 0.318035i	0.987276	−	0.159018i	$-0.0508327\pi$
$678$	− 195742.i	− 0.0163534i
$679$	1.68276e7	1.40071
$680$	0	0
$681$	−2.81137e7	−2.32300
$682$	464995.i	0.0382814i
$683$	1.12547e7i	0.923171i	0.887096	+	0.461586i	$0.152719\pi$
$683$	1.12547e7i	0.923171i	−0.887096	+	0.461586i	$0.847281\pi$
$684$	−1.41400e6	−0.115560
$685$	0	0
$686$	−517353.	−0.0419736
$687$	3.71169e6i	0.300041i
$688$	− 1.09975e7i	− 0.885770i
$689$	−6.06905e6	−0.487049
$690$	0	0
$691$	1.53091e7	1.21970	0.609852	−	0.792515i	$-0.291229\pi$
$691$	1.53091e7	1.21970	0.609852	+	0.792515i	$0.291229\pi$
$692$	− 6.70532e6i	− 0.532297i
$693$	− 2.65167e7i	− 2.09743i
$694$	−1.81801e6	−0.143284
$695$	0	0
$696$	226513.	0.0177243
$697$	164442.i	0.0128213i
$698$	− 163316.i	− 0.0126879i
$699$	−1.62660e7	−1.25918
$700$	0	0
$701$	−1.32613e7	−1.01928	−0.509639	−	0.860389i	$-0.670221\pi$
$701$	−1.32613e7	−1.01928	−0.509639	+	0.860389i	$0.670221\pi$
$702$	− 409032.i	− 0.0313267i
$703$	− 1.20116e6i	− 0.0916670i
$704$	−1.14104e7	−0.867698
$705$	0	0
$706$	438577.	0.0331157
$707$	499801.i	0.0376053i
$708$	1.34071e6i	0.100520i
$709$	6.03399e6	0.450805	0.225403	−	0.974266i	$-0.427630\pi$
$709$	6.03399e6	0.450805	0.225403	+	0.974266i	$0.427630\pi$
$710$	0	0
$711$	−3.42726e6	−0.254257
$712$	2.98532e6i	0.220694i
$713$	− 4.71930e6i	− 0.347659i
$714$	−2.96283e6	−0.217501
$715$	0	0
$716$	5.01006e6	0.365225
$717$	− 3.24395e7i	− 2.35655i
$718$	− 1.00014e6i	− 0.0724020i
$719$	2.60342e7	1.87812	0.939059	−	0.343757i	$-0.111700\pi$
$719$	2.60342e7	1.87812	0.939059	+	0.343757i	$0.111700\pi$
$720$	0	0
$721$	1.84565e7	1.32224
$722$	1.08140e6i	0.0772049i
$723$	1.93097e6i	0.137382i
$724$	−7.53146e6	−0.533990
$725$	0	0
$726$	−353863.	−0.0249169
$727$	− 1.17829e7i	− 0.826832i	−0.910542	−	0.413416i	$-0.864336\pi$
$727$	− 1.17829e7i	− 0.826832i	0.910542	−	0.413416i	$-0.135664\pi$
$728$	− 767396.i	− 0.0536650i
$729$	1.92484e7	1.34146
$730$	0	0
$731$	−1.72673e7	−1.19517
$732$	1.69060e7i	1.16617i
$733$	− 3.67552e6i	− 0.252673i	−0.991987	−	0.126336i	$-0.959678\pi$
$733$	− 3.67552e6i	− 0.252673i	0.991987	−	0.126336i	$-0.0403219\pi$
$734$	1.61646e6	0.110745
$735$	0	0
$736$	−2.14739e6	−0.146122
$737$	1.27739e7i	0.866273i
$738$	− 20656.0i	− 0.00139607i
$739$	−4.16486e6	−0.280536	−0.140268	−	0.990114i	$-0.544796\pi$
$739$	−4.16486e6	−0.280536	−0.140268	+	0.990114i	$0.544796\pi$
$740$	0	0
$741$	−437961.	−0.0293015
$742$	− 2.55560e6i	− 0.170405i
$743$	2.00724e6i	0.133391i	0.997773	+	0.0666955i	$0.0212456\pi$
$743$	2.00724e6i	0.133391i	−0.997773	+	0.0666955i	$0.978754\pi$
$744$	2.16625e6	0.143475
$745$	0	0
$746$	−1.04449e6	−0.0687162
$747$	− 412975.i	− 0.0270784i
$748$	1.81382e7i	1.18533i
$749$	−3.07516e7	−2.00292
$750$	0	0
$751$	6.47699e6	0.419057	0.209529	−	0.977803i	$-0.432807\pi$
$751$	6.47699e6	0.419057	0.209529	+	0.977803i	$0.432807\pi$
$752$	− 1.50327e7i	− 0.969374i
$753$	− 5.04827e7i	− 3.24456i
$754$	22751.3	0.00145740
$755$	0	0
$756$	−2.84989e7	−1.81352
$757$	1.80725e7i	1.14625i	0.819469	+	0.573123i	$0.194268\pi$
$757$	1.80725e7i	1.14625i	−0.819469	+	0.573123i	$0.805732\pi$
$758$	− 2.35192e6i	− 0.148679i
$759$	−1.53063e7	−0.964418
$760$	0	0
$761$	1.16338e7	0.728215	0.364107	−	0.931357i	$-0.381374\pi$
$761$	1.16338e7	0.728215	0.364107	+	0.931357i	$0.381374\pi$
$762$	2.87123e6i	0.179135i
$763$	− 1.58292e7i	− 0.984344i
$764$	−1.50456e7	−0.932557
$765$	0	0
$766$	925132.	0.0569681
$767$	270140.i	0.0165806i
$768$	2.60041e7i	1.59089i
$769$	1.28037e7	0.780764	0.390382	−	0.920653i	$-0.372343\pi$
$769$	1.28037e7	0.780764	0.390382	+	0.920653i	$0.372343\pi$
$770$	0	0
$771$	−1.36588e7	−0.827519
$772$	− 2.04926e7i	− 1.23753i
$773$	1.45066e7i	0.873205i	0.899655	+	0.436602i	$0.143818\pi$
$773$	1.45066e7i	0.873205i	−0.899655	+	0.436602i	$0.856182\pi$
$774$	2.16899e6	0.130138
$775$	0	0
$776$	2.90049e6	0.172908
$777$	− 5.23109e7i	− 3.10842i
$778$	− 192719.i	− 0.0114150i
$779$	−10235.6	−0.000604324 0
$780$	0	0
$781$	−9.44112e6	−0.553855
$782$	1.11256e6i	0.0650591i
$783$	− 1.69494e6i	− 0.0987987i
$784$	−9.59035e6	−0.557243
$785$	0	0
$786$	−1.73126e6	−0.0999551
$787$	− 792407.i	− 0.0456049i	−0.999740	−	0.0228024i	$-0.992741\pi$
$787$	− 792407.i	− 0.0456049i	0.999740	−	0.0228024i	$-0.00725887\pi$
$788$	7.68328e6i	0.440789i
$789$	1.22101e6	0.0698278
$790$	0	0
$791$	−2.74795e6	−0.156159
$792$	− 4.57055e6i	− 0.258914i
$793$	3.40640e6i	0.192359i
$794$	−1.95274e6	−0.109924
$795$	0	0
$796$	1.77691e7	0.993990
$797$	− 1.10193e7i	− 0.614483i	−0.951632	−	0.307242i	$-0.900594\pi$
$797$	− 1.10193e7i	− 0.614483i	0.951632	−	0.307242i	$-0.0994060\pi$
$798$	− 184420.i	− 0.0102518i
$799$	−2.36030e7	−1.30798
$800$	0	0
$801$	4.82687e7	2.65818
$802$	292202.i	0.0160416i
$803$	2.72570e7i	1.49172i
$804$	2.96649e7	1.61846
$805$	0	0
$806$	217582.	0.0117974
$807$	− 1.24769e7i	− 0.674408i
$808$	86148.3i	0.00464214i
$809$	−1.06902e7	−0.574266	−0.287133	−	0.957891i	$-0.592702\pi$
$809$	−1.06902e7	−0.574266	−0.287133	+	0.957891i	$0.592702\pi$
$810$	0	0
$811$	3.41511e7	1.82327	0.911637	−	0.410996i	$-0.134819\pi$
$811$	3.41511e7	1.82327	0.911637	+	0.410996i	$0.134819\pi$
$812$	− 1.58518e6i	− 0.0843699i
$813$	− 7.44495e6i	− 0.395035i
$814$	1.93545e6	0.102381
$815$	0	0
$816$	4.18664e7	2.20110
$817$	− 1.07479e6i	− 0.0563338i
$818$	− 1.26950e6i	− 0.0663359i
$819$	−1.24078e7	−0.646375
$820$	0	0
$821$	−2.06578e7	−1.06961	−0.534805	−	0.844976i	$-0.679615\pi$
$821$	−2.06578e7	−1.06961	−0.534805	+	0.844976i	$0.679615\pi$
$822$	− 3.10445e6i	− 0.160252i
$823$	− 2.30912e7i	− 1.18836i	−0.804334	−	0.594178i	$-0.797477\pi$
$823$	− 2.30912e7i	− 1.18836i	0.804334	−	0.594178i	$-0.202523\pi$
$824$	3.18125e6	0.163222
$825$	0	0
$826$	−113752.	−0.00580111
$827$	2.48313e7i	1.26251i	0.775574	+	0.631256i	$0.217460\pi$
$827$	2.48313e7i	1.26251i	−0.775574	+	0.631256i	$0.782540\pi$
$828$	2.31237e7i	1.17215i
$829$	−3.83083e6	−0.193601	−0.0968003	−	0.995304i	$-0.530861\pi$
$829$	−3.83083e6	−0.193601	−0.0968003	+	0.995304i	$0.530861\pi$
$830$	0	0
$831$	4.00519e6	0.201197
$832$	5.33918e6i	0.267403i
$833$	1.50579e7i	0.751888i
$834$	−2.08974e6	−0.104035
$835$	0	0
$836$	−1.12900e6	−0.0558701
$837$	− 1.62096e7i	− 0.799757i
$838$	− 2.01016e6i	− 0.0988829i
$839$	−1.73573e6	−0.0851290	−0.0425645	−	0.999094i	$-0.513553\pi$
$839$	−1.73573e6	−0.0851290	−0.0425645	+	0.999094i	$0.513553\pi$
$840$	0	0
$841$	−2.04169e7	−0.995404
$842$	1.19961e6i	0.0583123i
$843$	− 4.80913e7i	− 2.33076i
$844$	−8.13844e6	−0.393265
$845$	0	0
$846$	2.96484e6	0.142422
$847$	4.96777e6i	0.237932i
$848$	3.61120e7i	1.72450i
$849$	2.57886e7	1.22789
$850$	0	0
$851$	−1.96431e7	−0.929793
$852$	2.19252e7i	1.03477i
$853$	6.36325e6i	0.299438i	0.988729	+	0.149719i	$0.0478368\pi$
$853$	6.36325e6i	0.299438i	−0.988729	+	0.149719i	$0.952163\pi$
$854$	−1.43439e6	−0.0673012
$855$	0	0
$856$	−5.30050e6	−0.247248
$857$	− 5.21486e6i	− 0.242544i	−0.992619	−	0.121272i	$-0.961303\pi$
$857$	− 5.21486e6i	− 0.242544i	0.992619	−	0.121272i	$-0.0386973\pi$
$858$	− 705692.i	− 0.0327263i
$859$	2.08063e7	0.962081	0.481041	−	0.876698i	$-0.340259\pi$
$859$	2.08063e7	0.962081	0.481041	+	0.876698i	$0.340259\pi$
$860$	0	0
$861$	−445763.	−0.0204926
$862$	− 2.32261e6i	− 0.106466i
$863$	1.40008e7i	0.639919i	0.947431	+	0.319959i	$0.103669\pi$
$863$	1.40008e7i	0.639919i	−0.947431	+	0.319959i	$0.896331\pi$
$864$	−7.37571e6	−0.336140
$865$	0	0
$866$	609892.	0.0276349
$867$	− 2.82943e7i	− 1.27835i
$868$	− 1.51598e7i	− 0.682959i
$869$	−2.73649e6	−0.122926
$870$	0	0
$871$	5.97720e6	0.266964
$872$	− 2.72840e6i	− 0.121511i
$873$	− 4.68971e7i	− 2.08262i
$874$	−69251.0	−0.00306653
$875$	0	0
$876$	6.32991e7	2.78700
$877$	− 1.29936e6i	− 0.0570466i	−0.999593	−	0.0285233i	$-0.990920\pi$
$877$	− 1.29936e6i	− 0.0570466i	0.999593	−	0.0285233i	$-0.00908049\pi$
$878$	− 970794.i	− 0.0425002i
$879$	7.35580e7	3.21113
$880$	0	0
$881$	−2.42501e7	−1.05263	−0.526313	−	0.850291i	$-0.676426\pi$
$881$	−2.42501e7	−1.05263	−0.526313	+	0.850291i	$0.676426\pi$
$882$	− 1.89147e6i	− 0.0818707i
$883$	− 716859.i	− 0.0309408i	−0.999880	−	0.0154704i	$-0.995075\pi$
$883$	− 716859.i	− 0.0309408i	0.999880	−	0.0154704i	$-0.00492458\pi$
$884$	8.48728e6	0.365290
$885$	0	0
$886$	1.23632e6	0.0529111
$887$	2.39534e7i	1.02225i	0.859506	+	0.511126i	$0.170771\pi$
$887$	2.39534e7i	1.02225i	−0.859506	+	0.511126i	$0.829229\pi$
$888$	− 9.01656e6i	− 0.383715i
$889$	4.03082e7	1.71056
$890$	0	0
$891$	−1.28737e7	−0.543262
$892$	− 6.37895e6i	− 0.268434i
$893$	− 1.46916e6i	− 0.0616509i
$894$	5.44752e6	0.227958
$895$	0	0
$896$	−9.18800e6	−0.382341
$897$	7.16217e6i	0.297210i
$898$	3.20150e6i	0.132484i
$899$	901616.	0.0372068
$900$	0	0
$901$	5.67000e7	2.32686
$902$	− 16492.8i	0 0.000674958i
$903$	− 4.68075e7i	− 1.91028i
$904$	−473651.	−0.0192769
$905$	0	0
$906$	3.95977e6	0.160269
$907$	− 4.33061e7i	− 1.74796i	−0.485965	−	0.873978i	$-0.661532\pi$
$907$	− 4.33061e7i	− 1.74796i	0.485965	−	0.873978i	$-0.338468\pi$
$908$	3.39119e7i	1.36502i
$909$	1.39290e6	0.0559129
$910$	0	0
$911$	1.76763e7	0.705658	0.352829	−	0.935688i	$-0.385220\pi$
$911$	1.76763e7	0.705658	0.352829	+	0.935688i	$0.385220\pi$
$912$	2.60595e6i	0.103748i
$913$	− 329739.i	− 0.0130916i
$914$	−876731.	−0.0347137
$915$	0	0
$916$	4.47720e6	0.176306
$917$	2.43045e7i	0.954473i
$918$	3.82137e6i	0.149662i
$919$	−3.02645e7	−1.18208	−0.591038	−	0.806644i	$-0.701282\pi$
$919$	−3.02645e7	−1.18208	−0.591038	+	0.806644i	$0.701282\pi$
$920$	0	0
$921$	4.90121e6	0.190394
$922$	− 2.05532e6i	− 0.0796254i
$923$	4.41772e6i	0.170684i
$924$	−4.91684e7	−1.89455
$925$	0	0
$926$	−325874.	−0.0124888
$927$	− 5.14367e7i	− 1.96596i
$928$	− 410255.i	− 0.0156381i
$929$	6.53353e6	0.248376	0.124188	−	0.992259i	$-0.460367\pi$
$929$	6.53353e6	0.248376	0.124188	+	0.992259i	$0.460367\pi$
$930$	0	0
$931$	−937274.	−0.0354399
$932$	1.96207e7i	0.739902i
$933$	− 3.61436e7i	− 1.35934i
$934$	−218406.	−0.00819214
$935$	0	0
$936$	−2.13867e6	−0.0797910
$937$	4.85033e7i	1.80477i	0.430930	+	0.902385i	$0.358185\pi$
$937$	4.85033e7i	1.80477i	−0.430930	+	0.902385i	$0.641815\pi$
$938$	2.51693e6i	0.0934035i
$939$	6.70991e7	2.48343
$940$	0	0
$941$	4.81050e7	1.77099	0.885494	−	0.464650i	$-0.153820\pi$
$941$	4.81050e7	1.77099	0.885494	+	0.464650i	$0.153820\pi$
$942$	− 327053.i	− 0.0120085i
$943$	167387.i	0.00612976i
$944$	1.60738e6	0.0587069
$945$	0	0
$946$	1.73183e6	0.0629182
$947$	2.51097e7i	0.909843i	0.890532	+	0.454922i	$0.150333\pi$
$947$	2.51097e7i	0.909843i	−0.890532	+	0.454922i	$0.849667\pi$
$948$	6.35497e6i	0.229664i
$949$	1.27542e7	0.459713
$950$	0	0
$951$	−5.13566e7	−1.84139
$952$	7.16938e6i	0.256383i
$953$	4.59098e7i	1.63747i	0.574172	+	0.818734i	$0.305324\pi$
$953$	4.59098e7i	1.63747i	−0.574172	+	0.818734i	$0.694676\pi$
$954$	−7.12225e6	−0.253365
$955$	0	0
$956$	−3.91299e7	−1.38473
$957$	− 2.92425e6i	− 0.103213i
$958$	1.12164e6i	0.0394858i
$959$	−4.35823e7	−1.53025
$960$	0	0
$961$	−2.00066e7	−0.698818
$962$	− 905640.i	− 0.0315513i
$963$	8.57021e7i	2.97801i
$964$	2.32922e6	0.0807267
$965$	0	0
$966$	−3.01590e6	−0.103986
$967$	6.73270e6i	0.231538i	0.993276	+	0.115769i	$0.0369333\pi$
$967$	6.73270e6i	0.231538i	−0.993276	+	0.115769i	$0.963067\pi$
$968$	856269.i	0.0293712i
$969$	4.09164e6	0.139987
$970$	0	0
$971$	3.65302e6	0.124338	0.0621690	−	0.998066i	$-0.480198\pi$
$971$	3.65302e6	0.124338	0.0621690	+	0.998066i	$0.480198\pi$
$972$	− 1.27703e7i	− 0.433546i
$973$	2.93372e7i	0.993428i
$974$	−1.79062e6	−0.0604793
$975$	0	0
$976$	2.02687e7	0.681085
$977$	1.38794e7i	0.465194i	0.972573	+	0.232597i	$0.0747223\pi$
$977$	1.38794e7i	0.465194i	−0.972573	+	0.232597i	$0.925278\pi$
$978$	2.53776e6i	0.0848405i
$979$	3.85400e7	1.28515
$980$	0	0
$981$	−4.41146e7	−1.46356
$982$	1.38861e6i	0.0459518i
$983$	− 4.67481e7i	− 1.54305i	−0.636198	−	0.771526i	$-0.719494\pi$
$983$	− 4.67481e7i	− 1.54305i	0.636198	−	0.771526i	$-0.280506\pi$
$984$	−76834.0	−0.00252968
$985$	0	0
$986$	−212554.	−0.00696268
$987$	− 6.39822e7i	− 2.09058i
$988$	528287.i	0.0172178i
$989$	−1.75765e7	−0.571403
$990$	0	0
$991$	−1.60784e7	−0.520066	−0.260033	−	0.965600i	$-0.583733\pi$
$991$	−1.60784e7	−0.520066	−0.260033	+	0.965600i	$0.583733\pi$
$992$	− 3.92347e6i	− 0.126587i
$993$	7.39624e7i	2.38034i
$994$	−1.86025e6	−0.0597179
$995$	0	0
$996$	−765756.	−0.0244592
$997$	2.29803e7i	0.732180i	0.930579	+	0.366090i	$0.119304\pi$
$997$	2.29803e7i	0.732180i	−0.930579	+	0.366090i	$0.880696\pi$
$998$	2.59748e6i	0.0825517i
$999$	−6.74690e7	−2.13890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.b.274.2		4
5.2	odd	4		325.6.a.b.1.1		2
5.3	odd	4		13.6.a.a.1.2	✓	2
5.4	even	2	inner	325.6.b.b.274.3		4
15.8	even	4		117.6.a.c.1.1		2
20.3	even	4		208.6.a.h.1.2		2
35.13	even	4		637.6.a.a.1.2		2
40.3	even	4		832.6.a.i.1.1		2
40.13	odd	4		832.6.a.p.1.2		2
65.8	even	4		169.6.b.a.168.2		4
65.18	even	4		169.6.b.a.168.3		4
65.38	odd	4		169.6.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.2	✓	2	5.3	odd	4
117.6.a.c.1.1		2	15.8	even	4
169.6.a.a.1.1		2	65.38	odd	4
169.6.b.a.168.2		4	65.8	even	4
169.6.b.a.168.3		4	65.18	even	4
208.6.a.h.1.2		2	20.3	even	4
325.6.a.b.1.1		2	5.2	odd	4
325.6.b.b.274.2		4	1.1	even	1	trivial
325.6.b.b.274.3		4	5.4	even	2	inner
637.6.a.a.1.2		2	35.13	even	4
832.6.a.i.1.1		2	40.3	even	4
832.6.a.p.1.2		2	40.13	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.438447i q^{2} +26.3693i q^{3} +31.8078 q^{4} +11.5616 q^{6} -162.309i q^{7} -27.9763i q^{8} -452.341 q^{9} +O(q^{10})\)	\(q-0.438447i q^{2} +26.3693i q^{3} +31.8078 q^{4} +11.5616 q^{6} -162.309i q^{7} -27.9763i q^{8} -452.341 q^{9} -361.170 q^{11} +838.749i q^{12} +169.000i q^{13} -71.1638 q^{14} +1005.58 q^{16} -1578.88i q^{17} +198.328i q^{18} +98.2765 q^{19} +4279.97 q^{21} +158.354i q^{22} -1607.16i q^{23} +737.717 q^{24} +74.0976 q^{26} -5520.18i q^{27} -5162.68i q^{28} +307.045 q^{29} +2936.42 q^{31} -1336.14i q^{32} -9523.82i q^{33} -692.255 q^{34} -14388.0 q^{36} -12222.3i q^{37} -43.0891i q^{38} -4456.41 q^{39} -104.151 q^{41} -1876.54i q^{42} -10936.4i q^{43} -11488.0 q^{44} -704.654 q^{46} -14949.2i q^{47} +26516.5i q^{48} -9537.11 q^{49} +41634.0 q^{51} +5375.51i q^{52} +35911.5i q^{53} -2420.31 q^{54} -4540.80 q^{56} +2591.48i q^{57} -134.623i q^{58} +1598.46 q^{59} +20156.2 q^{61} -1287.47i q^{62} +73418.9i q^{63} +31592.8 q^{64} -4175.69 q^{66} -35368.1i q^{67} -50220.6i q^{68} +42379.7 q^{69} +26140.3 q^{71} +12654.8i q^{72} -75468.4i q^{73} -5358.81 q^{74} +3125.96 q^{76} +58621.1i q^{77} +1953.90i q^{78} +7576.72 q^{79} +35644.4 q^{81} +45.6648i q^{82} +912.974i q^{83} +136136. q^{84} -4795.04 q^{86} +8096.58i q^{87} +10104.2i q^{88} -106709. q^{89} +27430.2 q^{91} -51120.1i q^{92} +77431.5i q^{93} -6554.44 q^{94} +35233.0 q^{96} +103676. i q^{97} +4181.52i q^{98} +163372. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9}+O(q^{10}) \) 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9	\( 4 q + 86 q^{4} + 38 q^{6} - 424 q^{9} - 752 q^{11} + 1010 q^{14} + 930 q^{16} + 624 q^{19} + 8148 q^{21} + 2118 q^{24} + 1690 q^{26} + 1624 q^{29} + 15440 q^{31} - 10974 q^{34} - 23396 q^{36} - 9464 q^{39} + 15680 q^{41} - 23308 q^{44} + 37192 q^{46} - 17368 q^{49} + 86696 q^{51} + 2350 q^{54} + 40690 q^{56} + 77872 q^{59} + 3968 q^{61} - 6434 q^{64} - 8572 q^{66} + 70960 q^{69} + 134792 q^{71} - 53010 q^{74} + 11036 q^{76} + 110592 q^{79} + 185524 q^{81} + 267662 q^{84} + 68106 q^{86} - 233016 q^{89} + 12168 q^{91} + 214250 q^{94} + 82986 q^{96} + 319616 q^{99}+O(q^{100}) \) 4 * q + 86 * q^4 + 38 * q^6 - 424 * q^9 - 752 * q^11 + 1010 * q^14 + 930 * q^16 + 624 * q^19 + 8148 * q^21 + 2118 * q^24 + 1690 * q^26 + 1624 * q^29 + 15440 * q^31 - 10974 * q^34 - 23396 * q^36 - 9464 * q^39 + 15680 * q^41 - 23308 * q^44 + 37192 * q^46 - 17368 * q^49 + 86696 * q^51 + 2350 * q^54 + 40690 * q^56 + 77872 * q^59 + 3968 * q^61 - 6434 * q^64 - 8572 * q^66 + 70960 * q^69 + 134792 * q^71 - 53010 * q^74 + 11036 * q^76 + 110592 * q^79 + 185524 * q^81 + 267662 * q^84 + 68106 * q^86 - 233016 * q^89 + 12168 * q^91 + 214250 * q^94 + 82986 * q^96 + 319616 * q^99

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