LMFDB - Embedded newform 525.6.d.d.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.d (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:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.6.d.d.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +9.00000i q^{3} +31.0000 q^{4} -9.00000 q^{6} -49.0000i q^{7} +63.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +9.00000i q^{3} +31.0000 q^{4} -9.00000 q^{6} -49.0000i q^{7} +63.0000i q^{8} -81.0000 q^{9} -340.000 q^{11} +279.000i q^{12} -454.000i q^{13} +49.0000 q^{14} +929.000 q^{16} -798.000i q^{17} -81.0000i q^{18} -892.000 q^{19} +441.000 q^{21} -340.000i q^{22} +3192.00i q^{23} -567.000 q^{24} +454.000 q^{26} -729.000i q^{27} -1519.00i q^{28} +8242.00 q^{29} -2496.00 q^{31} +2945.00i q^{32} -3060.00i q^{33} +798.000 q^{34} -2511.00 q^{36} +9798.00i q^{37} -892.000i q^{38} +4086.00 q^{39} +19834.0 q^{41} +441.000i q^{42} +17236.0i q^{43} -10540.0 q^{44} -3192.00 q^{46} +8928.00i q^{47} +8361.00i q^{48} -2401.00 q^{49} +7182.00 q^{51} -14074.0i q^{52} -150.000i q^{53} +729.000 q^{54} +3087.00 q^{56} -8028.00i q^{57} +8242.00i q^{58} +42396.0 q^{59} +14758.0 q^{61} -2496.00i q^{62} +3969.00i q^{63} +26783.0 q^{64} +3060.00 q^{66} -1676.00i q^{67} -24738.0i q^{68} -28728.0 q^{69} +14568.0 q^{71} -5103.00i q^{72} -78378.0i q^{73} -9798.00 q^{74} -27652.0 q^{76} +16660.0i q^{77} +4086.00i q^{78} +2272.00 q^{79} +6561.00 q^{81} +19834.0i q^{82} +37764.0i q^{83} +13671.0 q^{84} -17236.0 q^{86} +74178.0i q^{87} -21420.0i q^{88} +117286. q^{89} -22246.0 q^{91} +98952.0i q^{92} -22464.0i q^{93} -8928.00 q^{94} -26505.0 q^{96} +10002.0i q^{97} -2401.00i q^{98} +27540.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 62 q^{4} - 18 q^{6} - 162 q^{9}+O(q^{10}) $ 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9	$ 2 q + 62 q^{4} - 18 q^{6} - 162 q^{9} - 680 q^{11} + 98 q^{14} + 1858 q^{16} - 1784 q^{19} + 882 q^{21} - 1134 q^{24} + 908 q^{26} + 16484 q^{29} - 4992 q^{31} + 1596 q^{34} - 5022 q^{36} + 8172 q^{39} + 39668 q^{41} - 21080 q^{44} - 6384 q^{46} - 4802 q^{49} + 14364 q^{51} + 1458 q^{54} + 6174 q^{56} + 84792 q^{59} + 29516 q^{61} + 53566 q^{64} + 6120 q^{66} - 57456 q^{69} + 29136 q^{71} - 19596 q^{74} - 55304 q^{76} + 4544 q^{79} + 13122 q^{81} + 27342 q^{84} - 34472 q^{86} + 234572 q^{89} - 44492 q^{91} - 17856 q^{94} - 53010 q^{96} + 55080 q^{99}+O(q^{100}) $ 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9 - 680 * q^11 + 98 * q^14 + 1858 * q^16 - 1784 * q^19 + 882 * q^21 - 1134 * q^24 + 908 * q^26 + 16484 * q^29 - 4992 * q^31 + 1596 * q^34 - 5022 * q^36 + 8172 * q^39 + 39668 * q^41 - 21080 * q^44 - 6384 * q^46 - 4802 * q^49 + 14364 * q^51 + 1458 * q^54 + 6174 * q^56 + 84792 * q^59 + 29516 * q^61 + 53566 * q^64 + 6120 * q^66 - 57456 * q^69 + 29136 * q^71 - 19596 * q^74 - 55304 * q^76 + 4544 * q^79 + 13122 * q^81 + 27342 * q^84 - 34472 * q^86 + 234572 * q^89 - 44492 * q^91 - 17856 * q^94 - 53010 * q^96 + 55080 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	1.00000i	0.176777i	0.996086	+	0.0883883i	$0.0281716\pi$
$2$	1.00000i	0.176777i	−0.996086	+	0.0883883i	$0.971828\pi$
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	31.0000	0.968750
$5$	0	0
$5$	0	0
$6$	−9.00000	−0.102062
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	63.0000i	0.348029i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−340.000	−0.847222	−0.423611	−	0.905844i	$-0.639238\pi$
$11$	−340.000	−0.847222	−0.423611	+	0.905844i	$0.639238\pi$
$12$	279.000i	0.559308i
$13$	− 454.000i	− 0.745071i	−0.928018	−	0.372535i	$-0.878489\pi$
$13$	− 454.000i	− 0.745071i	0.928018	−	0.372535i	$-0.121511\pi$
$14$	49.0000	0.0668153
$15$	0	0
$16$	929.000	0.907227
$17$	− 798.000i	− 0.669700i	−0.942271	−	0.334850i	$-0.891314\pi$
$17$	− 798.000i	− 0.669700i	0.942271	−	0.334850i	$-0.108686\pi$
$18$	− 81.0000i	− 0.0589256i
$19$	−892.000	−0.566867	−0.283433	−	0.958992i	$-0.591473\pi$
$19$	−892.000	−0.566867	−0.283433	+	0.958992i	$0.591473\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	− 340.000i	− 0.149769i
$23$	3192.00i	1.25818i	0.777332	+	0.629091i	$0.216573\pi$
$23$	3192.00i	1.25818i	−0.777332	+	0.629091i	$0.783427\pi$
$24$	−567.000	−0.200935
$25$	0	0
$26$	454.000	0.131711
$27$	− 729.000i	− 0.192450i
$28$	− 1519.00i	− 0.366153i
$29$	8242.00	1.81986	0.909929	−	0.414764i	$-0.136136\pi$
$29$	8242.00	1.81986	0.909929	+	0.414764i	$0.136136\pi$
$30$	0	0
$31$	−2496.00	−0.466488	−0.233244	−	0.972418i	$-0.574934\pi$
$31$	−2496.00	−0.466488	−0.233244	+	0.972418i	$0.574934\pi$
$32$	2945.00i	0.508406i
$33$	− 3060.00i	− 0.489144i
$34$	798.000	0.118387
$35$	0	0
$36$	−2511.00	−0.322917
$37$	9798.00i	1.17661i	0.808639	+	0.588306i	$0.200205\pi$
$37$	9798.00i	1.17661i	−0.808639	+	0.588306i	$0.799795\pi$
$38$	− 892.000i	− 0.100209i
$39$	4086.00	0.430167
$40$	0	0
$41$	19834.0	1.84268	0.921342	−	0.388754i	$-0.127094\pi$
$41$	19834.0	1.84268	0.921342	+	0.388754i	$0.127094\pi$
$42$	441.000i	0.0385758i
$43$	17236.0i	1.42156i	0.703414	+	0.710780i	$0.251658\pi$
$43$	17236.0i	1.42156i	−0.703414	+	0.710780i	$0.748342\pi$
$44$	−10540.0	−0.820746
$45$	0	0
$46$	−3192.00	−0.222417
$47$	8928.00i	0.589535i	0.955569	+	0.294767i	$0.0952422\pi$
$47$	8928.00i	0.589535i	−0.955569	+	0.294767i	$0.904758\pi$
$48$	8361.00i	0.523788i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	7182.00	0.386652
$52$	− 14074.0i	− 0.721787i
$53$	− 150.000i	− 0.00733502i	−0.999993	−	0.00366751i	$-0.998833\pi$
$53$	− 150.000i	− 0.00733502i	0.999993	−	0.00366751i	$-0.00116741\pi$
$54$	729.000	0.0340207
$55$	0	0
$56$	3087.00	0.131543
$57$	− 8028.00i	− 0.327281i
$58$	8242.00i	0.321709i
$59$	42396.0	1.58560	0.792802	−	0.609479i	$-0.208621\pi$
$59$	42396.0	1.58560	0.792802	+	0.609479i	$0.208621\pi$
$60$	0	0
$61$	14758.0	0.507812	0.253906	−	0.967229i	$-0.418285\pi$
$61$	14758.0	0.507812	0.253906	+	0.967229i	$0.418285\pi$
$62$	− 2496.00i	− 0.0824642i
$63$	3969.00i	0.125988i
$64$	26783.0	0.817352
$65$	0	0
$66$	3060.00	0.0864692
$67$	− 1676.00i	− 0.0456128i	−0.999740	−	0.0228064i	$-0.992740\pi$
$67$	− 1676.00i	− 0.0456128i	0.999740	−	0.0228064i	$-0.00726014\pi$
$68$	− 24738.0i	− 0.648772i
$69$	−28728.0	−0.726411
$70$	0	0
$71$	14568.0	0.342968	0.171484	−	0.985187i	$-0.445144\pi$
$71$	14568.0	0.342968	0.171484	+	0.985187i	$0.445144\pi$
$72$	− 5103.00i	− 0.116010i
$73$	− 78378.0i	− 1.72142i	−0.509095	−	0.860710i	$-0.670020\pi$
$73$	− 78378.0i	− 1.72142i	0.509095	−	0.860710i	$-0.329980\pi$
$74$	−9798.00	−0.207998
$75$	0	0
$76$	−27652.0	−0.549152
$77$	16660.0i	0.320220i
$78$	4086.00i	0.0760435i
$79$	2272.00	0.0409582	0.0204791	−	0.999790i	$-0.493481\pi$
$79$	2272.00	0.0409582	0.0204791	+	0.999790i	$0.493481\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	19834.0i	0.325743i
$83$	37764.0i	0.601704i	0.953671	+	0.300852i	$0.0972710\pi$
$83$	37764.0i	0.601704i	−0.953671	+	0.300852i	$0.902729\pi$
$84$	13671.0	0.211399
$85$	0	0
$86$	−17236.0	−0.251299
$87$	74178.0i	1.05070i
$88$	− 21420.0i	− 0.294858i
$89$	117286.	1.56954	0.784768	−	0.619790i	$-0.212782\pi$
$89$	117286.	1.56954	0.784768	+	0.619790i	$0.212782\pi$
$90$	0	0
$91$	−22246.0	−0.281610
$92$	98952.0i	1.21886i
$93$	− 22464.0i	− 0.269327i
$94$	−8928.00	−0.104216
$95$	0	0
$96$	−26505.0	−0.293528
$97$	10002.0i	0.107934i	0.998543	+	0.0539669i	$0.0171865\pi$
$97$	10002.0i	0.107934i	−0.998543	+	0.0539669i	$0.982813\pi$
$98$	− 2401.00i	− 0.0252538i
$99$	27540.0	0.282407
$100$	0	0
$101$	−108770.	−1.06098	−0.530488	−	0.847692i	$-0.677991\pi$
$101$	−108770.	−1.06098	−0.530488	+	0.847692i	$0.677991\pi$
$102$	7182.00i	0.0683510i
$103$	199192.i	1.85003i	0.379930	+	0.925015i	$0.375948\pi$
$103$	199192.i	1.85003i	−0.379930	+	0.925015i	$0.624052\pi$
$104$	28602.0	0.259306
$105$	0	0
$106$	150.000	0.00129666
$107$	− 79972.0i	− 0.675272i	−0.941277	−	0.337636i	$-0.890373\pi$
$107$	− 79972.0i	− 0.675272i	0.941277	−	0.337636i	$-0.109627\pi$
$108$	− 22599.0i	− 0.186436i
$109$	46098.0	0.371634	0.185817	−	0.982584i	$-0.440507\pi$
$109$	46098.0	0.371634	0.185817	+	0.982584i	$0.440507\pi$
$110$	0	0
$111$	−88182.0	−0.679317
$112$	− 45521.0i	− 0.342899i
$113$	− 262706.i	− 1.93541i	−0.252078	−	0.967707i	$-0.581114\pi$
$113$	− 262706.i	− 1.93541i	0.252078	−	0.967707i	$-0.418886\pi$
$114$	8028.00	0.0578556
$115$	0	0
$116$	255502.	1.76299
$117$	36774.0i	0.248357i
$118$	42396.0i	0.280298i
$119$	−39102.0	−0.253123
$120$	0	0
$121$	−45451.0	−0.282215
$122$	14758.0i	0.0897693i
$123$	178506.i	1.06387i
$124$	−77376.0	−0.451910
$125$	0	0
$126$	−3969.00	−0.0222718
$127$	196608.i	1.08166i	0.841131	+	0.540831i	$0.181890\pi$
$127$	196608.i	1.08166i	−0.841131	+	0.540831i	$0.818110\pi$
$128$	121023.i	0.652894i
$129$	−155124.	−0.820738
$130$	0	0
$131$	−77140.0	−0.392737	−0.196368	−	0.980530i	$-0.562915\pi$
$131$	−77140.0	−0.392737	−0.196368	+	0.980530i	$0.562915\pi$
$132$	− 94860.0i	− 0.473858i
$133$	43708.0i	0.214255i
$134$	1676.00	0.00806329
$135$	0	0
$136$	50274.0	0.233075
$137$	208170.i	0.947582i	0.880637	+	0.473791i	$0.157115\pi$
$137$	208170.i	0.947582i	−0.880637	+	0.473791i	$0.842885\pi$
$138$	− 28728.0i	− 0.128413i
$139$	275580.	1.20979	0.604896	−	0.796304i	$-0.293215\pi$
$139$	275580.	1.20979	0.604896	+	0.796304i	$0.293215\pi$
$140$	0	0
$141$	−80352.0	−0.340368
$142$	14568.0i	0.0606288i
$143$	154360.i	0.631240i
$144$	−75249.0	−0.302409
$145$	0	0
$146$	78378.0	0.304307
$147$	− 21609.0i	− 0.0824786i
$148$	303738.i	1.13984i
$149$	296106.	1.09265	0.546326	−	0.837573i	$-0.316026\pi$
$149$	296106.	1.09265	0.546326	+	0.837573i	$0.316026\pi$
$150$	0	0
$151$	−426472.	−1.52212	−0.761059	−	0.648683i	$-0.775320\pi$
$151$	−426472.	−1.52212	−0.761059	+	0.648683i	$0.775320\pi$
$152$	− 56196.0i	− 0.197286i
$153$	64638.0i	0.223233i
$154$	−16660.0	−0.0566074
$155$	0	0
$156$	126666.	0.416724
$157$	178486.i	0.577903i	0.957344	+	0.288952i	$0.0933067\pi$
$157$	178486.i	0.577903i	−0.957344	+	0.288952i	$0.906693\pi$
$158$	2272.00i	0.00724045i
$159$	1350.00	0.00423488
$160$	0	0
$161$	156408.	0.475548
$162$	6561.00i	0.0196419i
$163$	− 252772.i	− 0.745178i	−0.927996	−	0.372589i	$-0.878470\pi$
$163$	− 252772.i	− 0.745178i	0.927996	−	0.372589i	$-0.121530\pi$
$164$	614854.	1.78510
$165$	0	0
$166$	−37764.0	−0.106367
$167$	508088.i	1.40977i	0.709322	+	0.704884i	$0.249001\pi$
$167$	508088.i	1.40977i	−0.709322	+	0.704884i	$0.750999\pi$
$168$	27783.0i	0.0759462i
$169$	165177.	0.444870
$170$	0	0
$171$	72252.0	0.188956
$172$	534316.i	1.37714i
$173$	221834.i	0.563525i	0.959484	+	0.281762i	$0.0909190\pi$
$173$	221834.i	0.563525i	−0.959484	+	0.281762i	$0.909081\pi$
$174$	−74178.0	−0.185739
$175$	0	0
$176$	−315860.	−0.768622
$177$	381564.i	0.915449i
$178$	117286.i	0.277457i
$179$	113564.	0.264916	0.132458	−	0.991189i	$-0.457713\pi$
$179$	113564.	0.264916	0.132458	+	0.991189i	$0.457713\pi$
$180$	0	0
$181$	663118.	1.50451	0.752254	−	0.658873i	$-0.228967\pi$
$181$	663118.	1.50451	0.752254	+	0.658873i	$0.228967\pi$
$182$	− 22246.0i	− 0.0497821i
$183$	132822.i	0.293185i
$184$	−201096.	−0.437884
$185$	0	0
$186$	22464.0	0.0476107
$187$	271320.i	0.567385i
$188$	276768.i	0.571112i
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	505664.	1.00295	0.501474	−	0.865173i	$-0.332791\pi$
$191$	505664.	1.00295	0.501474	+	0.865173i	$0.332791\pi$
$192$	241047.i	0.471899i
$193$	432382.i	0.835554i	0.908550	+	0.417777i	$0.137191\pi$
$193$	432382.i	0.835554i	−0.908550	+	0.417777i	$0.862809\pi$
$194$	−10002.0	−0.0190802
$195$	0	0
$196$	−74431.0	−0.138393
$197$	− 131962.i	− 0.242261i	−0.992637	−	0.121130i	$-0.961348\pi$
$197$	− 131962.i	− 0.242261i	0.992637	−	0.121130i	$-0.0386519\pi$
$198$	27540.0i	0.0499230i
$199$	−298536.	−0.534397	−0.267199	−	0.963642i	$-0.586098\pi$
$199$	−298536.	−0.534397	−0.267199	+	0.963642i	$0.586098\pi$
$200$	0	0
$201$	15084.0	0.0263346
$202$	− 108770.i	− 0.187556i
$203$	− 403858.i	− 0.687842i
$204$	222642.	0.374569
$205$	0	0
$206$	−199192.	−0.327042
$207$	− 258552.i	− 0.419394i
$208$	− 421766.i	− 0.675948i
$209$	303280.	0.480262
$210$	0	0
$211$	−1.17062e6	−1.81013	−0.905065	−	0.425273i	$-0.860178\pi$
$211$	−1.17062e6	−1.81013	−0.905065	+	0.425273i	$0.860178\pi$
$212$	− 4650.00i	− 0.00710581i
$213$	131112.i	0.198013i
$214$	79972.0	0.119372
$215$	0	0
$216$	45927.0	0.0669782
$217$	122304.i	0.176316i
$218$	46098.0i	0.0656963i
$219$	705402.	0.993863
$220$	0	0
$221$	−362292.	−0.498974
$222$	− 88182.0i	− 0.120087i
$223$	− 399376.i	− 0.537799i	−0.963168	−	0.268899i	$-0.913340\pi$
$223$	− 399376.i	− 0.537799i	0.963168	−	0.268899i	$-0.0866599\pi$
$224$	144305.	0.192159
$225$	0	0
$226$	262706.	0.342136
$227$	707916.i	0.911837i	0.890022	+	0.455918i	$0.150689\pi$
$227$	707916.i	0.911837i	−0.890022	+	0.455918i	$0.849311\pi$
$228$	− 248868.i	− 0.317053i
$229$	735778.	0.927167	0.463584	−	0.886053i	$-0.346563\pi$
$229$	735778.	0.927167	0.463584	+	0.886053i	$0.346563\pi$
$230$	0	0
$231$	−149940.	−0.184879
$232$	519246.i	0.633364i
$233$	208758.i	0.251915i	0.992036	+	0.125957i	$0.0402002\pi$
$233$	208758.i	0.251915i	−0.992036	+	0.125957i	$0.959800\pi$
$234$	−36774.0	−0.0439037
$235$	0	0
$236$	1.31428e6	1.53605
$237$	20448.0i	0.0236472i
$238$	− 39102.0i	− 0.0447462i
$239$	−713376.	−0.807837	−0.403919	−	0.914795i	$-0.632352\pi$
$239$	−713376.	−0.807837	−0.403919	+	0.914795i	$0.632352\pi$
$240$	0	0
$241$	−505246.	−0.560351	−0.280176	−	0.959949i	$-0.590393\pi$
$241$	−505246.	−0.560351	−0.280176	+	0.959949i	$0.590393\pi$
$242$	− 45451.0i	− 0.0498890i
$243$	59049.0i	0.0641500i
$244$	457498.	0.491943
$245$	0	0
$246$	−178506.	−0.188068
$247$	404968.i	0.422356i
$248$	− 157248.i	− 0.162351i
$249$	−339876.	−0.347394
$250$	0	0
$251$	317108.	0.317704	0.158852	−	0.987302i	$-0.449221\pi$
$251$	317108.	0.317704	0.158852	+	0.987302i	$0.449221\pi$
$252$	123039.i	0.122051i
$253$	− 1.08528e6i	− 1.06596i
$254$	−196608.	−0.191213
$255$	0	0
$256$	736033.	0.701936
$257$	− 1.44285e6i	− 1.36266i	−0.731977	−	0.681329i	$-0.761402\pi$
$257$	− 1.44285e6i	− 1.36266i	0.731977	−	0.681329i	$-0.238598\pi$
$258$	− 155124.i	− 0.145087i
$259$	480102.	0.444717
$260$	0	0
$261$	−667602.	−0.606619
$262$	− 77140.0i	− 0.0694267i
$263$	− 271496.i	− 0.242033i	−0.992651	−	0.121016i	$-0.961385\pi$
$263$	− 271496.i	− 0.242033i	0.992651	−	0.121016i	$-0.0386153\pi$
$264$	192780.	0.170236
$265$	0	0
$266$	−43708.0	−0.0378754
$267$	1.05557e6i	0.906172i
$268$	− 51956.0i	− 0.0441874i
$269$	−850614.	−0.716724	−0.358362	−	0.933583i	$-0.616665\pi$
$269$	−850614.	−0.716724	−0.358362	+	0.933583i	$0.616665\pi$
$270$	0	0
$271$	−540128.	−0.446759	−0.223380	−	0.974732i	$-0.571709\pi$
$271$	−540128.	−0.446759	−0.223380	+	0.974732i	$0.571709\pi$
$272$	− 741342.i	− 0.607570i
$273$	− 200214.i	− 0.162588i
$274$	−208170.	−0.167510
$275$	0	0
$276$	−890568.	−0.703711
$277$	513574.i	0.402164i	0.979574	+	0.201082i	$0.0644458\pi$
$277$	513574.i	0.402164i	−0.979574	+	0.201082i	$0.935554\pi$
$278$	275580.i	0.213863i
$279$	202176.	0.155496
$280$	0	0
$281$	−1.35642e6	−1.02478	−0.512388	−	0.858754i	$-0.671239\pi$
$281$	−1.35642e6	−1.02478	−0.512388	+	0.858754i	$0.671239\pi$
$282$	− 80352.0i	− 0.0601692i
$283$	− 286756.i	− 0.212837i	−0.994321	−	0.106418i	$-0.966062\pi$
$283$	− 286756.i	− 0.212837i	0.994321	−	0.106418i	$-0.0339383\pi$
$284$	451608.	0.332251
$285$	0	0
$286$	−154360.	−0.111589
$287$	− 971866.i	− 0.696469i
$288$	− 238545.i	− 0.169469i
$289$	783053.	0.551501
$290$	0	0
$291$	−90018.0	−0.0623156
$292$	− 2.42972e6i	− 1.66763i
$293$	1.70727e6i	1.16180i	0.813974	+	0.580901i	$0.197300\pi$
$293$	1.70727e6i	1.16180i	−0.813974	+	0.580901i	$0.802700\pi$
$294$	21609.0	0.0145803
$295$	0	0
$296$	−617274.	−0.409495
$297$	247860.i	0.163048i
$298$	296106.i	0.193155i
$299$	1.44917e6	0.937434
$300$	0	0
$301$	844564.	0.537299
$302$	− 426472.i	− 0.269075i
$303$	− 978930.i	− 0.612555i
$304$	−828668.	−0.514276
$305$	0	0
$306$	−64638.0	−0.0394625
$307$	− 546788.i	− 0.331111i	−0.986201	−	0.165555i	$-0.947058\pi$
$307$	− 546788.i	− 0.331111i	0.986201	−	0.165555i	$-0.0529416\pi$
$308$	516460.i	0.310213i
$309$	−1.79273e6	−1.06812
$310$	0	0
$311$	3.23426e6	1.89616	0.948079	−	0.318035i	$-0.103023\pi$
$311$	3.23426e6	1.89616	0.948079	+	0.318035i	$0.103023\pi$
$312$	257418.i	0.149711i
$313$	− 1.81313e6i	− 1.04609i	−0.852306	−	0.523044i	$-0.824796\pi$
$313$	− 1.81313e6i	− 1.04609i	0.852306	−	0.523044i	$-0.175204\pi$
$314$	−178486.	−0.102160
$315$	0	0
$316$	70432.0	0.0396782
$317$	− 1.27658e6i	− 0.713509i	−0.934198	−	0.356754i	$-0.883883\pi$
$317$	− 1.27658e6i	− 0.713509i	0.934198	−	0.356754i	$-0.116117\pi$
$318$	1350.00i	0 0.000748628i
$319$	−2.80228e6	−1.54182
$320$	0	0
$321$	719748.	0.389868
$322$	156408.i	0.0840658i
$323$	711816.i	0.379631i
$324$	203391.	0.107639
$325$	0	0
$326$	252772.	0.131730
$327$	414882.i	0.214563i
$328$	1.24954e6i	0.641307i
$329$	437472.	0.222823
$330$	0	0
$331$	−1.73621e6	−0.871029	−0.435515	−	0.900182i	$-0.643434\pi$
$331$	−1.73621e6	−0.871029	−0.435515	+	0.900182i	$0.643434\pi$
$332$	1.17068e6i	0.582901i
$333$	− 793638.i	− 0.392204i
$334$	−508088.	−0.249214
$335$	0	0
$336$	409689.	0.197973
$337$	2.07215e6i	0.993907i	0.867777	+	0.496953i	$0.165548\pi$
$337$	2.07215e6i	0.993907i	−0.867777	+	0.496953i	$0.834452\pi$
$338$	165177.i	0.0786426i
$339$	2.36435e6	1.11741
$340$	0	0
$341$	848640.	0.395219
$342$	72252.0i	0.0334029i
$343$	117649.i	0.0539949i
$344$	−1.08587e6	−0.494744
$345$	0	0
$346$	−221834.	−0.0996180
$347$	− 1.65146e6i	− 0.736282i	−0.929770	−	0.368141i	$-0.879994\pi$
$347$	− 1.65146e6i	− 0.736282i	0.929770	−	0.368141i	$-0.120006\pi$
$348$	2.29952e6i	1.01786i
$349$	−1.26645e6	−0.556578	−0.278289	−	0.960497i	$-0.589767\pi$
$349$	−1.26645e6	−0.556578	−0.278289	+	0.960497i	$0.589767\pi$
$350$	0	0
$351$	−330966.	−0.143389
$352$	− 1.00130e6i	− 0.430732i
$353$	− 573218.i	− 0.244840i	−0.992478	−	0.122420i	$-0.960934\pi$
$353$	− 573218.i	− 0.244840i	0.992478	−	0.122420i	$-0.0390656\pi$
$354$	−381564.	−0.161830
$355$	0	0
$356$	3.63587e6	1.52049
$357$	− 351918.i	− 0.146141i
$358$	113564.i	0.0468310i
$359$	−4.46322e6	−1.82773	−0.913866	−	0.406016i	$-0.866918\pi$
$359$	−4.46322e6	−1.82773	−0.913866	+	0.406016i	$0.866918\pi$
$360$	0	0
$361$	−1.68044e6	−0.678662
$362$	663118.i	0.265962i
$363$	− 409059.i	− 0.162937i
$364$	−689626.	−0.272810
$365$	0	0
$366$	−132822.	−0.0518283
$367$	− 4.50797e6i	− 1.74709i	−0.486742	−	0.873546i	$-0.661815\pi$
$367$	− 4.50797e6i	− 1.74709i	0.486742	−	0.873546i	$-0.338185\pi$
$368$	2.96537e6i	1.14146i
$369$	−1.60655e6	−0.614228
$370$	0	0
$371$	−7350.00	−0.00277238
$372$	− 696384.i	− 0.260910i
$373$	− 1.66535e6i	− 0.619774i	−0.950773	−	0.309887i	$-0.899709\pi$
$373$	− 1.66535e6i	− 0.619774i	0.950773	−	0.309887i	$-0.100291\pi$
$374$	−271320.	−0.100300
$375$	0	0
$376$	−562464.	−0.205175
$377$	− 3.74187e6i	− 1.35592i
$378$	− 35721.0i	− 0.0128586i
$379$	2.53232e6	0.905568	0.452784	−	0.891620i	$-0.350431\pi$
$379$	2.53232e6	0.905568	0.452784	+	0.891620i	$0.350431\pi$
$380$	0	0
$381$	−1.76947e6	−0.624498
$382$	505664.i	0.177298i
$383$	− 796368.i	− 0.277407i	−0.990334	−	0.138703i	$-0.955707\pi$
$383$	− 796368.i	− 0.277407i	0.990334	−	0.138703i	$-0.0442934\pi$
$384$	−1.08921e6	−0.376949
$385$	0	0
$386$	−432382.	−0.147706
$387$	− 1.39612e6i	− 0.473853i
$388$	310062.i	0.104561i
$389$	−1.94799e6	−0.652699	−0.326349	−	0.945249i	$-0.605819\pi$
$389$	−1.94799e6	−0.652699	−0.326349	+	0.945249i	$0.605819\pi$
$390$	0	0
$391$	2.54722e6	0.842605
$392$	− 151263.i	− 0.0497184i
$393$	− 694260.i	− 0.226747i
$394$	131962.	0.0428261
$395$	0	0
$396$	853740.	0.273582
$397$	1.08116e6i	0.344281i	0.985072	+	0.172140i	$0.0550683\pi$
$397$	1.08116e6i	0.344281i	−0.985072	+	0.172140i	$0.944932\pi$
$398$	− 298536.i	− 0.0944689i
$399$	−393372.	−0.123700
$400$	0	0
$401$	2.76770e6	0.859524	0.429762	−	0.902942i	$-0.358598\pi$
$401$	2.76770e6	0.859524	0.429762	+	0.902942i	$0.358598\pi$
$402$	15084.0i	0.00465534i
$403$	1.13318e6i	0.347566i
$404$	−3.37187e6	−1.02782
$405$	0	0
$406$	403858.	0.121594
$407$	− 3.33132e6i	− 0.996851i
$408$	452466.i	0.134566i
$409$	−2.36350e6	−0.698630	−0.349315	−	0.937005i	$-0.613586\pi$
$409$	−2.36350e6	−0.698630	−0.349315	+	0.937005i	$0.613586\pi$
$410$	0	0
$411$	−1.87353e6	−0.547087
$412$	6.17495e6i	1.79222i
$413$	− 2.07740e6i	− 0.599302i
$414$	258552.	0.0741391
$415$	0	0
$416$	1.33703e6	0.378798
$417$	2.48022e6i	0.698474i
$418$	303280.i	0.0848991i
$419$	2.98669e6	0.831104	0.415552	−	0.909569i	$-0.363588\pi$
$419$	2.98669e6	0.831104	0.415552	+	0.909569i	$0.363588\pi$
$420$	0	0
$421$	−3.46331e6	−0.952326	−0.476163	−	0.879357i	$-0.657973\pi$
$421$	−3.46331e6	−0.952326	−0.476163	+	0.879357i	$0.657973\pi$
$422$	− 1.17062e6i	− 0.319989i
$423$	− 723168.i	− 0.196512i
$424$	9450.00	0.00255280
$425$	0	0
$426$	−131112.	−0.0350041
$427$	− 723142.i	− 0.191935i
$428$	− 2.47913e6i	− 0.654169i
$429$	−1.38924e6	−0.364447
$430$	0	0
$431$	2.33693e6	0.605971	0.302986	−	0.952995i	$-0.402017\pi$
$431$	2.33693e6	0.605971	0.302986	+	0.952995i	$0.402017\pi$
$432$	− 677241.i	− 0.174596i
$433$	3.50838e6i	0.899264i	0.893214	+	0.449632i	$0.148445\pi$
$433$	3.50838e6i	0.899264i	−0.893214	+	0.449632i	$0.851555\pi$
$434$	−122304.	−0.0311685
$435$	0	0
$436$	1.42904e6	0.360021
$437$	− 2.84726e6i	− 0.713221i
$438$	705402.i	0.175692i
$439$	−3.54833e6	−0.878744	−0.439372	−	0.898305i	$-0.644799\pi$
$439$	−3.54833e6	−0.878744	−0.439372	+	0.898305i	$0.644799\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	− 362292.i	− 0.0882070i
$443$	− 1.76833e6i	− 0.428109i	−0.976822	−	0.214055i	$-0.931333\pi$
$443$	− 1.76833e6i	− 0.428109i	0.976822	−	0.214055i	$-0.0686670\pi$
$444$	−2.73364e6	−0.658088
$445$	0	0
$446$	399376.	0.0950703
$447$	2.66495e6i	0.630842i
$448$	− 1.31237e6i	− 0.308930i
$449$	5.52579e6	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$449$	5.52579e6	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$450$	0	0
$451$	−6.74356e6	−1.56116
$452$	− 8.14389e6i	− 1.87493i
$453$	− 3.83825e6i	− 0.878795i
$454$	−707916.	−0.161191
$455$	0	0
$456$	505764.	0.113903
$457$	− 2.96226e6i	− 0.663488i	−0.943369	−	0.331744i	$-0.892363\pi$
$457$	− 2.96226e6i	− 0.663488i	0.943369	−	0.331744i	$-0.107637\pi$
$458$	735778.i	0.163902i
$459$	−581742.	−0.128884
$460$	0	0
$461$	2.11884e6	0.464350	0.232175	−	0.972674i	$-0.425416\pi$
$461$	2.11884e6	0.464350	0.232175	+	0.972674i	$0.425416\pi$
$462$	− 149940.i	− 0.0326823i
$463$	− 3.19226e6i	− 0.692062i	−0.938223	−	0.346031i	$-0.887529\pi$
$463$	− 3.19226e6i	− 0.692062i	0.938223	−	0.346031i	$-0.112471\pi$
$464$	7.65682e6	1.65102
$465$	0	0
$466$	−208758.	−0.0445326
$467$	− 7.42621e6i	− 1.57571i	−0.615863	−	0.787853i	$-0.711193\pi$
$467$	− 7.42621e6i	− 1.57571i	0.615863	−	0.787853i	$-0.288807\pi$
$468$	1.13999e6i	0.240596i
$469$	−82124.0	−0.0172400
$470$	0	0
$471$	−1.60637e6	−0.333653
$472$	2.67095e6i	0.551837i
$473$	− 5.86024e6i	− 1.20438i
$474$	−20448.0	−0.00418028
$475$	0	0
$476$	−1.21216e6	−0.245213
$477$	12150.0i	0.00244501i
$478$	− 713376.i	− 0.142807i
$479$	3.39685e6	0.676453	0.338226	−	0.941065i	$-0.390173\pi$
$479$	3.39685e6	0.676453	0.338226	+	0.941065i	$0.390173\pi$
$480$	0	0
$481$	4.44829e6	0.876659
$482$	− 505246.i	− 0.0990570i
$483$	1.40767e6i	0.274558i
$484$	−1.40898e6	−0.273396
$485$	0	0
$486$	−59049.0	−0.0113402
$487$	− 3.71382e6i	− 0.709574i	−0.934947	−	0.354787i	$-0.884553\pi$
$487$	− 3.71382e6i	− 0.709574i	0.934947	−	0.354787i	$-0.115447\pi$
$488$	929754.i	0.176733i
$489$	2.27495e6	0.430229
$490$	0	0
$491$	5.57494e6	1.04361	0.521803	−	0.853066i	$-0.325260\pi$
$491$	5.57494e6	1.04361	0.521803	+	0.853066i	$0.325260\pi$
$492$	5.53369e6i	1.03063i
$493$	− 6.57712e6i	− 1.21876i
$494$	−404968.	−0.0746626
$495$	0	0
$496$	−2.31878e6	−0.423210
$497$	− 713832.i	− 0.129630i
$498$	− 339876.i	− 0.0614111i
$499$	−3.92698e6	−0.706004	−0.353002	−	0.935623i	$-0.614839\pi$
$499$	−3.92698e6	−0.706004	−0.353002	+	0.935623i	$0.614839\pi$
$500$	0	0
$501$	−4.57279e6	−0.813930
$502$	317108.i	0.0561627i
$503$	− 6.42079e6i	− 1.13154i	−0.824564	−	0.565768i	$-0.808580\pi$
$503$	− 6.42079e6i	− 1.13154i	0.824564	−	0.565768i	$-0.191420\pi$
$504$	−250047.	−0.0438475
$505$	0	0
$506$	1.08528e6	0.188437
$507$	1.48659e6i	0.256846i
$508$	6.09485e6i	1.04786i
$509$	−146278.	−0.0250256	−0.0125128	−	0.999922i	$-0.503983\pi$
$509$	−146278.	−0.0250256	−0.0125128	+	0.999922i	$0.503983\pi$
$510$	0	0
$511$	−3.84052e6	−0.650636
$512$	4.60877e6i	0.776980i
$513$	650268.i	0.109094i
$514$	1.44285e6	0.240886
$515$	0	0
$516$	−4.80884e6	−0.795090
$517$	− 3.03552e6i	− 0.499467i
$518$	480102.i	0.0786157i
$519$	−1.99651e6	−0.325351
$520$	0	0
$521$	7.70937e6	1.24430	0.622149	−	0.782899i	$-0.286260\pi$
$521$	7.70937e6	1.24430	0.622149	+	0.782899i	$0.286260\pi$
$522$	− 667602.i	− 0.107236i
$523$	569420.i	0.0910287i	0.998964	+	0.0455144i	$0.0144927\pi$
$523$	569420.i	0.0910287i	−0.998964	+	0.0455144i	$0.985507\pi$
$524$	−2.39134e6	−0.380464
$525$	0	0
$526$	271496.	0.0427857
$527$	1.99181e6i	0.312407i
$528$	− 2.84274e6i	− 0.443764i
$529$	−3.75252e6	−0.583021
$530$	0	0
$531$	−3.43408e6	−0.528535
$532$	1.35495e6i	0.207560i
$533$	− 9.00464e6i	− 1.37293i
$534$	−1.05557e6	−0.160190
$535$	0	0
$536$	105588.	0.0158746
$537$	1.02208e6i	0.152949i
$538$	− 850614.i	− 0.126700i
$539$	816340.	0.121032
$540$	0	0
$541$	−9.44802e6	−1.38787	−0.693933	−	0.720040i	$-0.744124\pi$
$541$	−9.44802e6	−1.38787	−0.693933	+	0.720040i	$0.744124\pi$
$542$	− 540128.i	− 0.0789766i
$543$	5.96806e6i	0.868628i
$544$	2.35011e6	0.340479
$545$	0	0
$546$	200214.	0.0287417
$547$	− 1.35321e6i	− 0.193374i	−0.995315	−	0.0966869i	$-0.969175\pi$
$547$	− 1.35321e6i	− 0.193374i	0.995315	−	0.0966869i	$-0.0308245\pi$
$548$	6.45327e6i	0.917970i
$549$	−1.19540e6	−0.169271
$550$	0	0
$551$	−7.35186e6	−1.03162
$552$	− 1.80986e6i	− 0.252812i
$553$	− 111328.i	− 0.0154807i
$554$	−513574.	−0.0710933
$555$	0	0
$556$	8.54298e6	1.17199
$557$	8.19390e6i	1.11906i	0.828811	+	0.559529i	$0.189018\pi$
$557$	8.19390e6i	1.11906i	−0.828811	+	0.559529i	$0.810982\pi$
$558$	202176.i	0.0274881i
$559$	7.82514e6	1.05916
$560$	0	0
$561$	−2.44188e6	−0.327580
$562$	− 1.35642e6i	− 0.181157i
$563$	1.05796e7i	1.40669i	0.710847	+	0.703347i	$0.248312\pi$
$563$	1.05796e7i	1.40669i	−0.710847	+	0.703347i	$0.751688\pi$
$564$	−2.49091e6	−0.329732
$565$	0	0
$566$	286756.	0.0376246
$567$	− 321489.i	− 0.0419961i
$568$	917784.i	0.119363i
$569$	1.20205e7	1.55648	0.778238	−	0.627969i	$-0.216114\pi$
$569$	1.20205e7	1.55648	0.778238	+	0.627969i	$0.216114\pi$
$570$	0	0
$571$	−2.48948e6	−0.319534	−0.159767	−	0.987155i	$-0.551074\pi$
$571$	−2.48948e6	−0.319534	−0.159767	+	0.987155i	$0.551074\pi$
$572$	4.78516e6i	0.611514i
$573$	4.55098e6i	0.579053i
$574$	971866.	0.123119
$575$	0	0
$576$	−2.16942e6	−0.272451
$577$	8.21322e6i	1.02701i	0.858087	+	0.513504i	$0.171653\pi$
$577$	8.21322e6i	1.02701i	−0.858087	+	0.513504i	$0.828347\pi$
$578$	783053.i	0.0974926i
$579$	−3.89144e6	−0.482407
$580$	0	0
$581$	1.85044e6	0.227423
$582$	− 90018.0i	− 0.0110159i
$583$	51000.0i	0.00621439i
$584$	4.93781e6	0.599105
$585$	0	0
$586$	−1.70727e6	−0.205380
$587$	− 1.21827e6i	− 0.145931i	−0.997334	−	0.0729655i	$-0.976754\pi$
$587$	− 1.21827e6i	− 0.145931i	0.997334	−	0.0729655i	$-0.0232463\pi$
$588$	− 669879.i	− 0.0799012i
$589$	2.22643e6	0.264436
$590$	0	0
$591$	1.18766e6	0.139869
$592$	9.10234e6i	1.06745i
$593$	8.42379e6i	0.983718i	0.870675	+	0.491859i	$0.163683\pi$
$593$	8.42379e6i	0.983718i	−0.870675	+	0.491859i	$0.836317\pi$
$594$	−247860.	−0.0288231
$595$	0	0
$596$	9.17929e6	1.05851
$597$	− 2.68682e6i	− 0.308534i
$598$	1.44917e6i	0.165717i
$599$	−8.21254e6	−0.935212	−0.467606	−	0.883937i	$-0.654883\pi$
$599$	−8.21254e6	−0.935212	−0.467606	+	0.883937i	$0.654883\pi$
$600$	0	0
$601$	3.25478e6	0.367566	0.183783	−	0.982967i	$-0.441166\pi$
$601$	3.25478e6	0.367566	0.183783	+	0.982967i	$0.441166\pi$
$602$	844564.i	0.0949820i
$603$	135756.i	0.0152043i
$604$	−1.32206e7	−1.47455
$605$	0	0
$606$	978930.	0.108285
$607$	7.82101e6i	0.861571i	0.902454	+	0.430785i	$0.141763\pi$
$607$	7.82101e6i	0.861571i	−0.902454	+	0.430785i	$0.858237\pi$
$608$	− 2.62694e6i	− 0.288198i
$609$	3.63472e6	0.397126
$610$	0	0
$611$	4.05331e6	0.439245
$612$	2.00378e6i	0.216257i
$613$	9.51670e6i	1.02290i	0.859312	+	0.511452i	$0.170892\pi$
$613$	9.51670e6i	1.02290i	−0.859312	+	0.511452i	$0.829108\pi$
$614$	546788.	0.0585326
$615$	0	0
$616$	−1.04958e6	−0.111446
$617$	− 7.04895e6i	− 0.745438i	−0.927944	−	0.372719i	$-0.878426\pi$
$617$	− 7.04895e6i	− 0.745438i	0.927944	−	0.372719i	$-0.121574\pi$
$618$	− 1.79273e6i	− 0.188818i
$619$	6.32174e6	0.663147	0.331574	−	0.943429i	$-0.392420\pi$
$619$	6.32174e6	0.663147	0.331574	+	0.943429i	$0.392420\pi$
$620$	0	0
$621$	2.32697e6	0.242137
$622$	3.23426e6i	0.335197i
$623$	− 5.74701e6i	− 0.593229i
$624$	3.79589e6	0.390259
$625$	0	0
$626$	1.81313e6	0.184924
$627$	2.72952e6i	0.277279i
$628$	5.53307e6i	0.559844i
$629$	7.81880e6	0.787977
$630$	0	0
$631$	8.61236e6	0.861090	0.430545	−	0.902569i	$-0.358321\pi$
$631$	8.61236e6	0.861090	0.430545	+	0.902569i	$0.358321\pi$
$632$	143136.i	0.0142546i
$633$	− 1.05356e7i	− 1.04508i
$634$	1.27658e6	0.126132
$635$	0	0
$636$	41850.0	0.00410254
$637$	1.09005e6i	0.106439i
$638$	− 2.80228e6i	− 0.272559i
$639$	−1.18001e6	−0.114323
$640$	0	0
$641$	−5.22829e6	−0.502590	−0.251295	−	0.967910i	$-0.580857\pi$
$641$	−5.22829e6	−0.502590	−0.251295	+	0.967910i	$0.580857\pi$
$642$	719748.i	0.0689196i
$643$	− 1.61373e7i	− 1.53923i	−0.638508	−	0.769615i	$-0.720448\pi$
$643$	− 1.61373e7i	− 1.53923i	0.638508	−	0.769615i	$-0.279552\pi$
$644$	4.84865e6	0.460687
$645$	0	0
$646$	−711816.	−0.0671099
$647$	− 1.58749e7i	− 1.49090i	−0.666560	−	0.745451i	$-0.732234\pi$
$647$	− 1.58749e7i	− 1.49090i	0.666560	−	0.745451i	$-0.267766\pi$
$648$	413343.i	0.0386699i
$649$	−1.44146e7	−1.34336
$650$	0	0
$651$	−1.10074e6	−0.101796
$652$	− 7.83593e6i	− 0.721891i
$653$	5.94112e6i	0.545237i	0.962122	+	0.272619i	$0.0878897\pi$
$653$	5.94112e6i	0.545237i	−0.962122	+	0.272619i	$0.912110\pi$
$654$	−414882.	−0.0379298
$655$	0	0
$656$	1.84258e7	1.67173
$657$	6.34862e6i	0.573807i
$658$	437472.i	0.0393900i
$659$	7.64430e6	0.685684	0.342842	−	0.939393i	$-0.388610\pi$
$659$	7.64430e6	0.685684	0.342842	+	0.939393i	$0.388610\pi$
$660$	0	0
$661$	−7.58688e6	−0.675398	−0.337699	−	0.941254i	$-0.609649\pi$
$661$	−7.58688e6	−0.675398	−0.337699	+	0.941254i	$0.609649\pi$
$662$	− 1.73621e6i	− 0.153978i
$663$	− 3.26063e6i	− 0.288083i
$664$	−2.37913e6	−0.209410
$665$	0	0
$666$	793638.	0.0693325
$667$	2.63085e7i	2.28971i
$668$	1.57507e7i	1.36571i
$669$	3.59438e6	0.310498
$670$	0	0
$671$	−5.01772e6	−0.430229
$672$	1.29874e6i	0.110943i
$673$	2.06681e7i	1.75899i	0.475910	+	0.879494i	$0.342119\pi$
$673$	2.06681e7i	1.75899i	−0.475910	+	0.879494i	$0.657881\pi$
$674$	−2.07215e6	−0.175700
$675$	0	0
$676$	5.12049e6	0.430968
$677$	7.89541e6i	0.662068i	0.943619	+	0.331034i	$0.107398\pi$
$677$	7.89541e6i	0.662068i	−0.943619	+	0.331034i	$0.892602\pi$
$678$	2.36435e6i	0.197532i
$679$	490098.	0.0407951
$680$	0	0
$681$	−6.37124e6	−0.526449
$682$	848640.i	0.0698655i
$683$	1.96015e7i	1.60782i	0.594750	+	0.803911i	$0.297251\pi$
$683$	1.96015e7i	1.60782i	−0.594750	+	0.803911i	$0.702749\pi$
$684$	2.23981e6	0.183051
$685$	0	0
$686$	−117649.	−0.00954504
$687$	6.62200e6i	0.535300i
$688$	1.60122e7i	1.28968i
$689$	−68100.0	−0.00546511
$690$	0	0
$691$	−1.72710e7	−1.37601	−0.688005	−	0.725706i	$-0.741513\pi$
$691$	−1.72710e7	−1.37601	−0.688005	+	0.725706i	$0.741513\pi$
$692$	6.87685e6i	0.545914i
$693$	− 1.34946e6i	− 0.106740i
$694$	1.65146e6	0.130158
$695$	0	0
$696$	−4.67321e6	−0.365673
$697$	− 1.58275e7i	− 1.23405i
$698$	− 1.26645e6i	− 0.0983900i
$699$	−1.87882e6	−0.145443
$700$	0	0
$701$	−5.36344e6	−0.412238	−0.206119	−	0.978527i	$-0.566083\pi$
$701$	−5.36344e6	−0.412238	−0.206119	+	0.978527i	$0.566083\pi$
$702$	− 330966.i	− 0.0253478i
$703$	− 8.73982e6i	− 0.666982i
$704$	−9.10622e6	−0.692479
$705$	0	0
$706$	573218.	0.0432821
$707$	5.32973e6i	0.401011i
$708$	1.18285e7i	0.886841i
$709$	1.73733e7	1.29798	0.648988	−	0.760798i	$-0.275192\pi$
$709$	1.73733e7	1.29798	0.648988	+	0.760798i	$0.275192\pi$
$710$	0	0
$711$	−184032.	−0.0136527
$712$	7.38902e6i	0.546244i
$713$	− 7.96723e6i	− 0.586926i
$714$	351918.	0.0258343
$715$	0	0
$716$	3.52048e6	0.256637
$717$	− 6.42038e6i	− 0.466405i
$718$	− 4.46322e6i	− 0.323100i
$719$	−424608.	−0.0306313	−0.0153157	−	0.999883i	$-0.504875\pi$
$719$	−424608.	−0.0306313	−0.0153157	+	0.999883i	$0.504875\pi$
$720$	0	0
$721$	9.76041e6	0.699246
$722$	− 1.68044e6i	− 0.119972i
$723$	− 4.54721e6i	− 0.323519i
$724$	2.05567e7	1.45749
$725$	0	0
$726$	409059.	0.0288034
$727$	2.18290e7i	1.53179i	0.642968	+	0.765893i	$0.277703\pi$
$727$	2.18290e7i	1.53179i	−0.642968	+	0.765893i	$0.722297\pi$
$728$	− 1.40150e6i	− 0.0980086i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.37543e7	0.952020
$732$	4.11748e6i	0.284023i
$733$	− 2.17675e7i	− 1.49640i	−0.663470	−	0.748202i	$-0.730917\pi$
$733$	− 2.17675e7i	− 1.49640i	0.663470	−	0.748202i	$-0.269083\pi$
$734$	4.50797e6	0.308845
$735$	0	0
$736$	−9.40044e6	−0.639667
$737$	569840.i	0.0386442i
$738$	− 1.60655e6i	− 0.108581i
$739$	−6.21786e6	−0.418822	−0.209411	−	0.977828i	$-0.567155\pi$
$739$	−6.21786e6	−0.418822	−0.209411	+	0.977828i	$0.567155\pi$
$740$	0	0
$741$	−3.64471e6	−0.243847
$742$	− 7350.00i	0 0.000490092i
$743$	− 3.77647e6i	− 0.250966i	−0.992096	−	0.125483i	$-0.959952\pi$
$743$	− 3.77647e6i	− 0.250966i	0.992096	−	0.125483i	$-0.0400480\pi$
$744$	1.41523e6	0.0937336
$745$	0	0
$746$	1.66535e6	0.109562
$747$	− 3.05888e6i	− 0.200568i
$748$	8.41092e6i	0.549654i
$749$	−3.91863e6	−0.255229
$750$	0	0
$751$	−2.88795e6	−0.186849	−0.0934244	−	0.995626i	$-0.529781\pi$
$751$	−2.88795e6	−0.186849	−0.0934244	+	0.995626i	$0.529781\pi$
$752$	8.29411e6i	0.534842i
$753$	2.85397e6i	0.183427i
$754$	3.74187e6	0.239696
$755$	0	0
$756$	−1.10735e6	−0.0704662
$757$	1.25519e6i	0.0796104i	0.999207	+	0.0398052i	$0.0126737\pi$
$757$	1.25519e6i	0.0796104i	−0.999207	+	0.0398052i	$0.987326\pi$
$758$	2.53232e6i	0.160083i
$759$	9.76752e6	0.615432
$760$	0	0
$761$	−1.42623e7	−0.892746	−0.446373	−	0.894847i	$-0.647284\pi$
$761$	−1.42623e7	−0.892746	−0.446373	+	0.894847i	$0.647284\pi$
$762$	− 1.76947e6i	− 0.110397i
$763$	− 2.25880e6i	− 0.140465i
$764$	1.56756e7	0.971606
$765$	0	0
$766$	796368.	0.0490390
$767$	− 1.92478e7i	− 1.18139i
$768$	6.62430e6i	0.405263i
$769$	2.02261e7	1.23338	0.616689	−	0.787207i	$-0.288474\pi$
$769$	2.02261e7	1.23338	0.616689	+	0.787207i	$0.288474\pi$
$770$	0	0
$771$	1.29856e7	0.786732
$772$	1.34038e7i	0.809443i
$773$	− 2.62288e7i	− 1.57881i	−0.613872	−	0.789406i	$-0.710389\pi$
$773$	− 2.62288e7i	− 1.57881i	0.613872	−	0.789406i	$-0.289611\pi$
$774$	1.39612e6	0.0837663
$775$	0	0
$776$	−630126.	−0.0375641
$777$	4.32092e6i	0.256758i
$778$	− 1.94799e6i	− 0.115382i
$779$	−1.76919e7	−1.04456
$780$	0	0
$781$	−4.95312e6	−0.290570
$782$	2.54722e6i	0.148953i
$783$	− 6.00842e6i	− 0.350232i
$784$	−2.23053e6	−0.129604
$785$	0	0
$786$	694260.	0.0400835
$787$	− 9.92829e6i	− 0.571397i	−0.958320	−	0.285698i	$-0.907774\pi$
$787$	− 9.92829e6i	− 0.571397i	0.958320	−	0.285698i	$-0.0922255\pi$
$788$	− 4.09082e6i	− 0.234690i
$789$	2.44346e6	0.139738
$790$	0	0
$791$	−1.28726e7	−0.731518
$792$	1.73502e6i	0.0982860i
$793$	− 6.70013e6i	− 0.378356i
$794$	−1.08116e6	−0.0608608
$795$	0	0
$796$	−9.25462e6	−0.517697
$797$	1.09033e7i	0.608014i	0.952670	+	0.304007i	$0.0983246\pi$
$797$	1.09033e7i	0.608014i	−0.952670	+	0.304007i	$0.901675\pi$
$798$	− 393372.i	− 0.0218674i
$799$	7.12454e6	0.394812
$800$	0	0
$801$	−9.50017e6	−0.523179
$802$	2.76770e6i	0.151944i
$803$	2.66485e7i	1.45843i
$804$	467604.	0.0255116
$805$	0	0
$806$	−1.13318e6	−0.0614416
$807$	− 7.65553e6i	− 0.413801i
$808$	− 6.85251e6i	− 0.369251i
$809$	−6.06398e6	−0.325751	−0.162876	−	0.986647i	$-0.552077\pi$
$809$	−6.06398e6	−0.325751	−0.162876	+	0.986647i	$0.552077\pi$
$810$	0	0
$811$	−8.59438e6	−0.458841	−0.229421	−	0.973327i	$-0.573683\pi$
$811$	−8.59438e6	−0.458841	−0.229421	+	0.973327i	$0.573683\pi$
$812$	− 1.25196e7i	− 0.666347i
$813$	− 4.86115e6i	− 0.257937i
$814$	3.33132e6	0.176220
$815$	0	0
$816$	6.67208e6	0.350781
$817$	− 1.53745e7i	− 0.805835i
$818$	− 2.36350e6i	− 0.123501i
$819$	1.80193e6	0.0938701
$820$	0	0
$821$	−2.01396e6	−0.104278	−0.0521391	−	0.998640i	$-0.516604\pi$
$821$	−2.01396e6	−0.104278	−0.0521391	+	0.998640i	$0.516604\pi$
$822$	− 1.87353e6i	− 0.0967122i
$823$	2.64679e7i	1.36213i	0.732221	+	0.681067i	$0.238484\pi$
$823$	2.64679e7i	1.36213i	−0.732221	+	0.681067i	$0.761516\pi$
$824$	−1.25491e7	−0.643864
$825$	0	0
$826$	2.07740e6	0.105943
$827$	− 3.90229e6i	− 0.198407i	−0.995067	−	0.0992033i	$-0.968371\pi$
$827$	− 3.90229e6i	− 0.198407i	0.995067	−	0.0992033i	$-0.0316294\pi$
$828$	− 8.01511e6i	− 0.406288i
$829$	1.95595e7	0.988487	0.494244	−	0.869323i	$-0.335445\pi$
$829$	1.95595e7	0.988487	0.494244	+	0.869323i	$0.335445\pi$
$830$	0	0
$831$	−4.62217e6	−0.232190
$832$	− 1.21595e7i	− 0.608985i
$833$	1.91600e6i	0.0956715i
$834$	−2.48022e6	−0.123474
$835$	0	0
$836$	9.40168e6	0.465254
$837$	1.81958e6i	0.0897756i
$838$	2.98669e6i	0.146920i
$839$	2.45448e7	1.20380	0.601901	−	0.798570i	$-0.294410\pi$
$839$	2.45448e7	1.20380	0.601901	+	0.798570i	$0.294410\pi$
$840$	0	0
$841$	4.74194e7	2.31188
$842$	− 3.46331e6i	− 0.168349i
$843$	− 1.22078e7i	− 0.591655i
$844$	−3.62892e7	−1.75356
$845$	0	0
$846$	723168.	0.0347387
$847$	2.22710e6i	0.106667i
$848$	− 139350.i	− 0.00665453i
$849$	2.58080e6	0.122881
$850$	0	0
$851$	−3.12752e7	−1.48039
$852$	4.06447e6i	0.191825i
$853$	− 3.38305e7i	− 1.59197i	−0.605314	−	0.795987i	$-0.706952\pi$
$853$	− 3.38305e7i	− 1.59197i	0.605314	−	0.795987i	$-0.293048\pi$
$854$	723142.	0.0339296
$855$	0	0
$856$	5.03824e6	0.235014
$857$	3.18009e7i	1.47907i	0.673120	+	0.739534i	$0.264954\pi$
$857$	3.18009e7i	1.47907i	−0.673120	+	0.739534i	$0.735046\pi$
$858$	− 1.38924e6i	− 0.0644257i
$859$	−638420.	−0.0295205	−0.0147602	−	0.999891i	$-0.504699\pi$
$859$	−638420.	−0.0295205	−0.0147602	+	0.999891i	$0.504699\pi$
$860$	0	0
$861$	8.74679e6	0.402106
$862$	2.33693e6i	0.107122i
$863$	4.22256e6i	0.192996i	0.995333	+	0.0964981i	$0.0307642\pi$
$863$	4.22256e6i	0.192996i	−0.995333	+	0.0964981i	$0.969236\pi$
$864$	2.14690e6	0.0978427
$865$	0	0
$866$	−3.50838e6	−0.158969
$867$	7.04748e6i	0.318409i
$868$	3.79142e6i	0.170806i
$869$	−772480.	−0.0347007
$870$	0	0
$871$	−760904.	−0.0339848
$872$	2.90417e6i	0.129340i
$873$	− 810162.i	− 0.0359779i
$874$	2.84726e6	0.126081
$875$	0	0
$876$	2.18675e7	0.962805
$877$	− 2.45043e7i	− 1.07583i	−0.842999	−	0.537915i	$-0.819212\pi$
$877$	− 2.45043e7i	− 1.07583i	0.842999	−	0.537915i	$-0.180788\pi$
$878$	− 3.54833e6i	− 0.155341i
$879$	−1.53654e7	−0.670767
$880$	0	0
$881$	−2.77630e7	−1.20511	−0.602555	−	0.798078i	$-0.705850\pi$
$881$	−2.77630e7	−1.20511	−0.602555	+	0.798078i	$0.705850\pi$
$882$	194481.i	0.00841794i
$883$	− 3.30170e7i	− 1.42507i	−0.701638	−	0.712534i	$-0.747548\pi$
$883$	− 3.30170e7i	− 1.42507i	0.701638	−	0.712534i	$-0.252452\pi$
$884$	−1.12311e7	−0.483381
$885$	0	0
$886$	1.76833e6	0.0756797
$887$	4.34462e6i	0.185414i	0.995693	+	0.0927070i	$0.0295520\pi$
$887$	4.34462e6i	0.185414i	−0.995693	+	0.0927070i	$0.970448\pi$
$888$	− 5.55547e6i	− 0.236422i
$889$	9.63379e6	0.408830
$890$	0	0
$891$	−2.23074e6	−0.0941358
$892$	− 1.23807e7i	− 0.520993i
$893$	− 7.96378e6i	− 0.334188i
$894$	−2.66495e6	−0.111518
$895$	0	0
$896$	5.93013e6	0.246771
$897$	1.30425e7i	0.541228i
$898$	5.52579e6i	0.228667i
$899$	−2.05720e7	−0.848942
$900$	0	0
$901$	−119700.	−0.00491227
$902$	− 6.74356e6i	− 0.275977i
$903$	7.60108e6i	0.310210i
$904$	1.65505e7	0.673580
$905$	0	0
$906$	3.83825e6	0.155350
$907$	1.96499e7i	0.793128i	0.918007	+	0.396564i	$0.129797\pi$
$907$	1.96499e7i	0.793128i	−0.918007	+	0.396564i	$0.870203\pi$
$908$	2.19454e7i	0.883342i
$909$	8.81037e6	0.353659
$910$	0	0
$911$	−7.26518e6	−0.290035	−0.145018	−	0.989429i	$-0.546324\pi$
$911$	−7.26518e6	−0.290035	−0.145018	+	0.989429i	$0.546324\pi$
$912$	− 7.45801e6i	− 0.296918i
$913$	− 1.28398e7i	− 0.509777i
$914$	2.96226e6	0.117289
$915$	0	0
$916$	2.28091e7	0.898193
$917$	3.77986e6i	0.148440i
$918$	− 581742.i	− 0.0227837i
$919$	−9.82532e6	−0.383758	−0.191879	−	0.981419i	$-0.561458\pi$
$919$	−9.82532e6	−0.383758	−0.191879	+	0.981419i	$0.561458\pi$
$920$	0	0
$921$	4.92109e6	0.191167
$922$	2.11884e6i	0.0820863i
$923$	− 6.61387e6i	− 0.255536i
$924$	−4.64814e6	−0.179102
$925$	0	0
$926$	3.19226e6	0.122340
$927$	− 1.61346e7i	− 0.616677i
$928$	2.42727e7i	0.925226i
$929$	−2.71152e7	−1.03080	−0.515399	−	0.856951i	$-0.672356\pi$
$929$	−2.71152e7	−1.03080	−0.515399	+	0.856951i	$0.672356\pi$
$930$	0	0
$931$	2.14169e6	0.0809809
$932$	6.47150e6i	0.244042i
$933$	2.91084e7i	1.09475i
$934$	7.42621e6	0.278548
$935$	0	0
$936$	−2.31676e6	−0.0864354
$937$	− 4.53522e7i	− 1.68752i	−0.536720	−	0.843761i	$-0.680337\pi$
$937$	− 4.53522e7i	− 1.68752i	0.536720	−	0.843761i	$-0.319663\pi$
$938$	− 82124.0i	− 0.00304764i
$939$	1.63182e7	0.603959
$940$	0	0
$941$	4.65780e7	1.71477	0.857387	−	0.514672i	$-0.172086\pi$
$941$	4.65780e7	1.71477	0.857387	+	0.514672i	$0.172086\pi$
$942$	− 1.60637e6i	− 0.0589820i
$943$	6.33101e7i	2.31843i
$944$	3.93859e7	1.43850
$945$	0	0
$946$	5.86024e6	0.212906
$947$	2.53799e7i	0.919632i	0.888014	+	0.459816i	$0.152085\pi$
$947$	2.53799e7i	0.919632i	−0.888014	+	0.459816i	$0.847915\pi$
$948$	633888.i	0.0229082i
$949$	−3.55836e7	−1.28258
$950$	0	0
$951$	1.14892e7	0.411944
$952$	− 2.46343e6i	− 0.0880942i
$953$	− 1.52948e7i	− 0.545520i	−0.962082	−	0.272760i	$-0.912063\pi$
$953$	− 1.52948e7i	− 0.545520i	0.962082	−	0.272760i	$-0.0879365\pi$
$954$	−12150.0	−0.000432220 0
$955$	0	0
$956$	−2.21147e7	−0.782592
$957$	− 2.52205e7i	− 0.890173i
$958$	3.39685e6i	0.119581i
$959$	1.02003e7	0.358152
$960$	0	0
$961$	−2.23991e7	−0.782389
$962$	4.44829e6i	0.154973i
$963$	6.47773e6i	0.225091i
$964$	−1.56626e7	−0.542840
$965$	0	0
$966$	−1.40767e6	−0.0485354
$967$	− 5.71465e6i	− 0.196527i	−0.995160	−	0.0982637i	$-0.968671\pi$
$967$	− 5.71465e6i	− 0.196527i	0.995160	−	0.0982637i	$-0.0313289\pi$
$968$	− 2.86341e6i	− 0.0982190i
$969$	−6.40634e6	−0.219180
$970$	0	0
$971$	1.30250e7	0.443332	0.221666	−	0.975123i	$-0.428851\pi$
$971$	1.30250e7	0.443332	0.221666	+	0.975123i	$0.428851\pi$
$972$	1.83052e6i	0.0621453i
$973$	− 1.35034e7i	− 0.457258i
$974$	3.71382e6	0.125436
$975$	0	0
$976$	1.37102e7	0.460700
$977$	− 1.70360e7i	− 0.570992i	−0.958380	−	0.285496i	$-0.907842\pi$
$977$	− 1.70360e7i	− 0.570992i	0.958380	−	0.285496i	$-0.0921583\pi$
$978$	2.27495e6i	0.0760544i
$979$	−3.98772e7	−1.32974
$980$	0	0
$981$	−3.73394e6	−0.123878
$982$	5.57494e6i	0.184485i
$983$	1.36985e7i	0.452156i	0.974109	+	0.226078i	$0.0725903\pi$
$983$	1.36985e7i	0.452156i	−0.974109	+	0.226078i	$0.927410\pi$
$984$	−1.12459e7	−0.370259
$985$	0	0
$986$	6.57712e6	0.215448
$987$	3.93725e6i	0.128647i
$988$	1.25540e7i	0.409157i
$989$	−5.50173e7	−1.78858
$990$	0	0
$991$	−3.49088e7	−1.12915	−0.564574	−	0.825383i	$-0.690959\pi$
$991$	−3.49088e7	−1.12915	−0.564574	+	0.825383i	$0.690959\pi$
$992$	− 7.35072e6i	− 0.237165i
$993$	− 1.56259e7i	− 0.502889i
$994$	713832.	0.0229155
$995$	0	0
$996$	−1.05362e7	−0.336538
$997$	875662.i	0.0278996i	0.999903	+	0.0139498i	$0.00444051\pi$
$997$	875662.i	0.0278996i	−0.999903	+	0.0139498i	$0.995559\pi$
$998$	− 3.92698e6i	− 0.124805i
$999$	7.14274e6	0.226439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.6.d.d.274.2		2
5.2	odd	4		525.6.a.c.1.1		1
5.3	odd	4		21.6.a.b.1.1	✓	1
5.4	even	2	inner	525.6.d.d.274.1		2
15.8	even	4		63.6.a.c.1.1		1
20.3	even	4		336.6.a.l.1.1		1
35.3	even	12		147.6.e.e.79.1		2
35.13	even	4		147.6.a.e.1.1		1
35.18	odd	12		147.6.e.f.79.1		2
35.23	odd	12		147.6.e.f.67.1		2
35.33	even	12		147.6.e.e.67.1		2
60.23	odd	4		1008.6.a.t.1.1		1
105.83	odd	4		441.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.b.1.1	✓	1	5.3	odd	4
63.6.a.c.1.1		1	15.8	even	4
147.6.a.e.1.1		1	35.13	even	4
147.6.e.e.67.1		2	35.33	even	12
147.6.e.e.79.1		2	35.3	even	12
147.6.e.f.67.1		2	35.23	odd	12
147.6.e.f.79.1		2	35.18	odd	12
336.6.a.l.1.1		1	20.3	even	4
441.6.a.d.1.1		1	105.83	odd	4
525.6.a.c.1.1		1	5.2	odd	4
525.6.d.d.274.1		2	5.4	even	2	inner
525.6.d.d.274.2		2	1.1	even	1	trivial
1008.6.a.t.1.1		1	60.23	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +9.00000i q^{3} +31.0000 q^{4} -9.00000 q^{6} -49.0000i q^{7} +63.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +9.00000i q^{3} +31.0000 q^{4} -9.00000 q^{6} -49.0000i q^{7} +63.0000i q^{8} -81.0000 q^{9} -340.000 q^{11} +279.000i q^{12} -454.000i q^{13} +49.0000 q^{14} +929.000 q^{16} -798.000i q^{17} -81.0000i q^{18} -892.000 q^{19} +441.000 q^{21} -340.000i q^{22} +3192.00i q^{23} -567.000 q^{24} +454.000 q^{26} -729.000i q^{27} -1519.00i q^{28} +8242.00 q^{29} -2496.00 q^{31} +2945.00i q^{32} -3060.00i q^{33} +798.000 q^{34} -2511.00 q^{36} +9798.00i q^{37} -892.000i q^{38} +4086.00 q^{39} +19834.0 q^{41} +441.000i q^{42} +17236.0i q^{43} -10540.0 q^{44} -3192.00 q^{46} +8928.00i q^{47} +8361.00i q^{48} -2401.00 q^{49} +7182.00 q^{51} -14074.0i q^{52} -150.000i q^{53} +729.000 q^{54} +3087.00 q^{56} -8028.00i q^{57} +8242.00i q^{58} +42396.0 q^{59} +14758.0 q^{61} -2496.00i q^{62} +3969.00i q^{63} +26783.0 q^{64} +3060.00 q^{66} -1676.00i q^{67} -24738.0i q^{68} -28728.0 q^{69} +14568.0 q^{71} -5103.00i q^{72} -78378.0i q^{73} -9798.00 q^{74} -27652.0 q^{76} +16660.0i q^{77} +4086.00i q^{78} +2272.00 q^{79} +6561.00 q^{81} +19834.0i q^{82} +37764.0i q^{83} +13671.0 q^{84} -17236.0 q^{86} +74178.0i q^{87} -21420.0i q^{88} +117286. q^{89} -22246.0 q^{91} +98952.0i q^{92} -22464.0i q^{93} -8928.00 q^{94} -26505.0 q^{96} +10002.0i q^{97} -2401.00i q^{98} +27540.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 62 q^{4} - 18 q^{6} - 162 q^{9}+O(q^{10}) \) 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9	\( 2 q + 62 q^{4} - 18 q^{6} - 162 q^{9} - 680 q^{11} + 98 q^{14} + 1858 q^{16} - 1784 q^{19} + 882 q^{21} - 1134 q^{24} + 908 q^{26} + 16484 q^{29} - 4992 q^{31} + 1596 q^{34} - 5022 q^{36} + 8172 q^{39} + 39668 q^{41} - 21080 q^{44} - 6384 q^{46} - 4802 q^{49} + 14364 q^{51} + 1458 q^{54} + 6174 q^{56} + 84792 q^{59} + 29516 q^{61} + 53566 q^{64} + 6120 q^{66} - 57456 q^{69} + 29136 q^{71} - 19596 q^{74} - 55304 q^{76} + 4544 q^{79} + 13122 q^{81} + 27342 q^{84} - 34472 q^{86} + 234572 q^{89} - 44492 q^{91} - 17856 q^{94} - 53010 q^{96} + 55080 q^{99}+O(q^{100}) \) 2 * q + 62 * q^4 - 18 * q^6 - 162 * q^9 - 680 * q^11 + 98 * q^14 + 1858 * q^16 - 1784 * q^19 + 882 * q^21 - 1134 * q^24 + 908 * q^26 + 16484 * q^29 - 4992 * q^31 + 1596 * q^34 - 5022 * q^36 + 8172 * q^39 + 39668 * q^41 - 21080 * q^44 - 6384 * q^46 - 4802 * q^49 + 14364 * q^51 + 1458 * q^54 + 6174 * q^56 + 84792 * q^59 + 29516 * q^61 + 53566 * q^64 + 6120 * q^66 - 57456 * q^69 + 29136 * q^71 - 19596 * q^74 - 55304 * q^76 + 4544 * q^79 + 13122 * q^81 + 27342 * q^84 - 34472 * q^86 + 234572 * q^89 - 44492 * q^91 - 17856 * q^94 - 53010 * q^96 + 55080 * q^99

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