LMFDB - Embedded newform 325.6.b.c.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 201x^{4} + 10512x^{2} + 65536 $ x^6 + 201x^4 + 10512x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			$-10.6486i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.c.274.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.64858i q^{2} -10.2870i q^{3} -42.7979 q^{4} -88.9676 q^{6} +2.86088i q^{7} +93.3863i q^{8} +137.178 q^{9} +O(q^{10})$	\(q-8.64858i q^{2} -10.2870i q^{3} -42.7979 q^{4} -88.9676 q^{6} +2.86088i q^{7} +93.3863i q^{8} +137.178 q^{9} +571.711 q^{11} +440.260i q^{12} -169.000i q^{13} +24.7426 q^{14} -561.873 q^{16} -855.571i q^{17} -1186.40i q^{18} +2355.90 q^{19} +29.4298 q^{21} -4944.49i q^{22} -2509.66i q^{23} +960.662 q^{24} -1461.61 q^{26} -3910.88i q^{27} -122.440i q^{28} +5497.50 q^{29} -144.924 q^{31} +7847.77i q^{32} -5881.17i q^{33} -7399.47 q^{34} -5870.95 q^{36} -515.975i q^{37} -20375.2i q^{38} -1738.50 q^{39} -13928.9 q^{41} -254.526i q^{42} -7753.58i q^{43} -24468.0 q^{44} -21705.0 q^{46} +8344.77i q^{47} +5779.97i q^{48} +16798.8 q^{49} -8801.22 q^{51} +7232.84i q^{52} +5976.21i q^{53} -33823.6 q^{54} -267.168 q^{56} -24235.1i q^{57} -47545.6i q^{58} -2110.84 q^{59} -17394.2 q^{61} +1253.38i q^{62} +392.452i q^{63} +49892.1 q^{64} -50863.8 q^{66} -3121.05i q^{67} +36616.6i q^{68} -25816.8 q^{69} +43323.3 q^{71} +12810.6i q^{72} -6808.31i q^{73} -4462.45 q^{74} -100828. q^{76} +1635.60i q^{77} +15035.5i q^{78} +1280.80 q^{79} -6896.72 q^{81} +120465. i q^{82} -7148.44i q^{83} -1259.53 q^{84} -67057.5 q^{86} -56552.6i q^{87} +53390.0i q^{88} -107123. q^{89} +483.489 q^{91} +107408. i q^{92} +1490.82i q^{93} +72170.4 q^{94} +80729.7 q^{96} +42456.2i q^{97} -145286. i q^{98} +78426.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 242 q^{4} - 398 q^{6} + 382 q^{9}+O(q^{10}) $ 6 * q - 242 * q^4 - 398 * q^6 + 382 * q^9	$ 6 q - 242 q^{4} - 398 q^{6} + 382 q^{9} + 1112 q^{11} + 1586 q^{14} + 5570 q^{16} - 296 q^{19} + 2780 q^{21} + 26754 q^{24} + 2366 q^{26} + 17516 q^{29} - 5216 q^{31} - 31978 q^{34} - 9356 q^{36} - 2704 q^{39} - 21996 q^{41} - 99580 q^{44} - 6816 q^{46} + 89142 q^{49} + 26936 q^{51} - 111190 q^{54} - 98574 q^{56} - 127896 q^{59} - 25508 q^{61} - 234786 q^{64} - 224708 q^{66} + 40224 q^{69} + 155160 q^{71} + 263038 q^{74} - 199276 q^{76} + 123744 q^{79} - 216442 q^{81} - 209446 q^{84} - 139794 q^{86} - 67388 q^{89} - 20280 q^{91} - 145686 q^{94} - 487282 q^{96} + 120664 q^{99}+O(q^{100}) $ 6 * q - 242 * q^4 - 398 * q^6 + 382 * q^9 + 1112 * q^11 + 1586 * q^14 + 5570 * q^16 - 296 * q^19 + 2780 * q^21 + 26754 * q^24 + 2366 * q^26 + 17516 * q^29 - 5216 * q^31 - 31978 * q^34 - 9356 * q^36 - 2704 * q^39 - 21996 * q^41 - 99580 * q^44 - 6816 * q^46 + 89142 * q^49 + 26936 * q^51 - 111190 * q^54 - 98574 * q^56 - 127896 * q^59 - 25508 * q^61 - 234786 * q^64 - 224708 * q^66 + 40224 * q^69 + 155160 * q^71 + 263038 * q^74 - 199276 * q^76 + 123744 * q^79 - 216442 * q^81 - 209446 * q^84 - 139794 * q^86 - 67388 * q^89 - 20280 * q^91 - 145686 * q^94 - 487282 * q^96 + 120664 * 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 $

$2$	− 8.64858i	− 1.52887i	−0.644703	−	0.764433i	$-0.723019\pi$
$2$	− 8.64858i	− 1.52887i	0.644703	−	0.764433i	$-0.276981\pi$
$3$	− 10.2870i	− 0.659909i	−0.943997	−	0.329954i	$-0.892967\pi$
$3$	− 10.2870i	− 0.659909i	0.943997	−	0.329954i	$-0.107033\pi$
$4$	−42.7979	−1.33743
$5$	0	0
$5$	0	0
$6$	−88.9676	−1.00891
$7$	2.86088i	0.0220676i	0.999939	+	0.0110338i	$0.00351224\pi$
$7$	2.86088i	0.0220676i	−0.999939	+	0.0110338i	$0.996488\pi$
$8$	93.3863i	0.515892i
$9$	137.178	0.564520
$10$	0	0
$11$	571.711	1.42461	0.712304	−	0.701871i	$-0.247652\pi$
$11$	571.711	1.42461	0.712304	+	0.701871i	$0.247652\pi$
$12$	440.260i	0.882585i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	24.7426	0.0337384
$15$	0	0
$16$	−561.873	−0.548704
$17$	− 855.571i	− 0.718015i	−0.933335	−	0.359008i	$-0.883115\pi$
$17$	− 855.571i	− 0.718015i	0.933335	−	0.359008i	$-0.116885\pi$
$18$	− 1186.40i	− 0.863076i
$19$	2355.90	1.49718	0.748589	−	0.663034i	$-0.230732\pi$
$19$	2355.90	1.49718	0.748589	+	0.663034i	$0.230732\pi$
$20$	0	0
$21$	29.4298	0.0145626
$22$	− 4944.49i	− 2.17804i
$23$	− 2509.66i	− 0.989227i	−0.869113	−	0.494613i	$-0.835310\pi$
$23$	− 2509.66i	− 0.989227i	0.869113	−	0.494613i	$-0.164690\pi$
$24$	960.662	0.340441
$25$	0	0
$26$	−1461.61	−0.424031
$27$	− 3910.88i	− 1.03244i
$28$	− 122.440i	− 0.0295140i
$29$	5497.50	1.21387	0.606933	−	0.794753i	$-0.292400\pi$
$29$	5497.50	1.21387	0.606933	+	0.794753i	$0.292400\pi$
$30$	0	0
$31$	−144.924	−0.0270854	−0.0135427	−	0.999908i	$-0.504311\pi$
$31$	−144.924	−0.0270854	−0.0135427	+	0.999908i	$0.504311\pi$
$32$	7847.77i	1.35479i
$33$	− 5881.17i	− 0.940111i
$34$	−7399.47	−1.09775
$35$	0	0
$36$	−5870.95	−0.755009
$37$	− 515.975i	− 0.0619618i	−0.999520	−	0.0309809i	$-0.990137\pi$
$37$	− 515.975i	− 0.0619618i	0.999520	−	0.0309809i	$-0.00986311\pi$
$38$	− 20375.2i	− 2.28899i
$39$	−1738.50	−0.183026
$40$	0	0
$41$	−13928.9	−1.29407	−0.647033	−	0.762462i	$-0.723991\pi$
$41$	−13928.9	−1.29407	−0.647033	+	0.762462i	$0.723991\pi$
$42$	− 254.526i	− 0.0222643i
$43$	− 7753.58i	− 0.639486i	−0.947504	−	0.319743i	$-0.896403\pi$
$43$	− 7753.58i	− 0.639486i	0.947504	−	0.319743i	$-0.103597\pi$
$44$	−24468.0	−1.90532
$45$	0	0
$46$	−21705.0	−1.51240
$47$	8344.77i	0.551023i	0.961298	+	0.275512i	$0.0888472\pi$
$47$	8344.77i	0.551023i	−0.961298	+	0.275512i	$0.911153\pi$
$48$	5779.97i	0.362095i
$49$	16798.8	0.999513
$50$	0	0
$51$	−8801.22	−0.473825
$52$	7232.84i	0.370937i
$53$	5976.21i	0.292238i	0.989267	+	0.146119i	$0.0466782\pi$
$53$	5976.21i	0.292238i	−0.989267	+	0.146119i	$0.953322\pi$
$54$	−33823.6	−1.57846
$55$	0	0
$56$	−267.168	−0.0113845
$57$	− 24235.1i	− 0.988001i
$58$	− 47545.6i	− 1.85584i
$59$	−2110.84	−0.0789451	−0.0394726	−	0.999221i	$-0.512568\pi$
$59$	−2110.84	−0.0789451	−0.0394726	+	0.999221i	$0.512568\pi$
$60$	0	0
$61$	−17394.2	−0.598522	−0.299261	−	0.954171i	$-0.596740\pi$
$61$	−17394.2	−0.598522	−0.299261	+	0.954171i	$0.596740\pi$
$62$	1253.38i	0.0414099i
$63$	392.452i	0.0124576i
$64$	49892.1	1.52259
$65$	0	0
$66$	−50863.8	−1.43730
$67$	− 3121.05i	− 0.0849402i	−0.999098	−	0.0424701i	$-0.986477\pi$
$67$	− 3121.05i	− 0.0849402i	0.999098	−	0.0424701i	$-0.0135227\pi$
$68$	36616.6i	0.960298i
$69$	−25816.8	−0.652800
$70$	0	0
$71$	43323.3	1.01994	0.509971	−	0.860191i	$-0.329656\pi$
$71$	43323.3	1.01994	0.509971	+	0.860191i	$0.329656\pi$
$72$	12810.6i	0.291231i
$73$	− 6808.31i	− 0.149531i	−0.997201	−	0.0747657i	$-0.976179\pi$
$73$	− 6808.31i	− 0.149531i	0.997201	−	0.0747657i	$-0.0238209\pi$
$74$	−4462.45	−0.0947314
$75$	0	0
$76$	−100828.	−2.00238
$77$	1635.60i	0.0314377i
$78$	15035.5i	0.279822i
$79$	1280.80	0.0230895	0.0115447	−	0.999933i	$-0.496325\pi$
$79$	1280.80	0.0230895	0.0115447	+	0.999933i	$0.496325\pi$
$80$	0	0
$81$	−6896.72	−0.116797
$82$	120465.i	1.97846i
$83$	− 7148.44i	− 0.113898i	−0.998377	−	0.0569490i	$-0.981863\pi$
$83$	− 7148.44i	− 0.113898i	0.998377	−	0.0569490i	$-0.0181372\pi$
$84$	−1259.53	−0.0194765
$85$	0	0
$86$	−67057.5	−0.977690
$87$	− 56552.6i	− 0.801041i
$88$	53390.0i	0.734943i
$89$	−107123.	−1.43353	−0.716765	−	0.697315i	$-0.754378\pi$
$89$	−107123.	−1.43353	−0.716765	+	0.697315i	$0.754378\pi$
$90$	0	0
$91$	483.489	0.00612045
$92$	107408.i	1.32303i
$93$	1490.82i	0.0178739i
$94$	72170.4	0.842441
$95$	0	0
$96$	80729.7	0.894036
$97$	42456.2i	0.458155i	0.973408	+	0.229077i	$0.0735709\pi$
$97$	42456.2i	0.458155i	−0.973408	+	0.229077i	$0.926429\pi$
$98$	− 145286.i	− 1.52812i
$99$	78426.5	0.804220
$100$	0	0
$101$	−87947.3	−0.857866	−0.428933	−	0.903336i	$-0.641110\pi$
$101$	−87947.3	−0.857866	−0.428933	+	0.903336i	$0.641110\pi$
$102$	76118.1i	0.724415i
$103$	− 96885.8i	− 0.899844i	−0.893068	−	0.449922i	$-0.851452\pi$
$103$	− 96885.8i	− 0.899844i	0.893068	−	0.449922i	$-0.148548\pi$
$104$	15782.3	0.143083
$105$	0	0
$106$	51685.7	0.446792
$107$	106509.i	0.899346i	0.893193	+	0.449673i	$0.148460\pi$
$107$	106509.i	0.899346i	−0.893193	+	0.449673i	$0.851540\pi$
$108$	167377.i	1.38082i
$109$	142520.	1.14897	0.574485	−	0.818515i	$-0.305202\pi$
$109$	142520.	1.14897	0.574485	+	0.818515i	$0.305202\pi$
$110$	0	0
$111$	−5307.81	−0.0408892
$112$	− 1607.45i	− 0.0121086i
$113$	209540.i	1.54373i	0.635787	+	0.771864i	$0.280676\pi$
$113$	209540.i	1.54373i	−0.635787	+	0.771864i	$0.719324\pi$
$114$	−209599.	−1.51052
$115$	0	0
$116$	−235282.	−1.62346
$117$	− 23183.2i	− 0.156570i
$118$	18255.8i	0.120697i
$119$	2447.69	0.0158449
$120$	0	0
$121$	165803.	1.02951
$122$	150435.i	0.915061i
$123$	143286.i	0.853966i
$124$	6202.42	0.0362249
$125$	0	0
$126$	3394.15	0.0190460
$127$	− 54468.2i	− 0.299663i	−0.988712	−	0.149832i	$-0.952127\pi$
$127$	− 54468.2i	− 0.299663i	0.988712	−	0.149832i	$-0.0478732\pi$
$128$	− 180367.i	− 0.973043i
$129$	−79760.8	−0.422003
$130$	0	0
$131$	258252.	1.31482	0.657408	−	0.753535i	$-0.271653\pi$
$131$	258252.	1.31482	0.657408	+	0.753535i	$0.271653\pi$
$132$	251702.i	1.25734i
$133$	6739.97i	0.0330391i
$134$	−26992.6	−0.129862
$135$	0	0
$136$	79898.6	0.370418
$137$	293754.i	1.33716i	0.743641	+	0.668580i	$0.233097\pi$
$137$	293754.i	1.33716i	−0.743641	+	0.668580i	$0.766903\pi$
$138$	223279.i	0.998044i
$139$	−180400.	−0.791955	−0.395977	−	0.918260i	$-0.629594\pi$
$139$	−180400.	−0.791955	−0.395977	+	0.918260i	$0.629594\pi$
$140$	0	0
$141$	85842.4	0.363625
$142$	− 374685.i	− 1.55936i
$143$	− 96619.2i	− 0.395115i
$144$	−77076.9	−0.309755
$145$	0	0
$146$	−58882.2	−0.228614
$147$	− 172809.i	− 0.659588i
$148$	22082.6i	0.0828698i
$149$	−516009.	−1.90411	−0.952054	−	0.305930i	$-0.901032\pi$
$149$	−516009.	−1.90411	−0.952054	+	0.305930i	$0.901032\pi$
$150$	0	0
$151$	−333770.	−1.19126	−0.595628	−	0.803260i	$-0.703097\pi$
$151$	−333770.	−1.19126	−0.595628	+	0.803260i	$0.703097\pi$
$152$	220009.i	0.772381i
$153$	− 117366.i	− 0.405334i
$154$	14145.6	0.0480640
$155$	0	0
$156$	74404.0	0.244785
$157$	− 265860.i	− 0.860804i	−0.902637	−	0.430402i	$-0.858372\pi$
$157$	− 265860.i	− 0.860804i	0.902637	−	0.430402i	$-0.141628\pi$
$158$	− 11077.1i	− 0.0353007i
$159$	61477.0	0.192850
$160$	0	0
$161$	7179.86	0.0218299
$162$	59646.8i	0.178566i
$163$	231417.i	0.682223i	0.940023	+	0.341112i	$0.110803\pi$
$163$	231417.i	0.682223i	−0.940023	+	0.341112i	$0.889197\pi$
$164$	596127.	1.73073
$165$	0	0
$166$	−61823.8	−0.174135
$167$	− 656226.i	− 1.82080i	−0.413730	−	0.910400i	$-0.635774\pi$
$167$	− 656226.i	− 1.82080i	0.413730	−	0.910400i	$-0.364226\pi$
$168$	2748.34i	0.00751273i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	323179.	0.845187
$172$	331837.i	0.855271i
$173$	− 45111.6i	− 0.114597i	−0.998357	−	0.0572985i	$-0.981751\pi$
$173$	− 45111.6i	− 0.114597i	0.998357	−	0.0572985i	$-0.0182487\pi$
$174$	−489100.	−1.22468
$175$	0	0
$176$	−321229.	−0.781688
$177$	21714.1i	0.0520966i
$178$	926460.i	2.19168i
$179$	−521846.	−1.21733	−0.608667	−	0.793426i	$-0.708295\pi$
$179$	−521846.	−1.21733	−0.608667	+	0.793426i	$0.708295\pi$
$180$	0	0
$181$	−122780.	−0.278569	−0.139284	−	0.990252i	$-0.544480\pi$
$181$	−122780.	−0.278569	−0.139284	+	0.990252i	$0.544480\pi$
$182$	− 4181.50i	− 0.00935736i
$183$	178934.i	0.394970i
$184$	234368.	0.510334
$185$	0	0
$186$	12893.5	0.0273268
$187$	− 489140.i	− 1.02289i
$188$	− 357139.i	− 0.736957i
$189$	11188.6	0.0227835
$190$	0	0
$191$	350208.	0.694613	0.347306	−	0.937752i	$-0.387096\pi$
$191$	350208.	0.694613	0.347306	+	0.937752i	$0.387096\pi$
$192$	− 513238.i	− 1.00477i
$193$	− 876316.i	− 1.69343i	−0.532046	−	0.846716i	$-0.678577\pi$
$193$	− 876316.i	− 1.69343i	0.532046	−	0.846716i	$-0.321423\pi$
$194$	367186.	0.700457
$195$	0	0
$196$	−718954.	−1.33678
$197$	306582.i	0.562835i	0.959585	+	0.281418i	$0.0908046\pi$
$197$	306582.i	0.562835i	−0.959585	+	0.281418i	$0.909195\pi$
$198$	− 678277.i	− 1.22954i
$199$	327849.	0.586869	0.293434	−	0.955979i	$-0.405202\pi$
$199$	327849.	0.586869	0.293434	+	0.955979i	$0.405202\pi$
$200$	0	0
$201$	−32106.1	−0.0560528
$202$	760619.i	1.31156i
$203$	15727.7i	0.0267871i
$204$	376674.	0.633709
$205$	0	0
$206$	−837924.	−1.37574
$207$	− 344272.i	− 0.558439i
$208$	94956.6i	0.152183i
$209$	1.34690e6	2.13289
$210$	0	0
$211$	−509483.	−0.787814	−0.393907	−	0.919150i	$-0.628877\pi$
$211$	−509483.	−0.787814	−0.393907	+	0.919150i	$0.628877\pi$
$212$	− 255769.i	− 0.390849i
$213$	− 445665.i	− 0.673069i
$214$	921151.	1.37498
$215$	0	0
$216$	365223.	0.532627
$217$	− 414.610i	0 0.000597709i
$218$	− 1.23259e6i	− 1.75662i
$219$	−70036.8	−0.0986771
$220$	0	0
$221$	−144591.	−0.199142
$222$	45905.0i	0.0625141i
$223$	15061.4i	0.0202816i	0.999949	+	0.0101408i	$0.00322798\pi$
$223$	15061.4i	0.0202816i	−0.999949	+	0.0101408i	$0.996772\pi$
$224$	−22451.6	−0.0298969
$225$	0	0
$226$	1.81222e6	2.36016
$227$	273713.i	0.352558i	0.984340	+	0.176279i	$0.0564061\pi$
$227$	273713.i	0.352558i	−0.984340	+	0.176279i	$0.943594\pi$
$228$	1.03721e6i	1.32139i
$229$	−743056.	−0.936339	−0.468169	−	0.883639i	$-0.655086\pi$
$229$	−743056.	−0.936339	−0.468169	+	0.883639i	$0.655086\pi$
$230$	0	0
$231$	16825.4	0.0207460
$232$	513392.i	0.626223i
$233$	− 1.12122e6i	− 1.35301i	−0.736437	−	0.676506i	$-0.763493\pi$
$233$	− 1.12122e6i	− 1.35301i	0.736437	−	0.676506i	$-0.236507\pi$
$234$	−200501.	−0.239374
$235$	0	0
$236$	90339.5	0.105584
$237$	− 13175.6i	− 0.0152369i
$238$	− 21169.0i	− 0.0242247i
$239$	−279838.	−0.316892	−0.158446	−	0.987368i	$-0.550648\pi$
$239$	−279838.	−0.316892	−0.158446	+	0.987368i	$0.550648\pi$
$240$	0	0
$241$	816889.	0.905983	0.452992	−	0.891515i	$-0.350357\pi$
$241$	816889.	0.905983	0.452992	+	0.891515i	$0.350357\pi$
$242$	− 1.43396e6i	− 1.57398i
$243$	− 879398.i	− 0.955366i
$244$	744436.	0.800484
$245$	0	0
$246$	1.23922e6	1.30560
$247$	− 398148.i	− 0.415242i
$248$	− 13533.9i	− 0.0139731i
$249$	−73535.7	−0.0751623
$250$	0	0
$251$	−766697.	−0.768139	−0.384069	−	0.923304i	$-0.625478\pi$
$251$	−766697.	−0.768139	−0.384069	+	0.923304i	$0.625478\pi$
$252$	− 16796.1i	− 0.0166612i
$253$	− 1.43480e6i	− 1.40926i
$254$	−471072.	−0.458145
$255$	0	0
$256$	36629.4	0.0349325
$257$	− 1.82413e6i	− 1.72276i	−0.507963	−	0.861379i	$-0.669601\pi$
$257$	− 1.82413e6i	− 1.72276i	0.507963	−	0.861379i	$-0.330399\pi$
$258$	689817.i	0.645186i
$259$	1476.14	0.00136735
$260$	0	0
$261$	754139.	0.685252
$262$	− 2.23351e6i	− 2.01018i
$263$	294257.i	0.262323i	0.991361	+	0.131162i	$0.0418707\pi$
$263$	294257.i	0.262323i	−0.991361	+	0.131162i	$0.958129\pi$
$264$	549221.	0.484995
$265$	0	0
$266$	58291.1	0.0505124
$267$	1.10197e6i	0.945999i
$268$	133574.i	0.113602i
$269$	−289738.	−0.244132	−0.122066	−	0.992522i	$-0.538952\pi$
$269$	−289738.	−0.244132	−0.122066	+	0.992522i	$0.538952\pi$
$270$	0	0
$271$	1.63179e6	1.34971	0.674855	−	0.737950i	$-0.264206\pi$
$271$	1.63179e6	1.34971	0.674855	+	0.737950i	$0.264206\pi$
$272$	480722.i	0.393978i
$273$	− 4973.64i	− 0.00403894i
$274$	2.54056e6	2.04434
$275$	0	0
$276$	1.10490e6	0.873076
$277$	398868.i	0.312341i	0.987730	+	0.156171i	$0.0499150\pi$
$277$	398868.i	0.312341i	−0.987730	+	0.156171i	$0.950085\pi$
$278$	1.56021e6i	1.21079i
$279$	−19880.4	−0.0152902
$280$	0	0
$281$	274611.	0.207468	0.103734	−	0.994605i	$-0.466921\pi$
$281$	274611.	0.207468	0.103734	+	0.994605i	$0.466921\pi$
$282$	− 742414.i	− 0.555934i
$283$	794683.i	0.589831i	0.955523	+	0.294916i	$0.0952916\pi$
$283$	794683.i	0.589831i	−0.955523	+	0.294916i	$0.904708\pi$
$284$	−1.85415e6	−1.36411
$285$	0	0
$286$	−835619.	−0.604078
$287$	− 39848.9i	− 0.0285570i
$288$	1.07654e6i	0.764805i
$289$	687856.	0.484454
$290$	0	0
$291$	436746.	0.302340
$292$	291381.i	0.199988i
$293$	1.55323e6i	1.05698i	0.848940	+	0.528490i	$0.177241\pi$
$293$	1.55323e6i	1.05698i	−0.848940	+	0.528490i	$0.822759\pi$
$294$	−1.49455e6	−1.00842
$295$	0	0
$296$	48185.0	0.0319656
$297$	− 2.23590e6i	− 1.47082i
$298$	4.46274e6i	2.91113i
$299$	−424133.	−0.274362
$300$	0	0
$301$	22182.1	0.0141119
$302$	2.88664e6i	1.82127i
$303$	904711.i	0.566113i
$304$	−1.32372e6	−0.821508
$305$	0	0
$306$	−1.01505e6	−0.619702
$307$	2.07136e6i	1.25433i	0.778888	+	0.627163i	$0.215784\pi$
$307$	2.07136e6i	1.25433i	−0.778888	+	0.627163i	$0.784216\pi$
$308$	− 70000.2i	− 0.0420458i
$309$	−996660.	−0.593815
$310$	0	0
$311$	−2.50827e6	−1.47053	−0.735263	−	0.677782i	$-0.762941\pi$
$311$	−2.50827e6	−1.47053	−0.735263	+	0.677782i	$0.762941\pi$
$312$	− 162352.i	− 0.0944215i
$313$	− 832384.i	− 0.480245i	−0.970743	−	0.240123i	$-0.922812\pi$
$313$	− 832384.i	− 0.480245i	0.970743	−	0.240123i	$-0.0771876\pi$
$314$	−2.29931e6	−1.31606
$315$	0	0
$316$	−54815.6	−0.0308806
$317$	− 1.76537e6i	− 0.986708i	−0.869829	−	0.493354i	$-0.835771\pi$
$317$	− 1.76537e6i	− 0.986708i	0.869829	−	0.493354i	$-0.164229\pi$
$318$	− 531689.i	− 0.294842i
$319$	3.14299e6	1.72928
$320$	0	0
$321$	1.09565e6	0.593487
$322$	− 62095.5i	− 0.0333750i
$323$	− 2.01564e6i	− 1.07500i
$324$	295165.	0.156208
$325$	0	0
$326$	2.00143e6	1.04303
$327$	− 1.46609e6i	− 0.758215i
$328$	− 1.30077e6i	− 0.667598i
$329$	−23873.4	−0.0121598
$330$	0	0
$331$	−285068.	−0.143014	−0.0715070	−	0.997440i	$-0.522781\pi$
$331$	−285068.	−0.143014	−0.0715070	+	0.997440i	$0.522781\pi$
$332$	305938.i	0.152331i
$333$	− 70780.6i	− 0.0349787i
$334$	−5.67542e6	−2.78376
$335$	0	0
$336$	−16535.8	−0.00799057
$337$	1.96932e6i	0.944584i	0.881442	+	0.472292i	$0.156573\pi$
$337$	1.96932e6i	0.944584i	−0.881442	+	0.472292i	$0.843427\pi$
$338$	247012.i	0.117605i
$339$	2.15553e6	1.01872
$340$	0	0
$341$	−82854.5	−0.0385860
$342$	− 2.79504e6i	− 1.29218i
$343$	96142.4i	0.0441245i
$344$	724079.	0.329906
$345$	0	0
$346$	−390151.	−0.175204
$347$	1.89753e6i	0.845988i	0.906133	+	0.422994i	$0.139021\pi$
$347$	1.89753e6i	0.845988i	−0.906133	+	0.422994i	$0.860979\pi$
$348$	2.42033e6i	1.07134i
$349$	365808.	0.160764	0.0803822	−	0.996764i	$-0.474386\pi$
$349$	365808.	0.160764	0.0803822	+	0.996764i	$0.474386\pi$
$350$	0	0
$351$	−660939.	−0.286348
$352$	4.48666e6i	1.93004i
$353$	− 1.56672e6i	− 0.669196i	−0.942361	−	0.334598i	$-0.891399\pi$
$353$	− 1.56672e6i	− 0.669196i	0.942361	−	0.334598i	$-0.108601\pi$
$354$	187796.	0.0796487
$355$	0	0
$356$	4.58463e6	1.91725
$357$	− 25179.3i	− 0.0104562i
$358$	4.51322e6i	1.86114i
$359$	4.03307e6	1.65158	0.825789	−	0.563979i	$-0.190730\pi$
$359$	4.03307e6	1.65158	0.825789	+	0.563979i	$0.190730\pi$
$360$	0	0
$361$	3.07418e6	1.24154
$362$	1.06187e6i	0.425894i
$363$	− 1.70561e6i	− 0.679380i
$364$	−20692.3	−0.00818570
$365$	0	0
$366$	1.54752e6	0.603857
$367$	683844.i	0.265028i	0.991181	+	0.132514i	$0.0423050\pi$
$367$	683844.i	0.265028i	−0.991181	+	0.132514i	$0.957695\pi$
$368$	1.41011e6i	0.542793i
$369$	−1.91074e6	−0.730527
$370$	0	0
$371$	−17097.2	−0.00644899
$372$	− 63804.1i	− 0.0239051i
$373$	3.09158e6i	1.15056i	0.817957	+	0.575279i	$0.195107\pi$
$373$	3.09158e6i	1.15056i	−0.817957	+	0.575279i	$0.804893\pi$
$374$	−4.23036e6	−1.56386
$375$	0	0
$376$	−779288.	−0.284268
$377$	− 929078.i	− 0.336666i
$378$	− 96765.3i	− 0.0348329i
$379$	5.24600e6	1.87599	0.937995	−	0.346649i	$-0.112681\pi$
$379$	5.24600e6	1.87599	0.937995	+	0.346649i	$0.112681\pi$
$380$	0	0
$381$	−560312.	−0.197751
$382$	− 3.02880e6i	− 1.06197i
$383$	− 1.57009e6i	− 0.546926i	−0.961883	−	0.273463i	$-0.911831\pi$
$383$	− 1.57009e6i	− 0.546926i	0.961883	−	0.273463i	$-0.0881691\pi$
$384$	−1.85543e6	−0.642120
$385$	0	0
$386$	−7.57889e6	−2.58903
$387$	− 1.06362e6i	− 0.361003i
$388$	− 1.81704e6i	− 0.612751i
$389$	−1.54482e6	−0.517611	−0.258805	−	0.965929i	$-0.583329\pi$
$389$	−1.54482e6	−0.517611	−0.258805	+	0.965929i	$0.583329\pi$
$390$	0	0
$391$	−2.14719e6	−0.710280
$392$	1.56878e6i	0.515640i
$393$	− 2.65663e6i	− 0.867659i
$394$	2.65150e6	0.860500
$395$	0	0
$396$	−3.35649e6	−1.07559
$397$	5.06711e6i	1.61356i	0.590855	+	0.806778i	$0.298790\pi$
$397$	5.06711e6i	1.61356i	−0.590855	+	0.806778i	$0.701210\pi$
$398$	− 2.83543e6i	− 0.897244i
$399$	69333.8	0.0218028
$400$	0	0
$401$	−138120.	−0.0428939	−0.0214469	−	0.999770i	$-0.506827\pi$
$401$	−138120.	−0.0428939	−0.0214469	+	0.999770i	$0.506827\pi$
$402$	277672.i	0.0856973i
$403$	24492.1i	0.00751213i
$404$	3.76396e6	1.14734
$405$	0	0
$406$	136022.	0.0409539
$407$	− 294989.i	− 0.0882713i
$408$	− 821914.i	− 0.244442i
$409$	4.13317e6	1.22173	0.610865	−	0.791735i	$-0.290822\pi$
$409$	4.13317e6	1.22173	0.610865	+	0.791735i	$0.290822\pi$
$410$	0	0
$411$	3.02184e6	0.882403
$412$	4.14651e6i	1.20348i
$413$	− 6038.87i	− 0.00174213i
$414$	−2.97746e6	−0.853778
$415$	0	0
$416$	1.32627e6	0.375750
$417$	1.85577e6i	0.522618i
$418$	− 1.16487e7i	− 3.26091i
$419$	2.96703e6	0.825633	0.412817	−	0.910814i	$-0.364545\pi$
$419$	2.96703e6	0.825633	0.412817	+	0.910814i	$0.364545\pi$
$420$	0	0
$421$	−2.98125e6	−0.819773	−0.409887	−	0.912137i	$-0.634432\pi$
$421$	−2.98125e6	−0.819773	−0.409887	+	0.912137i	$0.634432\pi$
$422$	4.40630e6i	1.20446i
$423$	1.14472e6i	0.311064i
$424$	−558096.	−0.150763
$425$	0	0
$426$	−3.85437e6	−1.02903
$427$	− 49762.8i	− 0.0132080i
$428$	− 4.55836e6i	− 1.20282i
$429$	−993918.	−0.260740
$430$	0	0
$431$	−3.34790e6	−0.868119	−0.434059	−	0.900884i	$-0.642919\pi$
$431$	−3.34790e6	−0.868119	−0.434059	+	0.900884i	$0.642919\pi$
$432$	2.19742e6i	0.566505i
$433$	3.76782e6i	0.965764i	0.875686	+	0.482882i	$0.160410\pi$
$433$	3.76782e6i	0.965764i	−0.875686	+	0.482882i	$0.839590\pi$
$434$	−3585.78	−0.000913818 0
$435$	0	0
$436$	−6.09954e6	−1.53667
$437$	− 5.91252e6i	− 1.48105i
$438$	605719.i	0.150864i
$439$	−5.63075e6	−1.39446	−0.697228	−	0.716849i	$-0.745584\pi$
$439$	−5.63075e6	−1.39446	−0.697228	+	0.716849i	$0.745584\pi$
$440$	0	0
$441$	2.30444e6	0.564245
$442$	1.25051e6i	0.304461i
$443$	− 3.69803e6i	− 0.895284i	−0.894213	−	0.447642i	$-0.852264\pi$
$443$	− 3.69803e6i	− 0.895284i	0.894213	−	0.447642i	$-0.147736\pi$
$444$	227163.	0.0546865
$445$	0	0
$446$	130260.	0.0310079
$447$	5.30816e6i	1.25654i
$448$	142735.i	0.0335998i
$449$	505514.	0.118336	0.0591681	−	0.998248i	$-0.481155\pi$
$449$	505514.	0.118336	0.0591681	+	0.998248i	$0.481155\pi$
$450$	0	0
$451$	−7.96330e6	−1.84354
$452$	− 8.96787e6i	− 2.06464i
$453$	3.43348e6i	0.786121i
$454$	2.36723e6	0.539015
$455$	0	0
$456$	2.26323e6	0.509701
$457$	783627.i	0.175517i	0.996142	+	0.0877584i	$0.0279704\pi$
$457$	783627.i	0.175517i	−0.996142	+	0.0877584i	$0.972030\pi$
$458$	6.42638e6i	1.43154i
$459$	−3.34603e6	−0.741308
$460$	0	0
$461$	4.64856e6	1.01875	0.509373	−	0.860546i	$-0.329877\pi$
$461$	4.64856e6	1.01875	0.509373	+	0.860546i	$0.329877\pi$
$462$	− 145515.i	− 0.0317179i
$463$	− 2.64872e6i	− 0.574226i	−0.957897	−	0.287113i	$-0.907304\pi$
$463$	− 2.64872e6i	− 0.574226i	0.957897	−	0.287113i	$-0.0926956\pi$
$464$	−3.08890e6	−0.666053
$465$	0	0
$466$	−9.69698e6	−2.06858
$467$	− 2.42172e6i	− 0.513845i	−0.966432	−	0.256923i	$-0.917291\pi$
$467$	− 2.42172e6i	− 0.513845i	0.966432	−	0.256923i	$-0.0827085\pi$
$468$	992190.i	0.209402i
$469$	8928.96	0.00187443
$470$	0	0
$471$	−2.73490e6	−0.568052
$472$	− 197124.i	− 0.0407271i
$473$	− 4.43281e6i	− 0.911017i
$474$	−113950.	−0.0232953
$475$	0	0
$476$	−104756.	−0.0211915
$477$	819807.i	0.164974i
$478$	2.42020e6i	0.484486i
$479$	−5.58315e6	−1.11184	−0.555918	−	0.831237i	$-0.687633\pi$
$479$	−5.58315e6	−1.11184	−0.555918	+	0.831237i	$0.687633\pi$
$480$	0	0
$481$	−87199.7	−0.0171851
$482$	− 7.06492e6i	− 1.38513i
$483$	− 73858.9i	− 0.0144057i
$484$	−7.09602e6	−1.37690
$485$	0	0
$486$	−7.60554e6	−1.46063
$487$	7.25183e6i	1.38556i	0.721149	+	0.692780i	$0.243614\pi$
$487$	7.25183e6i	1.38556i	−0.721149	+	0.692780i	$0.756386\pi$
$488$	− 1.62438e6i	− 0.308773i
$489$	2.38058e6	0.450205
$490$	0	0
$491$	4.17410e6	0.781373	0.390687	−	0.920524i	$-0.372238\pi$
$491$	4.17410e6	0.781373	0.390687	+	0.920524i	$0.372238\pi$
$492$	− 6.13233e6i	− 1.14212i
$493$	− 4.70350e6i	− 0.871574i
$494$	−3.44341e6	−0.634850
$495$	0	0
$496$	81428.7	0.0148619
$497$	123943.i	0.0225077i
$498$	635979.i	0.114913i
$499$	7.83551e6	1.40869	0.704345	−	0.709857i	$-0.251241\pi$
$499$	7.83551e6	1.40869	0.704345	+	0.709857i	$0.251241\pi$
$500$	0	0
$501$	−6.75057e6	−1.20156
$502$	6.63084e6i	1.17438i
$503$	3.72420e6i	0.656315i	0.944623	+	0.328158i	$0.106428\pi$
$503$	3.72420e6i	0.656315i	−0.944623	+	0.328158i	$0.893572\pi$
$504$	−36649.6	−0.00642678
$505$	0	0
$506$	−1.24090e7	−2.15457
$507$	293806.i	0.0507622i
$508$	2.33112e6i	0.400780i
$509$	6.76675e6	1.15767	0.578836	−	0.815444i	$-0.303507\pi$
$509$	6.76675e6	1.15767	0.578836	+	0.815444i	$0.303507\pi$
$510$	0	0
$511$	19477.8	0.00329980
$512$	− 6.08853e6i	− 1.02645i
$513$	− 9.21366e6i	− 1.54575i
$514$	−1.57762e7	−2.63387
$515$	0	0
$516$	3.41359e6	0.564401
$517$	4.77080e6i	0.784992i
$518$	− 12766.5i	− 0.00209049i
$519$	−464062.	−0.0756236
$520$	0	0
$521$	9.08380e6	1.46613	0.733066	−	0.680157i	$-0.238088\pi$
$521$	9.08380e6	1.46613	0.733066	+	0.680157i	$0.238088\pi$
$522$	− 6.52223e6i	− 1.04766i
$523$	1.09284e7i	1.74703i	0.486796	+	0.873515i	$0.338165\pi$
$523$	1.09284e7i	1.74703i	−0.486796	+	0.873515i	$0.661835\pi$
$524$	−1.10526e7	−1.75848
$525$	0	0
$526$	2.54490e6	0.401057
$527$	123992.i	0.0194477i
$528$	3.30447e6i	0.515843i
$529$	137934.	0.0214305
$530$	0	0
$531$	−289562.	−0.0445661
$532$	− 288456.i	− 0.0441877i
$533$	2.35398e6i	0.358910i
$534$	9.53045e6	1.44631
$535$	0	0
$536$	291463.	0.0438200
$537$	5.36821e6i	0.803329i
$538$	2.50582e6i	0.373245i
$539$	9.60408e6	1.42391
$540$	0	0
$541$	1.17853e6	0.173120	0.0865598	−	0.996247i	$-0.472413\pi$
$541$	1.17853e6	0.173120	0.0865598	+	0.996247i	$0.472413\pi$
$542$	− 1.41126e7i	− 2.06353i
$543$	1.26304e6i	0.183830i
$544$	6.71432e6	0.972758
$545$	0	0
$546$	−43014.9	−0.00617500
$547$	9.57702e6i	1.36855i	0.729222	+	0.684277i	$0.239882\pi$
$547$	9.57702e6i	1.36855i	−0.729222	+	0.684277i	$0.760118\pi$
$548$	− 1.25721e7i	− 1.78836i
$549$	−2.38611e6	−0.337878
$550$	0	0
$551$	1.29516e7	1.81737
$552$	− 2.41094e6i	− 0.336774i
$553$	3664.22i	0 0.000509529i
$554$	3.44964e6	0.477528
$555$	0	0
$556$	7.72075e6	1.05919
$557$	2.46873e6i	0.337159i	0.985688	+	0.168580i	$0.0539181\pi$
$557$	2.46873e6i	0.337159i	−0.985688	+	0.168580i	$0.946082\pi$
$558$	171937.i	0.0233767i
$559$	−1.31036e6	−0.177362
$560$	0	0
$561$	−5.03176e6	−0.675014
$562$	− 2.37499e6i	− 0.317191i
$563$	− 7.81681e6i	− 1.03934i	−0.854367	−	0.519671i	$-0.826055\pi$
$563$	− 7.81681e6i	− 1.03934i	0.854367	−	0.519671i	$-0.173945\pi$
$564$	−3.67387e6	−0.486325
$565$	0	0
$566$	6.87288e6	0.901774
$567$	− 19730.7i	− 0.00257742i
$568$	4.04581e6i	0.526180i
$569$	9.64303e6	1.24863	0.624314	−	0.781174i	$-0.285379\pi$
$569$	9.64303e6	1.24863	0.624314	+	0.781174i	$0.285379\pi$
$570$	0	0
$571$	1.12378e7	1.44242	0.721211	−	0.692715i	$-0.243586\pi$
$571$	1.12378e7	1.44242	0.721211	+	0.692715i	$0.243586\pi$
$572$	4.13510e6i	0.528440i
$573$	− 3.60258e6i	− 0.458381i
$574$	−344636.	−0.0436598
$575$	0	0
$576$	6.84412e6	0.859530
$577$	− 5.44674e6i	− 0.681078i	−0.940230	−	0.340539i	$-0.889390\pi$
$577$	− 5.44674e6i	− 0.681078i	0.940230	−	0.340539i	$-0.110610\pi$
$578$	− 5.94897e6i	− 0.740666i
$579$	−9.01463e6	−1.11751
$580$	0	0
$581$	20450.9	0.00251346
$582$	− 3.77723e6i	− 0.462238i
$583$	3.41667e6i	0.416324i
$584$	635803.	0.0771420
$585$	0	0
$586$	1.34332e7	1.61598
$587$	9.96240e6i	1.19335i	0.802482	+	0.596676i	$0.203512\pi$
$587$	9.96240e6i	1.19335i	−0.802482	+	0.596676i	$0.796488\pi$
$588$	7.39585e6i	0.882155i
$589$	−341426.	−0.0405516
$590$	0	0
$591$	3.15380e6	0.371420
$592$	289912.i	0.0339987i
$593$	3.17929e6i	0.371273i	0.982619	+	0.185636i	$0.0594346\pi$
$593$	3.17929e6i	0.371273i	−0.982619	+	0.185636i	$0.940565\pi$
$594$	−1.93373e7	−2.24869
$595$	0	0
$596$	2.20841e7	2.54662
$597$	− 3.37257e6i	− 0.387280i
$598$	3.66815e6i	0.419463i
$599$	1.66017e7	1.89054	0.945271	−	0.326285i	$-0.105797\pi$
$599$	1.66017e7	1.89054	0.945271	+	0.326285i	$0.105797\pi$
$600$	0	0
$601$	1.05771e6	0.119449	0.0597244	−	0.998215i	$-0.480978\pi$
$601$	1.05771e6	0.119449	0.0597244	+	0.998215i	$0.480978\pi$
$602$	− 191844.i	− 0.0215753i
$603$	− 428141.i	− 0.0479505i
$604$	1.42847e7	1.59323
$605$	0	0
$606$	7.82446e6	0.865512
$607$	2.42026e6i	0.266619i	0.991074	+	0.133309i	$0.0425604\pi$
$607$	2.42026e6i	0.266619i	−0.991074	+	0.133309i	$0.957440\pi$
$608$	1.84886e7i	2.02836i
$609$	161790.	0.0176770
$610$	0	0
$611$	1.41027e6	0.152826
$612$	5.02301e6i	0.542108i
$613$	− 110410.i	− 0.0118675i	−0.999982	−	0.00593373i	$-0.998111\pi$
$613$	− 110410.i	− 0.0118675i	0.999982	−	0.00593373i	$-0.00188877\pi$
$614$	1.79144e7	1.91770
$615$	0	0
$616$	−152743.	−0.0162184
$617$	− 1.06544e7i	− 1.12672i	−0.826211	−	0.563361i	$-0.809508\pi$
$617$	− 1.06544e7i	− 1.12672i	0.826211	−	0.563361i	$-0.190492\pi$
$618$	8.61969e6i	0.907864i
$619$	1.30456e7	1.36848	0.684240	−	0.729256i	$-0.260134\pi$
$619$	1.30456e7	1.36848	0.684240	+	0.729256i	$0.260134\pi$
$620$	0	0
$621$	−9.81499e6	−1.02132
$622$	2.16929e7i	2.24824i
$623$	− 306466.i	− 0.0316346i
$624$	976815.	0.100427
$625$	0	0
$626$	−7.19894e6	−0.734231
$627$	− 1.38555e7i	− 1.40751i
$628$	1.13783e7i	1.15127i
$629$	−441453.	−0.0444895
$630$	0	0
$631$	−1.53156e6	−0.153130	−0.0765651	−	0.997065i	$-0.524395\pi$
$631$	−1.53156e6	−0.153130	−0.0765651	+	0.997065i	$0.524395\pi$
$632$	119609.i	0.0119117i
$633$	5.24103e6i	0.519885i
$634$	−1.52680e7	−1.50855
$635$	0	0
$636$	−2.63109e6	−0.257924
$637$	− 2.83900e6i	− 0.277215i
$638$	− 2.71824e7i	− 2.64384i
$639$	5.94302e6	0.575778
$640$	0	0
$641$	8.09995e6	0.778641	0.389321	−	0.921102i	$-0.372710\pi$
$641$	8.09995e6	0.778641	0.389321	+	0.921102i	$0.372710\pi$
$642$	− 9.47585e6i	− 0.907362i
$643$	5.29175e6i	0.504745i	0.967630	+	0.252373i	$0.0812108\pi$
$643$	5.29175e6i	0.504745i	−0.967630	+	0.252373i	$0.918789\pi$
$644$	−307283.	−0.0291960
$645$	0	0
$646$	−1.74324e7	−1.64353
$647$	1.80626e7i	1.69637i	0.529700	+	0.848185i	$0.322305\pi$
$647$	1.80626e7i	1.69637i	−0.529700	+	0.848185i	$0.677695\pi$
$648$	− 644060.i	− 0.0602544i
$649$	−1.20679e6	−0.112466
$650$	0	0
$651$	−4265.07	−0.000394434 0
$652$	− 9.90416e6i	− 0.912428i
$653$	7.10212e6i	0.651786i	0.945407	+	0.325893i	$0.105665\pi$
$653$	7.10212e6i	0.651786i	−0.945407	+	0.325893i	$0.894335\pi$
$654$	−1.26796e7	−1.15921
$655$	0	0
$656$	7.82627e6	0.710060
$657$	− 933954.i	− 0.0844135i
$658$	206471.i	0.0185907i
$659$	−1.41430e7	−1.26861	−0.634303	−	0.773084i	$-0.718713\pi$
$659$	−1.41430e7	−1.26861	−0.634303	+	0.773084i	$0.718713\pi$
$660$	0	0
$661$	−675807.	−0.0601615	−0.0300808	−	0.999547i	$-0.509576\pi$
$661$	−675807.	−0.0601615	−0.0300808	+	0.999547i	$0.509576\pi$
$662$	2.46543e6i	0.218649i
$663$	1.48741e6i	0.131415i
$664$	667566.	0.0587590
$665$	0	0
$666$	−612152.	−0.0534778
$667$	− 1.37969e7i	− 1.20079i
$668$	2.80851e7i	2.43520i
$669$	154936.	0.0133840
$670$	0	0
$671$	−9.94447e6	−0.852659
$672$	230958.i	0.0197292i
$673$	− 2.66714e6i	− 0.226991i	−0.993539	−	0.113496i	$-0.963795\pi$
$673$	− 2.66714e6i	− 0.226991i	0.993539	−	0.113496i	$-0.0362048\pi$
$674$	1.70318e7	1.44414
$675$	0	0
$676$	1.22235e6	0.102880
$677$	− 1.20319e7i	− 1.00893i	−0.863432	−	0.504465i	$-0.831690\pi$
$677$	− 1.20319e7i	− 1.00893i	0.863432	−	0.504465i	$-0.168310\pi$
$678$	− 1.86423e7i	− 1.55749i
$679$	−121462.	−0.0101104
$680$	0	0
$681$	2.81568e6	0.232656
$682$	716573.i	0.0589929i
$683$	1.83758e7i	1.50728i	0.657287	+	0.753640i	$0.271704\pi$
$683$	1.83758e7i	1.50728i	−0.657287	+	0.753640i	$0.728296\pi$
$684$	−1.38314e7	−1.13038
$685$	0	0
$686$	831495.	0.0674604
$687$	7.64379e6i	0.617898i
$688$	4.35653e6i	0.350889i
$689$	1.00998e6	0.0810521
$690$	0	0
$691$	1.33412e7	1.06292	0.531459	−	0.847084i	$-0.321644\pi$
$691$	1.33412e7	1.06292	0.531459	+	0.847084i	$0.321644\pi$
$692$	1.93068e6i	0.153266i
$693$	224369.i	0.0177472i
$694$	1.64109e7	1.29340
$695$	0	0
$696$	5.28124e6	0.413250
$697$	1.19171e7i	0.929159i
$698$	− 3.16372e6i	− 0.245787i
$699$	−1.15340e7	−0.892865
$700$	0	0
$701$	2.51952e6	0.193652	0.0968262	−	0.995301i	$-0.469131\pi$
$701$	2.51952e6	0.193652	0.0968262	+	0.995301i	$0.469131\pi$
$702$	5.71618e6i	0.437787i
$703$	− 1.21559e6i	− 0.0927679i
$704$	2.85239e7	2.16909
$705$	0	0
$706$	−1.35499e7	−1.02311
$707$	− 251607.i	− 0.0189310i
$708$	− 929319.i	− 0.0696757i
$709$	−2.51441e7	−1.87854	−0.939272	−	0.343174i	$-0.888498\pi$
$709$	−2.51441e7	−1.87854	−0.939272	+	0.343174i	$0.888498\pi$
$710$	0	0
$711$	175698.	0.0130345
$712$	− 1.00038e7i	− 0.739546i
$713$	363709.i	0.0267936i
$714$	−217765.	−0.0159861
$715$	0	0
$716$	2.23339e7	1.62810
$717$	2.87868e6i	0.209120i
$718$	− 3.48803e7i	− 2.52504i
$719$	−415865.	−0.0300006	−0.0150003	−	0.999887i	$-0.504775\pi$
$719$	−415865.	−0.0300006	−0.0150003	+	0.999887i	$0.504775\pi$
$720$	0	0
$721$	277179.	0.0198574
$722$	− 2.65873e7i	− 1.89815i
$723$	− 8.40330e6i	− 0.597866i
$724$	5.25474e6	0.372567
$725$	0	0
$726$	−1.47511e7	−1.03868
$727$	− 1.53987e7i	− 1.08056i	−0.841486	−	0.540278i	$-0.818319\pi$
$727$	− 1.53987e7i	− 1.08056i	0.841486	−	0.540278i	$-0.181681\pi$
$728$	45151.3i	0.00315749i
$729$	−1.07222e7	−0.747251
$730$	0	0
$731$	−6.63374e6	−0.459161
$732$	− 7.65798e6i	− 0.528246i
$733$	1.82125e7i	1.25202i	0.779817	+	0.626008i	$0.215312\pi$
$733$	1.82125e7i	1.25202i	−0.779817	+	0.626008i	$0.784688\pi$
$734$	5.91428e6	0.405193
$735$	0	0
$736$	1.96953e7	1.34019
$737$	− 1.78434e6i	− 0.121007i
$738$	1.65252e7i	1.11688i
$739$	1.88107e7	1.26705	0.633524	−	0.773723i	$-0.281608\pi$
$739$	1.88107e7	1.26705	0.633524	+	0.773723i	$0.281608\pi$
$740$	0	0
$741$	−4.09573e6	−0.274022
$742$	147867.i	0.00985964i
$743$	− 2.61292e7i	− 1.73642i	−0.496199	−	0.868209i	$-0.665271\pi$
$743$	− 2.61292e7i	− 1.73642i	0.496199	−	0.868209i	$-0.334729\pi$
$744$	−139223.	−0.00922098
$745$	0	0
$746$	2.67378e7	1.75905
$747$	− 980611.i	− 0.0642977i
$748$	2.09341e7i	1.36805i
$749$	−304710.	−0.0198464
$750$	0	0
$751$	−1.90282e7	−1.23111	−0.615557	−	0.788092i	$-0.711069\pi$
$751$	−1.90282e7	−1.23111	−0.615557	+	0.788092i	$0.711069\pi$
$752$	− 4.68871e6i	− 0.302349i
$753$	7.88699e6i	0.506902i
$754$	−8.03520e6	−0.514717
$755$	0	0
$756$	−478847.	−0.0304714
$757$	− 1.91639e7i	− 1.21547i	−0.794140	−	0.607734i	$-0.792079\pi$
$757$	− 1.91639e7i	− 1.21547i	0.794140	−	0.607734i	$-0.207921\pi$
$758$	− 4.53705e7i	− 2.86814i
$759$	−1.47598e7	−0.929983
$760$	0	0
$761$	−468091.	−0.0293001	−0.0146500	−	0.999893i	$-0.504663\pi$
$761$	−468091.	−0.0293001	−0.0146500	+	0.999893i	$0.504663\pi$
$762$	4.84590e6i	0.302334i
$763$	407732.i	0.0253550i
$764$	−1.49882e7	−0.928999
$765$	0	0
$766$	−1.35791e7	−0.836177
$767$	356732.i	0.0218954i
$768$	− 376805.i	− 0.0230523i
$769$	1.92729e7	1.17525	0.587627	−	0.809132i	$-0.300062\pi$
$769$	1.92729e7	1.17525	0.587627	+	0.809132i	$0.300062\pi$
$770$	0	0
$771$	−1.87648e7	−1.13686
$772$	3.75045e7i	2.26485i
$773$	− 2.42083e7i	− 1.45719i	−0.684947	−	0.728593i	$-0.740175\pi$
$773$	− 2.42083e7i	− 1.45719i	0.684947	−	0.728593i	$-0.259825\pi$
$774$	−9.19884e6	−0.551926
$775$	0	0
$776$	−3.96483e6	−0.236358
$777$	− 15185.0i	0 0.000902326i
$778$	1.33605e7i	0.791358i
$779$	−3.28151e7	−1.93745
$780$	0	0
$781$	2.47684e7	1.45302
$782$	1.85702e7i	1.08592i
$783$	− 2.15001e7i	− 1.25324i
$784$	−9.43881e6	−0.548437
$785$	0	0
$786$	−2.29760e7	−1.32653
$787$	− 1.48087e7i	− 0.852276i	−0.904658	−	0.426138i	$-0.859874\pi$
$787$	− 1.48087e7i	− 0.852276i	0.904658	−	0.426138i	$-0.140126\pi$
$788$	− 1.31211e7i	− 0.752755i
$789$	3.02701e6	0.173110
$790$	0	0
$791$	−599470.	−0.0340664
$792$	7.32396e6i	0.414890i
$793$	2.93962e6i	0.166000i
$794$	4.38233e7	2.46691
$795$	0	0
$796$	−1.40312e7	−0.784898
$797$	− 2.90994e7i	− 1.62270i	−0.584560	−	0.811351i	$-0.698733\pi$
$797$	− 2.90994e7i	− 1.62270i	0.584560	−	0.811351i	$-0.301267\pi$
$798$	− 599639.i	− 0.0333336i
$799$	7.13954e6	0.395643
$800$	0	0
$801$	−1.46949e7	−0.809257
$802$	1.19454e6i	0.0655790i
$803$	− 3.89239e6i	− 0.213023i
$804$	1.37407e6	0.0749669
$805$	0	0
$806$	211822.	0.0114850
$807$	2.98052e6i	0.161105i
$808$	− 8.21308e6i	− 0.442566i
$809$	−3.37792e7	−1.81459	−0.907295	−	0.420496i	$-0.861856\pi$
$809$	−3.37792e7	−1.81459	−0.907295	+	0.420496i	$0.861856\pi$
$810$	0	0
$811$	−2.42986e7	−1.29727	−0.648633	−	0.761101i	$-0.724659\pi$
$811$	−2.42986e7	−1.29727	−0.648633	+	0.761101i	$0.724659\pi$
$812$	− 673113.i	− 0.0358260i
$813$	− 1.67861e7i	− 0.890686i
$814$	−2.55123e6	−0.134955
$815$	0	0
$816$	4.94517e6	0.259990
$817$	− 1.82667e7i	− 0.957425i
$818$	− 3.57460e7i	− 1.86786i
$819$	66324.3	0.00345512
$820$	0	0
$821$	−7.42442e6	−0.384419	−0.192209	−	0.981354i	$-0.561565\pi$
$821$	−7.42442e6	−0.384419	−0.192209	+	0.981354i	$0.561565\pi$
$822$	− 2.61346e7i	− 1.34908i
$823$	2.98540e7i	1.53640i	0.640212	+	0.768198i	$0.278847\pi$
$823$	2.98540e7i	1.53640i	−0.640212	+	0.768198i	$0.721153\pi$
$824$	9.04781e6	0.464222
$825$	0	0
$826$	−52227.6	−0.00266348
$827$	2.47154e7i	1.25662i	0.777962	+	0.628311i	$0.216253\pi$
$827$	2.47154e7i	1.25662i	−0.777962	+	0.628311i	$0.783747\pi$
$828$	1.47341e7i	0.746875i
$829$	1.14800e7	0.580169	0.290084	−	0.957001i	$-0.406317\pi$
$829$	1.14800e7	0.580169	0.290084	+	0.957001i	$0.406317\pi$
$830$	0	0
$831$	4.10314e6	0.206117
$832$	− 8.43176e6i	− 0.422289i
$833$	− 1.43726e7i	− 0.717665i
$834$	1.60498e7	0.799013
$835$	0	0
$836$	−5.76443e7	−2.85260
$837$	566779.i	0.0279640i
$838$	− 2.56606e7i	− 1.26228i
$839$	−2.41188e7	−1.18291	−0.591454	−	0.806339i	$-0.701446\pi$
$839$	−2.41188e7	−1.18291	−0.591454	+	0.806339i	$0.701446\pi$
$840$	0	0
$841$	9.71140e6	0.473469
$842$	2.57836e7i	1.25332i
$843$	− 2.82491e6i	− 0.136910i
$844$	2.18048e7	1.05365
$845$	0	0
$846$	9.90022e6	0.475575
$847$	474343.i	0.0227187i
$848$	− 3.35787e6i	− 0.160352i
$849$	8.17487e6	0.389235
$850$	0	0
$851$	−1.29492e6	−0.0612943
$852$	1.90735e7i	0.900186i
$853$	2.02681e6i	0.0953764i	0.998862	+	0.0476882i	$0.0151854\pi$
$853$	2.02681e6i	0.0953764i	−0.998862	+	0.0476882i	$0.984815\pi$
$854$	−430378.	−0.0201932
$855$	0	0
$856$	−9.94649e6	−0.463965
$857$	1.52859e7i	0.710950i	0.934686	+	0.355475i	$0.115681\pi$
$857$	1.52859e7i	0.710950i	−0.934686	+	0.355475i	$0.884319\pi$
$858$	8.59598e6i	0.398637i
$859$	−1.78567e7	−0.825693	−0.412846	−	0.910801i	$-0.635465\pi$
$859$	−1.78567e7	−0.825693	−0.412846	+	0.910801i	$0.635465\pi$
$860$	0	0
$861$	−409924.	−0.0188450
$862$	2.89546e7i	1.32724i
$863$	1.12315e7i	0.513347i	0.966498	+	0.256673i	$0.0826264\pi$
$863$	1.12315e7i	0.513347i	−0.966498	+	0.256673i	$0.917374\pi$
$864$	3.06917e7	1.39874
$865$	0	0
$866$	3.25863e7	1.47652
$867$	− 7.07595e6i	− 0.319696i
$868$	17744.4i	0 0.000799397i
$869$	732249.	0.0328934
$870$	0	0
$871$	−527457.	−0.0235582
$872$	1.33094e7i	0.592744i
$873$	5.82408e6i	0.258638i
$874$	−5.11349e7	−2.26433
$875$	0	0
$876$	2.99743e6	0.131974
$877$	1.75233e6i	0.0769336i	0.999260	+	0.0384668i	$0.0122474\pi$
$877$	1.75233e6i	0.0769336i	−0.999260	+	0.0384668i	$0.987753\pi$
$878$	4.86980e7i	2.13194i
$879$	1.59780e7	0.697510
$880$	0	0
$881$	−1.04009e7	−0.451474	−0.225737	−	0.974188i	$-0.572479\pi$
$881$	−1.04009e7	−0.451474	−0.225737	+	0.974188i	$0.572479\pi$
$882$	− 1.99301e7i	− 0.862656i
$883$	− 4.25645e7i	− 1.83715i	−0.395241	−	0.918577i	$-0.629339\pi$
$883$	− 4.25645e7i	− 1.83715i	0.395241	−	0.918577i	$-0.370661\pi$
$884$	6.18821e6	0.266339
$885$	0	0
$886$	−3.19827e7	−1.36877
$887$	201500.i	0.00859935i	0.999991	+	0.00429968i	$0.00136863\pi$
$887$	201500.i	0.00859935i	−0.999991	+	0.00429968i	$0.998631\pi$
$888$	− 495677.i	− 0.0210944i
$889$	155827.	0.00661285
$890$	0	0
$891$	−3.94294e6	−0.166389
$892$	− 644596.i	− 0.0271253i
$893$	1.96595e7i	0.824980i
$894$	4.59081e7	1.92108
$895$	0	0
$896$	516009.	0.0214727
$897$	4.36304e6i	0.181054i
$898$	− 4.37198e6i	− 0.180920i
$899$	−796718.	−0.0328780
$900$	0	0
$901$	5.11307e6	0.209831
$902$	6.88712e7i	2.81852i
$903$	− 228186.i	− 0.00931259i
$904$	−1.95682e7	−0.796397
$905$	0	0
$906$	2.96947e7	1.20187
$907$	1.47930e7i	0.597088i	0.954396	+	0.298544i	$0.0965010\pi$
$907$	1.47930e7i	0.597088i	−0.954396	+	0.298544i	$0.903499\pi$
$908$	− 1.17143e7i	− 0.471523i
$909$	−1.20645e7	−0.484283
$910$	0	0
$911$	−3.50070e7	−1.39752	−0.698762	−	0.715354i	$-0.746265\pi$
$911$	−3.50070e7	−1.39752	−0.698762	+	0.715354i	$0.746265\pi$
$912$	1.36170e7i	0.542121i
$913$	− 4.08684e6i	− 0.162260i
$914$	6.77726e6	0.268342
$915$	0	0
$916$	3.18012e7	1.25229
$917$	738828.i	0.0290148i
$918$	2.89384e7i	1.13336i
$919$	−4.24526e7	−1.65812	−0.829060	−	0.559160i	$-0.811124\pi$
$919$	−4.24526e7	−1.65812	−0.829060	+	0.559160i	$0.811124\pi$
$920$	0	0
$921$	2.13080e7	0.827741
$922$	− 4.02034e7i	− 1.55753i
$923$	− 7.32164e6i	− 0.282881i
$924$	−720090.	−0.0277464
$925$	0	0
$926$	−2.29076e7	−0.877916
$927$	− 1.32906e7i	− 0.507980i
$928$	4.31431e7i	1.64453i
$929$	3.95487e7	1.50347	0.751733	−	0.659468i	$-0.229218\pi$
$929$	3.95487e7	1.50347	0.751733	+	0.659468i	$0.229218\pi$
$930$	0	0
$931$	3.95764e7	1.49645
$932$	4.79859e7i	1.80957i
$933$	2.58024e7i	0.970413i
$934$	−2.09445e7	−0.785601
$935$	0	0
$936$	2.16499e6	0.0807730
$937$	− 4.62850e7i	− 1.72223i	−0.508409	−	0.861116i	$-0.669766\pi$
$937$	− 4.62850e7i	− 1.72223i	0.508409	−	0.861116i	$-0.330234\pi$
$938$	− 77222.8i	− 0.00286575i
$939$	−8.56270e6	−0.316918
$940$	0	0
$941$	5.74465e6	0.211490	0.105745	−	0.994393i	$-0.466277\pi$
$941$	5.74465e6	0.211490	0.105745	+	0.994393i	$0.466277\pi$
$942$	2.36530e7i	0.868477i
$943$	3.49568e7i	1.28013i
$944$	1.18602e6	0.0433175
$945$	0	0
$946$	−3.83375e7	−1.39282
$947$	5.15966e7i	1.86959i	0.355187	+	0.934795i	$0.384417\pi$
$947$	5.15966e7i	1.86959i	−0.355187	+	0.934795i	$0.615583\pi$
$948$	563886.i	0.0203784i
$949$	−1.15060e6	−0.0414725
$950$	0	0
$951$	−1.81603e7	−0.651137
$952$	228581.i	0.00817424i
$953$	2.18333e7i	0.778729i	0.921084	+	0.389365i	$0.127305\pi$
$953$	2.18333e7i	0.778729i	−0.921084	+	0.389365i	$0.872695\pi$
$954$	7.09017e6	0.252223
$955$	0	0
$956$	1.19765e7	0.423822
$957$	− 3.23318e7i	− 1.14117i
$958$	4.82863e7i	1.69985i
$959$	−840398.	−0.0295079
$960$	0	0
$961$	−2.86081e7	−0.999266
$962$	754154.i	0.0262738i
$963$	1.46107e7i	0.507699i
$964$	−3.49611e7	−1.21169
$965$	0	0
$966$	−638774.	−0.0220244
$967$	− 7.30509e6i	− 0.251223i	−0.992080	−	0.125611i	$-0.959911\pi$
$967$	− 7.30509e6i	− 0.251223i	0.992080	−	0.125611i	$-0.0400893\pi$
$968$	1.54837e7i	0.531114i
$969$	−2.07348e7	−0.709400
$970$	0	0
$971$	−1.62933e6	−0.0554576	−0.0277288	−	0.999615i	$-0.508827\pi$
$971$	−1.62933e6	−0.0554576	−0.0277288	+	0.999615i	$0.508827\pi$
$972$	3.76364e7i	1.27774i
$973$	− 516105.i	− 0.0174765i
$974$	6.27180e7	2.11834
$975$	0	0
$976$	9.77335e6	0.328412
$977$	− 2.55515e7i	− 0.856407i	−0.903682	−	0.428204i	$-0.859147\pi$
$977$	− 2.55515e7i	− 0.856407i	0.903682	−	0.428204i	$-0.140853\pi$
$978$	− 2.05886e7i	− 0.688304i
$979$	−6.12433e7	−2.04222
$980$	0	0
$981$	1.95506e7	0.648617
$982$	− 3.61000e7i	− 1.19462i
$983$	8.48095e6i	0.279937i	0.990156	+	0.139969i	$0.0447002\pi$
$983$	8.48095e6i	0.279937i	−0.990156	+	0.139969i	$0.955300\pi$
$984$	−1.33809e7	−0.440554
$985$	0	0
$986$	−4.06786e7	−1.33252
$987$	245585.i	0.00802434i
$988$	1.70399e7i	0.555359i
$989$	−1.94589e7	−0.632597
$990$	0	0
$991$	−2.74483e7	−0.887833	−0.443916	−	0.896068i	$-0.646411\pi$
$991$	−2.74483e7	−0.887833	−0.443916	+	0.896068i	$0.646411\pi$
$992$	− 1.13733e6i	− 0.0366949i
$993$	2.93248e6i	0.0943762i
$994$	1.07193e6	0.0344113
$995$	0	0
$996$	3.14717e6	0.100525
$997$	− 2.26765e7i	− 0.722501i	−0.932469	−	0.361251i	$-0.882350\pi$
$997$	− 2.26765e7i	− 0.722501i	0.932469	−	0.361251i	$-0.117650\pi$
$998$	− 6.77660e7i	− 2.15370i
$999$	−2.01792e6	−0.0639719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.c.274.2		6
5.2	odd	4		325.6.a.c.1.3		3
5.3	odd	4		13.6.a.b.1.1	✓	3
5.4	even	2	inner	325.6.b.c.274.5		6
15.8	even	4		117.6.a.d.1.3		3
20.3	even	4		208.6.a.j.1.2		3
35.13	even	4		637.6.a.b.1.1		3
40.3	even	4		832.6.a.t.1.2		3
40.13	odd	4		832.6.a.s.1.2		3
65.8	even	4		169.6.b.b.168.2		6
65.18	even	4		169.6.b.b.168.5		6
65.38	odd	4		169.6.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.b.1.1	✓	3	5.3	odd	4
117.6.a.d.1.3		3	15.8	even	4
169.6.a.b.1.3		3	65.38	odd	4
169.6.b.b.168.2		6	65.8	even	4
169.6.b.b.168.5		6	65.18	even	4
208.6.a.j.1.2		3	20.3	even	4
325.6.a.c.1.3		3	5.2	odd	4
325.6.b.c.274.2		6	1.1	even	1	trivial
325.6.b.c.274.5		6	5.4	even	2	inner
637.6.a.b.1.1		3	35.13	even	4
832.6.a.s.1.2		3	40.13	odd	4
832.6.a.t.1.2		3	40.3	even	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)

\(f(q)\)	\(=\)	\(q-8.64858i q^{2} -10.2870i q^{3} -42.7979 q^{4} -88.9676 q^{6} +2.86088i q^{7} +93.3863i q^{8} +137.178 q^{9} +O(q^{10})\)	\(q-8.64858i q^{2} -10.2870i q^{3} -42.7979 q^{4} -88.9676 q^{6} +2.86088i q^{7} +93.3863i q^{8} +137.178 q^{9} +571.711 q^{11} +440.260i q^{12} -169.000i q^{13} +24.7426 q^{14} -561.873 q^{16} -855.571i q^{17} -1186.40i q^{18} +2355.90 q^{19} +29.4298 q^{21} -4944.49i q^{22} -2509.66i q^{23} +960.662 q^{24} -1461.61 q^{26} -3910.88i q^{27} -122.440i q^{28} +5497.50 q^{29} -144.924 q^{31} +7847.77i q^{32} -5881.17i q^{33} -7399.47 q^{34} -5870.95 q^{36} -515.975i q^{37} -20375.2i q^{38} -1738.50 q^{39} -13928.9 q^{41} -254.526i q^{42} -7753.58i q^{43} -24468.0 q^{44} -21705.0 q^{46} +8344.77i q^{47} +5779.97i q^{48} +16798.8 q^{49} -8801.22 q^{51} +7232.84i q^{52} +5976.21i q^{53} -33823.6 q^{54} -267.168 q^{56} -24235.1i q^{57} -47545.6i q^{58} -2110.84 q^{59} -17394.2 q^{61} +1253.38i q^{62} +392.452i q^{63} +49892.1 q^{64} -50863.8 q^{66} -3121.05i q^{67} +36616.6i q^{68} -25816.8 q^{69} +43323.3 q^{71} +12810.6i q^{72} -6808.31i q^{73} -4462.45 q^{74} -100828. q^{76} +1635.60i q^{77} +15035.5i q^{78} +1280.80 q^{79} -6896.72 q^{81} +120465. i q^{82} -7148.44i q^{83} -1259.53 q^{84} -67057.5 q^{86} -56552.6i q^{87} +53390.0i q^{88} -107123. q^{89} +483.489 q^{91} +107408. i q^{92} +1490.82i q^{93} +72170.4 q^{94} +80729.7 q^{96} +42456.2i q^{97} -145286. i q^{98} +78426.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 242 q^{4} - 398 q^{6} + 382 q^{9}+O(q^{10}) \) 6 * q - 242 * q^4 - 398 * q^6 + 382 * q^9	\( 6 q - 242 q^{4} - 398 q^{6} + 382 q^{9} + 1112 q^{11} + 1586 q^{14} + 5570 q^{16} - 296 q^{19} + 2780 q^{21} + 26754 q^{24} + 2366 q^{26} + 17516 q^{29} - 5216 q^{31} - 31978 q^{34} - 9356 q^{36} - 2704 q^{39} - 21996 q^{41} - 99580 q^{44} - 6816 q^{46} + 89142 q^{49} + 26936 q^{51} - 111190 q^{54} - 98574 q^{56} - 127896 q^{59} - 25508 q^{61} - 234786 q^{64} - 224708 q^{66} + 40224 q^{69} + 155160 q^{71} + 263038 q^{74} - 199276 q^{76} + 123744 q^{79} - 216442 q^{81} - 209446 q^{84} - 139794 q^{86} - 67388 q^{89} - 20280 q^{91} - 145686 q^{94} - 487282 q^{96} + 120664 q^{99}+O(q^{100}) \) 6 * q - 242 * q^4 - 398 * q^6 + 382 * q^9 + 1112 * q^11 + 1586 * q^14 + 5570 * q^16 - 296 * q^19 + 2780 * q^21 + 26754 * q^24 + 2366 * q^26 + 17516 * q^29 - 5216 * q^31 - 31978 * q^34 - 9356 * q^36 - 2704 * q^39 - 21996 * q^41 - 99580 * q^44 - 6816 * q^46 + 89142 * q^49 + 26936 * q^51 - 111190 * q^54 - 98574 * q^56 - 127896 * q^59 - 25508 * q^61 - 234786 * q^64 - 224708 * q^66 + 40224 * q^69 + 155160 * q^71 + 263038 * q^74 - 199276 * q^76 + 123744 * q^79 - 216442 * q^81 - 209446 * q^84 - 139794 * q^86 - 67388 * q^89 - 20280 * q^91 - 145686 * q^94 - 487282 * q^96 + 120664 * q^99

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