LMFDB - Embedded newform 3450.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.a (trivial)

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:	yes
Analytic conductor:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.12311 q^{11} +1.00000 q^{12} -2.00000 q^{13} -5.12311 q^{14} +1.00000 q^{16} -7.12311 q^{17} -1.00000 q^{18} +4.00000 q^{19} +5.12311 q^{21} -5.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{32} +5.12311 q^{33} +7.12311 q^{34} +1.00000 q^{36} +7.12311 q^{37} -4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{41} -5.12311 q^{42} +5.12311 q^{44} +1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +19.2462 q^{49} -7.12311 q^{51} -2.00000 q^{52} +4.24621 q^{53} -1.00000 q^{54} -5.12311 q^{56} +4.00000 q^{57} -2.00000 q^{58} -14.2462 q^{59} +0.876894 q^{61} +5.12311 q^{63} +1.00000 q^{64} -5.12311 q^{66} +8.00000 q^{67} -7.12311 q^{68} -1.00000 q^{69} +6.24621 q^{71} -1.00000 q^{72} -12.2462 q^{73} -7.12311 q^{74} +4.00000 q^{76} +26.2462 q^{77} +2.00000 q^{78} -5.12311 q^{79} +1.00000 q^{81} -2.00000 q^{82} -11.3693 q^{83} +5.12311 q^{84} +2.00000 q^{87} -5.12311 q^{88} +3.12311 q^{89} -10.2462 q^{91} -1.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} -0.246211 q^{97} -19.2462 q^{98} +5.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} + 2 q^{28} + 4 q^{29} - 2 q^{32} + 2 q^{33} + 6 q^{34} + 2 q^{36} + 6 q^{37} - 8 q^{38} - 4 q^{39} + 4 q^{41} - 2 q^{42} + 2 q^{44} + 2 q^{46} + 16 q^{47} + 2 q^{48} + 22 q^{49} - 6 q^{51} - 4 q^{52} - 8 q^{53} - 2 q^{54} - 2 q^{56} + 8 q^{57} - 4 q^{58} - 12 q^{59} + 10 q^{61} + 2 q^{63} + 2 q^{64} - 2 q^{66} + 16 q^{67} - 6 q^{68} - 2 q^{69} - 4 q^{71} - 2 q^{72} - 8 q^{73} - 6 q^{74} + 8 q^{76} + 36 q^{77} + 4 q^{78} - 2 q^{79} + 2 q^{81} - 4 q^{82} + 2 q^{83} + 2 q^{84} + 4 q^{87} - 2 q^{88} - 2 q^{89} - 4 q^{91} - 2 q^{92} - 16 q^{94} - 2 q^{96} + 16 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^11 + 2 * q^12 - 4 * q^13 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 2 * q^18 + 8 * q^19 + 2 * q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 + 4 * q^26 + 2 * q^27 + 2 * q^28 + 4 * q^29 - 2 * q^32 + 2 * q^33 + 6 * q^34 + 2 * q^36 + 6 * q^37 - 8 * q^38 - 4 * q^39 + 4 * q^41 - 2 * q^42 + 2 * q^44 + 2 * q^46 + 16 * q^47 + 2 * q^48 + 22 * q^49 - 6 * q^51 - 4 * q^52 - 8 * q^53 - 2 * q^54 - 2 * q^56 + 8 * q^57 - 4 * q^58 - 12 * q^59 + 10 * q^61 + 2 * q^63 + 2 * q^64 - 2 * q^66 + 16 * q^67 - 6 * q^68 - 2 * q^69 - 4 * q^71 - 2 * q^72 - 8 * q^73 - 6 * q^74 + 8 * q^76 + 36 * q^77 + 4 * q^78 - 2 * q^79 + 2 * q^81 - 4 * q^82 + 2 * q^83 + 2 * q^84 + 4 * q^87 - 2 * q^88 - 2 * q^89 - 4 * q^91 - 2 * q^92 - 16 * q^94 - 2 * q^96 + 16 * q^97 - 22 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	5.12311	1.93635	0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311	1.93635	0.968176	+	0.250270i	$0.0805195\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−5.12311	−1.36921
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.12311	−1.72761	−0.863803	−	0.503829i	$-0.831924\pi$
$17$	−7.12311	−1.72761	−0.863803	+	0.503829i	$0.831924\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	5.12311	1.11795
$22$	−5.12311	−1.09225
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	5.12311	0.968176
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	5.12311	0.891818
$34$	7.12311	1.22160
$35$	0	0
$36$	1.00000	0.166667
$37$	7.12311	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−5.12311	−0.790512
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	5.12311	0.772337
$45$	0	0
$46$	1.00000	0.147442
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	19.2462	2.74946
$50$	0	0
$51$	−7.12311	−0.997434
$52$	−2.00000	−0.277350
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−5.12311	−0.684604
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	−14.2462	−1.85470	−0.927349	−	0.374197i	$-0.877918\pi$
$59$	−14.2462	−1.85470	−0.927349	+	0.374197i	$0.877918\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	5.12311	0.645451
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.12311	−0.630611
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−7.12311	−0.863803
$69$	−1.00000	−0.120386
$70$	0	0
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	−1.00000	−0.117851
$73$	−12.2462	−1.43331	−0.716655	−	0.697428i	$-0.754328\pi$
$73$	−12.2462	−1.43331	−0.716655	+	0.697428i	$0.754328\pi$
$74$	−7.12311	−0.828044
$75$	0	0
$76$	4.00000	0.458831
$77$	26.2462	2.99103
$78$	2.00000	0.226455
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−11.3693	−1.24794	−0.623972	−	0.781446i	$-0.714482\pi$
$83$	−11.3693	−1.24794	−0.623972	+	0.781446i	$0.714482\pi$
$84$	5.12311	0.558977
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	−5.12311	−0.546125
$89$	3.12311	0.331049	0.165524	−	0.986206i	$-0.447068\pi$
$89$	3.12311	0.331049	0.165524	+	0.986206i	$0.447068\pi$
$90$	0	0
$91$	−10.2462	−1.07409
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−0.246211	−0.0249990	−0.0124995	−	0.999922i	$-0.503979\pi$
$97$	−0.246211	−0.0249990	−0.0124995	+	0.999922i	$0.503979\pi$
$98$	−19.2462	−1.94416
$99$	5.12311	0.514891
$100$	0	0
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	7.12311	0.705293
$103$	−15.3693	−1.51438	−0.757192	−	0.653192i	$-0.773429\pi$
$103$	−15.3693	−1.51438	−0.757192	+	0.653192i	$0.773429\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−4.24621	−0.412428
$107$	−11.3693	−1.09911	−0.549557	−	0.835456i	$-0.685203\pi$
$107$	−11.3693	−1.09911	−0.549557	+	0.835456i	$0.685203\pi$
$108$	1.00000	0.0962250
$109$	11.1231	1.06540	0.532700	−	0.846304i	$-0.321177\pi$
$109$	11.1231	1.06540	0.532700	+	0.846304i	$0.321177\pi$
$110$	0	0
$111$	7.12311	0.676095
$112$	5.12311	0.484088
$113$	0.876894	0.0824913	0.0412456	−	0.999149i	$-0.486867\pi$
$113$	0.876894	0.0824913	0.0412456	+	0.999149i	$0.486867\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	14.2462	1.31147
$119$	−36.4924	−3.34525
$120$	0	0
$121$	15.2462	1.38602
$122$	−0.876894	−0.0793903
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	−5.12311	−0.456403
$127$	22.2462	1.97403	0.987016	−	0.160622i	$-0.0513500\pi$
$127$	22.2462	1.97403	0.987016	+	0.160622i	$0.0513500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	5.12311	0.445909
$133$	20.4924	1.77692
$134$	−8.00000	−0.691095
$135$	0	0
$136$	7.12311	0.610801
$137$	19.1231	1.63380	0.816899	−	0.576781i	$-0.195692\pi$
$137$	19.1231	1.63380	0.816899	+	0.576781i	$0.195692\pi$
$138$	1.00000	0.0851257
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−6.24621	−0.524170
$143$	−10.2462	−0.856831
$144$	1.00000	0.0833333
$145$	0	0
$146$	12.2462	1.01350
$147$	19.2462	1.58740
$148$	7.12311	0.585516
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−10.2462	−0.833825	−0.416912	−	0.908947i	$-0.636888\pi$
$151$	−10.2462	−0.833825	−0.416912	+	0.908947i	$0.636888\pi$
$152$	−4.00000	−0.324443
$153$	−7.12311	−0.575869
$154$	−26.2462	−2.11498
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−0.876894	−0.0699838	−0.0349919	−	0.999388i	$-0.511141\pi$
$157$	−0.876894	−0.0699838	−0.0349919	+	0.999388i	$0.511141\pi$
$158$	5.12311	0.407572
$159$	4.24621	0.336746
$160$	0	0
$161$	−5.12311	−0.403757
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	11.3693	0.882430
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−5.12311	−0.395256
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	−3.75379	−0.285395	−0.142698	−	0.989766i	$-0.545578\pi$
$173$	−3.75379	−0.285395	−0.142698	+	0.989766i	$0.545578\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	5.12311	0.386169
$177$	−14.2462	−1.07081
$178$	−3.12311	−0.234087
$179$	24.4924	1.83065	0.915325	−	0.402716i	$-0.131934\pi$
$179$	24.4924	1.83065	0.915325	+	0.402716i	$0.131934\pi$
$180$	0	0
$181$	19.1231	1.42141	0.710705	−	0.703491i	$-0.248376\pi$
$181$	19.1231	1.42141	0.710705	+	0.703491i	$0.248376\pi$
$182$	10.2462	0.759500
$183$	0.876894	0.0648219
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	−36.4924	−2.66859
$188$	8.00000	0.583460
$189$	5.12311	0.372651
$190$	0	0
$191$	−20.4924	−1.48278	−0.741390	−	0.671075i	$-0.765833\pi$
$191$	−20.4924	−1.48278	−0.741390	+	0.671075i	$0.765833\pi$
$192$	1.00000	0.0721688
$193$	8.24621	0.593575	0.296788	−	0.954944i	$-0.404085\pi$
$193$	8.24621	0.593575	0.296788	+	0.954944i	$0.404085\pi$
$194$	0.246211	0.0176769
$195$	0	0
$196$	19.2462	1.37473
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−5.12311	−0.364083
$199$	10.8769	0.771043	0.385521	−	0.922699i	$-0.374022\pi$
$199$	10.8769	0.771043	0.385521	+	0.922699i	$0.374022\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	16.2462	1.14308
$203$	10.2462	0.719143
$204$	−7.12311	−0.498717
$205$	0	0
$206$	15.3693	1.07083
$207$	−1.00000	−0.0695048
$208$	−2.00000	−0.138675
$209$	20.4924	1.41749
$210$	0	0
$211$	16.4924	1.13539	0.567693	−	0.823241i	$-0.307836\pi$
$211$	16.4924	1.13539	0.567693	+	0.823241i	$0.307836\pi$
$212$	4.24621	0.291631
$213$	6.24621	0.427983
$214$	11.3693	0.777191
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−11.1231	−0.753352
$219$	−12.2462	−0.827522
$220$	0	0
$221$	14.2462	0.958304
$222$	−7.12311	−0.478072
$223$	−6.24621	−0.418277	−0.209139	−	0.977886i	$-0.567066\pi$
$223$	−6.24621	−0.418277	−0.209139	+	0.977886i	$0.567066\pi$
$224$	−5.12311	−0.342302
$225$	0	0
$226$	−0.876894	−0.0583301
$227$	−19.3693	−1.28559	−0.642793	−	0.766040i	$-0.722225\pi$
$227$	−19.3693	−1.28559	−0.642793	+	0.766040i	$0.722225\pi$
$228$	4.00000	0.264906
$229$	−1.36932	−0.0904870	−0.0452435	−	0.998976i	$-0.514406\pi$
$229$	−1.36932	−0.0904870	−0.0452435	+	0.998976i	$0.514406\pi$
$230$	0	0
$231$	26.2462	1.72687
$232$	−2.00000	−0.131306
$233$	20.7386	1.35863	0.679317	−	0.733845i	$-0.262276\pi$
$233$	20.7386	1.35863	0.679317	+	0.733845i	$0.262276\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−14.2462	−0.927349
$237$	−5.12311	−0.332781
$238$	36.4924	2.36545
$239$	−1.75379	−0.113443	−0.0567216	−	0.998390i	$-0.518065\pi$
$239$	−1.75379	−0.113443	−0.0567216	+	0.998390i	$0.518065\pi$
$240$	0	0
$241$	−2.49242	−0.160551	−0.0802755	−	0.996773i	$-0.525580\pi$
$241$	−2.49242	−0.160551	−0.0802755	+	0.996773i	$0.525580\pi$
$242$	−15.2462	−0.980064
$243$	1.00000	0.0641500
$244$	0.876894	0.0561374
$245$	0	0
$246$	−2.00000	−0.127515
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−11.3693	−0.720501
$250$	0	0
$251$	−7.36932	−0.465147	−0.232574	−	0.972579i	$-0.574715\pi$
$251$	−7.36932	−0.465147	−0.232574	+	0.972579i	$0.574715\pi$
$252$	5.12311	0.322725
$253$	−5.12311	−0.322087
$254$	−22.2462	−1.39585
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.24621	0.514385	0.257192	−	0.966360i	$-0.417203\pi$
$257$	8.24621	0.514385	0.257192	+	0.966360i	$0.417203\pi$
$258$	0	0
$259$	36.4924	2.26753
$260$	0	0
$261$	2.00000	0.123797
$262$	4.00000	0.247121
$263$	−10.2462	−0.631808	−0.315904	−	0.948791i	$-0.602308\pi$
$263$	−10.2462	−0.631808	−0.315904	+	0.948791i	$0.602308\pi$
$264$	−5.12311	−0.315305
$265$	0	0
$266$	−20.4924	−1.25647
$267$	3.12311	0.191131
$268$	8.00000	0.488678
$269$	−16.2462	−0.990549	−0.495274	−	0.868737i	$-0.664933\pi$
$269$	−16.2462	−0.990549	−0.495274	+	0.868737i	$0.664933\pi$
$270$	0	0
$271$	10.2462	0.622413	0.311207	−	0.950342i	$-0.399267\pi$
$271$	10.2462	0.622413	0.311207	+	0.950342i	$0.399267\pi$
$272$	−7.12311	−0.431902
$273$	−10.2462	−0.620129
$274$	−19.1231	−1.15527
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−20.2462	−1.21648	−0.608238	−	0.793754i	$-0.708124\pi$
$277$	−20.2462	−1.21648	−0.608238	+	0.793754i	$0.708124\pi$
$278$	16.4924	0.989150
$279$	0	0
$280$	0	0
$281$	−12.8769	−0.768171	−0.384086	−	0.923298i	$-0.625483\pi$
$281$	−12.8769	−0.768171	−0.384086	+	0.923298i	$0.625483\pi$
$282$	−8.00000	−0.476393
$283$	2.24621	0.133523	0.0667617	−	0.997769i	$-0.478733\pi$
$283$	2.24621	0.133523	0.0667617	+	0.997769i	$0.478733\pi$
$284$	6.24621	0.370644
$285$	0	0
$286$	10.2462	0.605871
$287$	10.2462	0.604815
$288$	−1.00000	−0.0589256
$289$	33.7386	1.98463
$290$	0	0
$291$	−0.246211	−0.0144332
$292$	−12.2462	−0.716655
$293$	−11.7538	−0.686664	−0.343332	−	0.939214i	$-0.611556\pi$
$293$	−11.7538	−0.686664	−0.343332	+	0.939214i	$0.611556\pi$
$294$	−19.2462	−1.12246
$295$	0	0
$296$	−7.12311	−0.414022
$297$	5.12311	0.297273
$298$	10.0000	0.579284
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	10.2462	0.589603
$303$	−16.2462	−0.933320
$304$	4.00000	0.229416
$305$	0	0
$306$	7.12311	0.407201
$307$	−0.492423	−0.0281040	−0.0140520	−	0.999901i	$-0.504473\pi$
$307$	−0.492423	−0.0281040	−0.0140520	+	0.999901i	$0.504473\pi$
$308$	26.2462	1.49552
$309$	−15.3693	−0.874330
$310$	0	0
$311$	26.7386	1.51621	0.758104	−	0.652133i	$-0.226126\pi$
$311$	26.7386	1.51621	0.758104	+	0.652133i	$0.226126\pi$
$312$	2.00000	0.113228
$313$	−10.4924	−0.593067	−0.296533	−	0.955022i	$-0.595831\pi$
$313$	−10.4924	−0.593067	−0.296533	+	0.955022i	$0.595831\pi$
$314$	0.876894	0.0494860
$315$	0	0
$316$	−5.12311	−0.288197
$317$	−20.7386	−1.16480	−0.582399	−	0.812903i	$-0.697886\pi$
$317$	−20.7386	−1.16480	−0.582399	+	0.812903i	$0.697886\pi$
$318$	−4.24621	−0.238116
$319$	10.2462	0.573678
$320$	0	0
$321$	−11.3693	−0.634573
$322$	5.12311	0.285500
$323$	−28.4924	−1.58536
$324$	1.00000	0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	11.1231	0.615109
$328$	−2.00000	−0.110432
$329$	40.9848	2.25957
$330$	0	0
$331$	−32.4924	−1.78595	−0.892973	−	0.450111i	$-0.851384\pi$
$331$	−32.4924	−1.78595	−0.892973	+	0.450111i	$0.851384\pi$
$332$	−11.3693	−0.623972
$333$	7.12311	0.390344
$334$	−8.00000	−0.437741
$335$	0	0
$336$	5.12311	0.279488
$337$	6.49242	0.353665	0.176832	−	0.984241i	$-0.443415\pi$
$337$	6.49242	0.353665	0.176832	+	0.984241i	$0.443415\pi$
$338$	9.00000	0.489535
$339$	0.876894	0.0476264
$340$	0	0
$341$	0	0
$342$	−4.00000	−0.216295
$343$	62.7386	3.38757
$344$	0	0
$345$	0	0
$346$	3.75379	0.201805
$347$	34.7386	1.86487	0.932434	−	0.361341i	$-0.117681\pi$
$347$	34.7386	1.86487	0.932434	+	0.361341i	$0.117681\pi$
$348$	2.00000	0.107211
$349$	−24.7386	−1.32423	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	−24.7386	−1.32423	−0.662114	+	0.749403i	$0.730341\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−5.12311	−0.273062
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	14.2462	0.757178
$355$	0	0
$356$	3.12311	0.165524
$357$	−36.4924	−1.93138
$358$	−24.4924	−1.29446
$359$	−12.4924	−0.659325	−0.329662	−	0.944099i	$-0.606935\pi$
$359$	−12.4924	−0.659325	−0.329662	+	0.944099i	$0.606935\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−19.1231	−1.00509
$363$	15.2462	0.800219
$364$	−10.2462	−0.537047
$365$	0	0
$366$	−0.876894	−0.0458360
$367$	−13.1231	−0.685021	−0.342510	−	0.939514i	$-0.611277\pi$
$367$	−13.1231	−0.685021	−0.342510	+	0.939514i	$0.611277\pi$
$368$	−1.00000	−0.0521286
$369$	2.00000	0.104116
$370$	0	0
$371$	21.7538	1.12940
$372$	0	0
$373$	−13.3693	−0.692237	−0.346118	−	0.938191i	$-0.612500\pi$
$373$	−13.3693	−0.692237	−0.346118	+	0.938191i	$0.612500\pi$
$374$	36.4924	1.88698
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−4.00000	−0.206010
$378$	−5.12311	−0.263504
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	22.2462	1.13971
$382$	20.4924	1.04848
$383$	−13.7538	−0.702786	−0.351393	−	0.936228i	$-0.614292\pi$
$383$	−13.7538	−0.702786	−0.351393	+	0.936228i	$0.614292\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−8.24621	−0.419721
$387$	0	0
$388$	−0.246211	−0.0124995
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	7.12311	0.360231
$392$	−19.2462	−0.972080
$393$	−4.00000	−0.201773
$394$	−10.0000	−0.503793
$395$	0	0
$396$	5.12311	0.257446
$397$	34.4924	1.73113	0.865563	−	0.500801i	$-0.166961\pi$
$397$	34.4924	1.73113	0.865563	+	0.500801i	$0.166961\pi$
$398$	−10.8769	−0.545209
$399$	20.4924	1.02590
$400$	0	0
$401$	−17.3693	−0.867382	−0.433691	−	0.901062i	$-0.642789\pi$
$401$	−17.3693	−0.867382	−0.433691	+	0.901062i	$0.642789\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	−16.2462	−0.808279
$405$	0	0
$406$	−10.2462	−0.508511
$407$	36.4924	1.80886
$408$	7.12311	0.352646
$409$	−16.2462	−0.803323	−0.401662	−	0.915788i	$-0.631567\pi$
$409$	−16.2462	−0.803323	−0.401662	+	0.915788i	$0.631567\pi$
$410$	0	0
$411$	19.1231	0.943273
$412$	−15.3693	−0.757192
$413$	−72.9848	−3.59135
$414$	1.00000	0.0491473
$415$	0	0
$416$	2.00000	0.0980581
$417$	−16.4924	−0.807637
$418$	−20.4924	−1.00232
$419$	−1.61553	−0.0789237	−0.0394619	−	0.999221i	$-0.512564\pi$
$419$	−1.61553	−0.0789237	−0.0394619	+	0.999221i	$0.512564\pi$
$420$	0	0
$421$	13.3693	0.651581	0.325790	−	0.945442i	$-0.394370\pi$
$421$	13.3693	0.651581	0.325790	+	0.945442i	$0.394370\pi$
$422$	−16.4924	−0.802839
$423$	8.00000	0.388973
$424$	−4.24621	−0.206214
$425$	0	0
$426$	−6.24621	−0.302630
$427$	4.49242	0.217404
$428$	−11.3693	−0.549557
$429$	−10.2462	−0.494692
$430$	0	0
$431$	26.2462	1.26424	0.632118	−	0.774872i	$-0.282186\pi$
$431$	26.2462	1.26424	0.632118	+	0.774872i	$0.282186\pi$
$432$	1.00000	0.0481125
$433$	−10.4924	−0.504234	−0.252117	−	0.967697i	$-0.581127\pi$
$433$	−10.4924	−0.504234	−0.252117	+	0.967697i	$0.581127\pi$
$434$	0	0
$435$	0	0
$436$	11.1231	0.532700
$437$	−4.00000	−0.191346
$438$	12.2462	0.585147
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	19.2462	0.916486
$442$	−14.2462	−0.677623
$443$	24.4924	1.16367	0.581835	−	0.813307i	$-0.302335\pi$
$443$	24.4924	1.16367	0.581835	+	0.813307i	$0.302335\pi$
$444$	7.12311	0.338048
$445$	0	0
$446$	6.24621	0.295767
$447$	−10.0000	−0.472984
$448$	5.12311	0.242044
$449$	−10.4924	−0.495168	−0.247584	−	0.968866i	$-0.579637\pi$
$449$	−10.4924	−0.495168	−0.247584	+	0.968866i	$0.579637\pi$
$450$	0	0
$451$	10.2462	0.482475
$452$	0.876894	0.0412456
$453$	−10.2462	−0.481409
$454$	19.3693	0.909047
$455$	0	0
$456$	−4.00000	−0.187317
$457$	32.7386	1.53145	0.765724	−	0.643169i	$-0.222381\pi$
$457$	32.7386	1.53145	0.765724	+	0.643169i	$0.222381\pi$
$458$	1.36932	0.0639840
$459$	−7.12311	−0.332478
$460$	0	0
$461$	32.7386	1.52479	0.762395	−	0.647112i	$-0.224023\pi$
$461$	32.7386	1.52479	0.762395	+	0.647112i	$0.224023\pi$
$462$	−26.2462	−1.22108
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−20.7386	−0.960699
$467$	3.36932	0.155913	0.0779567	−	0.996957i	$-0.475160\pi$
$467$	3.36932	0.155913	0.0779567	+	0.996957i	$0.475160\pi$
$468$	−2.00000	−0.0924500
$469$	40.9848	1.89250
$470$	0	0
$471$	−0.876894	−0.0404052
$472$	14.2462	0.655735
$473$	0	0
$474$	5.12311	0.235312
$475$	0	0
$476$	−36.4924	−1.67263
$477$	4.24621	0.194421
$478$	1.75379	0.0802164
$479$	−20.4924	−0.936323	−0.468161	−	0.883643i	$-0.655083\pi$
$479$	−20.4924	−0.936323	−0.468161	+	0.883643i	$0.655083\pi$
$480$	0	0
$481$	−14.2462	−0.649571
$482$	2.49242	0.113527
$483$	−5.12311	−0.233109
$484$	15.2462	0.693010
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−40.4924	−1.83489	−0.917443	−	0.397866i	$-0.869751\pi$
$487$	−40.4924	−1.83489	−0.917443	+	0.397866i	$0.869751\pi$
$488$	−0.876894	−0.0396951
$489$	12.0000	0.542659
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	2.00000	0.0901670
$493$	−14.2462	−0.641617
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	32.0000	1.43540
$498$	11.3693	0.509471
$499$	−28.9848	−1.29754	−0.648770	−	0.760985i	$-0.724716\pi$
$499$	−28.9848	−1.29754	−0.648770	+	0.760985i	$0.724716\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	7.36932	0.328909
$503$	6.73863	0.300461	0.150230	−	0.988651i	$-0.451998\pi$
$503$	6.73863	0.300461	0.150230	+	0.988651i	$0.451998\pi$
$504$	−5.12311	−0.228201
$505$	0	0
$506$	5.12311	0.227750
$507$	−9.00000	−0.399704
$508$	22.2462	0.987016
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	−62.7386	−2.77539
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−8.24621	−0.363725
$515$	0	0
$516$	0	0
$517$	40.9848	1.80251
$518$	−36.4924	−1.60338
$519$	−3.75379	−0.164773
$520$	0	0
$521$	8.87689	0.388904	0.194452	−	0.980912i	$-0.437707\pi$
$521$	8.87689	0.388904	0.194452	+	0.980912i	$0.437707\pi$
$522$	−2.00000	−0.0875376
$523$	28.4924	1.24589	0.622943	−	0.782267i	$-0.285937\pi$
$523$	28.4924	1.24589	0.622943	+	0.782267i	$0.285937\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	10.2462	0.446756
$527$	0	0
$528$	5.12311	0.222955
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.2462	−0.618233
$532$	20.4924	0.888459
$533$	−4.00000	−0.173259
$534$	−3.12311	−0.135150
$535$	0	0
$536$	−8.00000	−0.345547
$537$	24.4924	1.05693
$538$	16.2462	0.700424
$539$	98.6004	4.24702
$540$	0	0
$541$	−15.7538	−0.677308	−0.338654	−	0.940911i	$-0.609972\pi$
$541$	−15.7538	−0.677308	−0.338654	+	0.940911i	$0.609972\pi$
$542$	−10.2462	−0.440112
$543$	19.1231	0.820651
$544$	7.12311	0.305401
$545$	0	0
$546$	10.2462	0.438497
$547$	−0.492423	−0.0210545	−0.0105272	−	0.999945i	$-0.503351\pi$
$547$	−0.492423	−0.0210545	−0.0105272	+	0.999945i	$0.503351\pi$
$548$	19.1231	0.816899
$549$	0.876894	0.0374249
$550$	0	0
$551$	8.00000	0.340811
$552$	1.00000	0.0425628
$553$	−26.2462	−1.11610
$554$	20.2462	0.860179
$555$	0	0
$556$	−16.4924	−0.699435
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−36.4924	−1.54071
$562$	12.8769	0.543179
$563$	−11.3693	−0.479160	−0.239580	−	0.970877i	$-0.577010\pi$
$563$	−11.3693	−0.479160	−0.239580	+	0.970877i	$0.577010\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	−2.24621	−0.0944153
$567$	5.12311	0.215150
$568$	−6.24621	−0.262085
$569$	27.1231	1.13706	0.568530	−	0.822663i	$-0.307512\pi$
$569$	27.1231	1.13706	0.568530	+	0.822663i	$0.307512\pi$
$570$	0	0
$571$	34.7386	1.45377	0.726883	−	0.686761i	$-0.240968\pi$
$571$	34.7386	1.45377	0.726883	+	0.686761i	$0.240968\pi$
$572$	−10.2462	−0.428416
$573$	−20.4924	−0.856083
$574$	−10.2462	−0.427669
$575$	0	0
$576$	1.00000	0.0416667
$577$	−14.4924	−0.603327	−0.301664	−	0.953414i	$-0.597542\pi$
$577$	−14.4924	−0.603327	−0.301664	+	0.953414i	$0.597542\pi$
$578$	−33.7386	−1.40334
$579$	8.24621	0.342701
$580$	0	0
$581$	−58.2462	−2.41646
$582$	0.246211	0.0102058
$583$	21.7538	0.900950
$584$	12.2462	0.506752
$585$	0	0
$586$	11.7538	0.485545
$587$	9.75379	0.402582	0.201291	−	0.979531i	$-0.435486\pi$
$587$	9.75379	0.402582	0.201291	+	0.979531i	$0.435486\pi$
$588$	19.2462	0.793700
$589$	0	0
$590$	0	0
$591$	10.0000	0.411345
$592$	7.12311	0.292758
$593$	−7.75379	−0.318410	−0.159205	−	0.987246i	$-0.550893\pi$
$593$	−7.75379	−0.318410	−0.159205	+	0.987246i	$0.550893\pi$
$594$	−5.12311	−0.210204
$595$	0	0
$596$	−10.0000	−0.409616
$597$	10.8769	0.445162
$598$	−2.00000	−0.0817861
$599$	−6.24621	−0.255213	−0.127607	−	0.991825i	$-0.540730\pi$
$599$	−6.24621	−0.255213	−0.127607	+	0.991825i	$0.540730\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	−10.2462	−0.416912
$605$	0	0
$606$	16.2462	0.659957
$607$	−26.7386	−1.08529	−0.542644	−	0.839963i	$-0.682577\pi$
$607$	−26.7386	−1.08529	−0.542644	+	0.839963i	$0.682577\pi$
$608$	−4.00000	−0.162221
$609$	10.2462	0.415197
$610$	0	0
$611$	−16.0000	−0.647291
$612$	−7.12311	−0.287934
$613$	−43.1231	−1.74173	−0.870863	−	0.491526i	$-0.836439\pi$
$613$	−43.1231	−1.74173	−0.870863	+	0.491526i	$0.836439\pi$
$614$	0.492423	0.0198726
$615$	0	0
$616$	−26.2462	−1.05749
$617$	−40.1080	−1.61469	−0.807343	−	0.590083i	$-0.799095\pi$
$617$	−40.1080	−1.61469	−0.807343	+	0.590083i	$0.799095\pi$
$618$	15.3693	0.618245
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−26.7386	−1.07212
$623$	16.0000	0.641026
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.4924	0.419362
$627$	20.4924	0.818389
$628$	−0.876894	−0.0349919
$629$	−50.7386	−2.02308
$630$	0	0
$631$	−39.3693	−1.56727	−0.783634	−	0.621223i	$-0.786636\pi$
$631$	−39.3693	−1.56727	−0.783634	+	0.621223i	$0.786636\pi$
$632$	5.12311	0.203786
$633$	16.4924	0.655515
$634$	20.7386	0.823636
$635$	0	0
$636$	4.24621	0.168373
$637$	−38.4924	−1.52513
$638$	−10.2462	−0.405651
$639$	6.24621	0.247096
$640$	0	0
$641$	−9.36932	−0.370066	−0.185033	−	0.982732i	$-0.559239\pi$
$641$	−9.36932	−0.370066	−0.185033	+	0.982732i	$0.559239\pi$
$642$	11.3693	0.448711
$643$	−36.4924	−1.43912	−0.719560	−	0.694430i	$-0.755657\pi$
$643$	−36.4924	−1.43912	−0.719560	+	0.694430i	$0.755657\pi$
$644$	−5.12311	−0.201879
$645$	0	0
$646$	28.4924	1.12102
$647$	−11.5076	−0.452410	−0.226205	−	0.974080i	$-0.572632\pi$
$647$	−11.5076	−0.452410	−0.226205	+	0.974080i	$0.572632\pi$
$648$	−1.00000	−0.0392837
$649$	−72.9848	−2.86491
$650$	0	0
$651$	0	0
$652$	12.0000	0.469956
$653$	−10.4924	−0.410600	−0.205300	−	0.978699i	$-0.565817\pi$
$653$	−10.4924	−0.410600	−0.205300	+	0.978699i	$0.565817\pi$
$654$	−11.1231	−0.434948
$655$	0	0
$656$	2.00000	0.0780869
$657$	−12.2462	−0.477770
$658$	−40.9848	−1.59776
$659$	−7.36932	−0.287068	−0.143534	−	0.989645i	$-0.545847\pi$
$659$	−7.36932	−0.287068	−0.143534	+	0.989645i	$0.545847\pi$
$660$	0	0
$661$	33.8617	1.31707	0.658535	−	0.752551i	$-0.271177\pi$
$661$	33.8617	1.31707	0.658535	+	0.752551i	$0.271177\pi$
$662$	32.4924	1.26285
$663$	14.2462	0.553277
$664$	11.3693	0.441215
$665$	0	0
$666$	−7.12311	−0.276015
$667$	−2.00000	−0.0774403
$668$	8.00000	0.309529
$669$	−6.24621	−0.241492
$670$	0	0
$671$	4.49242	0.173428
$672$	−5.12311	−0.197628
$673$	46.9848	1.81113	0.905566	−	0.424205i	$-0.139446\pi$
$673$	46.9848	1.81113	0.905566	+	0.424205i	$0.139446\pi$
$674$	−6.49242	−0.250079
$675$	0	0
$676$	−9.00000	−0.346154
$677$	46.4924	1.78685	0.893424	−	0.449213i	$-0.148296\pi$
$677$	46.4924	1.78685	0.893424	+	0.449213i	$0.148296\pi$
$678$	−0.876894	−0.0336769
$679$	−1.26137	−0.0484068
$680$	0	0
$681$	−19.3693	−0.742234
$682$	0	0
$683$	−17.7538	−0.679330	−0.339665	−	0.940547i	$-0.610314\pi$
$683$	−17.7538	−0.679330	−0.339665	+	0.940547i	$0.610314\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−62.7386	−2.39537
$687$	−1.36932	−0.0522427
$688$	0	0
$689$	−8.49242	−0.323536
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−3.75379	−0.142698
$693$	26.2462	0.997011
$694$	−34.7386	−1.31866
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	−14.2462	−0.539614
$698$	24.7386	0.936371
$699$	20.7386	0.784407
$700$	0	0
$701$	−36.2462	−1.36900	−0.684500	−	0.729013i	$-0.739980\pi$
$701$	−36.2462	−1.36900	−0.684500	+	0.729013i	$0.739980\pi$
$702$	2.00000	0.0754851
$703$	28.4924	1.07461
$704$	5.12311	0.193084
$705$	0	0
$706$	−14.0000	−0.526897
$707$	−83.2311	−3.13023
$708$	−14.2462	−0.535405
$709$	13.3693	0.502095	0.251048	−	0.967975i	$-0.419225\pi$
$709$	13.3693	0.502095	0.251048	+	0.967975i	$0.419225\pi$
$710$	0	0
$711$	−5.12311	−0.192131
$712$	−3.12311	−0.117043
$713$	0	0
$714$	36.4924	1.36569
$715$	0	0
$716$	24.4924	0.915325
$717$	−1.75379	−0.0654964
$718$	12.4924	0.466213
$719$	−9.75379	−0.363755	−0.181877	−	0.983321i	$-0.558217\pi$
$719$	−9.75379	−0.363755	−0.181877	+	0.983321i	$0.558217\pi$
$720$	0	0
$721$	−78.7386	−2.93238
$722$	3.00000	0.111648
$723$	−2.49242	−0.0926942
$724$	19.1231	0.710705
$725$	0	0
$726$	−15.2462	−0.565840
$727$	−27.8617	−1.03333	−0.516667	−	0.856186i	$-0.672828\pi$
$727$	−27.8617	−1.03333	−0.516667	+	0.856186i	$0.672828\pi$
$728$	10.2462	0.379750
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0.876894	0.0324109
$733$	13.8617	0.511995	0.255998	−	0.966677i	$-0.417596\pi$
$733$	13.8617	0.511995	0.255998	+	0.966677i	$0.417596\pi$
$734$	13.1231	0.484383
$735$	0	0
$736$	1.00000	0.0368605
$737$	40.9848	1.50970
$738$	−2.00000	−0.0736210
$739$	40.4924	1.48954	0.744769	−	0.667322i	$-0.232560\pi$
$739$	40.4924	1.48954	0.744769	+	0.667322i	$0.232560\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	−21.7538	−0.798607
$743$	14.7386	0.540708	0.270354	−	0.962761i	$-0.412859\pi$
$743$	14.7386	0.540708	0.270354	+	0.962761i	$0.412859\pi$
$744$	0	0
$745$	0	0
$746$	13.3693	0.489485
$747$	−11.3693	−0.415982
$748$	−36.4924	−1.33430
$749$	−58.2462	−2.12827
$750$	0	0
$751$	43.8617	1.60054	0.800269	−	0.599641i	$-0.204690\pi$
$751$	43.8617	1.60054	0.800269	+	0.599641i	$0.204690\pi$
$752$	8.00000	0.291730
$753$	−7.36932	−0.268553
$754$	4.00000	0.145671
$755$	0	0
$756$	5.12311	0.186326
$757$	−9.86174	−0.358431	−0.179216	−	0.983810i	$-0.557356\pi$
$757$	−9.86174	−0.358431	−0.179216	+	0.983810i	$0.557356\pi$
$758$	−4.00000	−0.145287
$759$	−5.12311	−0.185957
$760$	0	0
$761$	−48.2462	−1.74892	−0.874462	−	0.485094i	$-0.838785\pi$
$761$	−48.2462	−1.74892	−0.874462	+	0.485094i	$0.838785\pi$
$762$	−22.2462	−0.805895
$763$	56.9848	2.06299
$764$	−20.4924	−0.741390
$765$	0	0
$766$	13.7538	0.496945
$767$	28.4924	1.02880
$768$	1.00000	0.0360844
$769$	−20.7386	−0.747854	−0.373927	−	0.927458i	$-0.621989\pi$
$769$	−20.7386	−0.747854	−0.373927	+	0.927458i	$0.621989\pi$
$770$	0	0
$771$	8.24621	0.296980
$772$	8.24621	0.296788
$773$	36.2462	1.30369	0.651843	−	0.758354i	$-0.273996\pi$
$773$	36.2462	1.30369	0.651843	+	0.758354i	$0.273996\pi$
$774$	0	0
$775$	0	0
$776$	0.246211	0.00883847
$777$	36.4924	1.30916
$778$	18.0000	0.645331
$779$	8.00000	0.286630
$780$	0	0
$781$	32.0000	1.14505
$782$	−7.12311	−0.254722
$783$	2.00000	0.0714742
$784$	19.2462	0.687365
$785$	0	0
$786$	4.00000	0.142675
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	10.0000	0.356235
$789$	−10.2462	−0.364775
$790$	0	0
$791$	4.49242	0.159732
$792$	−5.12311	−0.182042
$793$	−1.75379	−0.0622789
$794$	−34.4924	−1.22409
$795$	0	0
$796$	10.8769	0.385521
$797$	22.4924	0.796722	0.398361	−	0.917229i	$-0.369579\pi$
$797$	22.4924	0.796722	0.398361	+	0.917229i	$0.369579\pi$
$798$	−20.4924	−0.725424
$799$	−56.9848	−2.01598
$800$	0	0
$801$	3.12311	0.110350
$802$	17.3693	0.613332
$803$	−62.7386	−2.21400
$804$	8.00000	0.282138
$805$	0	0
$806$	0	0
$807$	−16.2462	−0.571894
$808$	16.2462	0.571540
$809$	4.24621	0.149289	0.0746444	−	0.997210i	$-0.476218\pi$
$809$	4.24621	0.149289	0.0746444	+	0.997210i	$0.476218\pi$
$810$	0	0
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	10.2462	0.359572
$813$	10.2462	0.359350
$814$	−36.4924	−1.27906
$815$	0	0
$816$	−7.12311	−0.249359
$817$	0	0
$818$	16.2462	0.568035
$819$	−10.2462	−0.358032
$820$	0	0
$821$	−19.7538	−0.689412	−0.344706	−	0.938711i	$-0.612021\pi$
$821$	−19.7538	−0.689412	−0.344706	+	0.938711i	$0.612021\pi$
$822$	−19.1231	−0.666995
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	15.3693	0.535416
$825$	0	0
$826$	72.9848	2.53947
$827$	−17.1231	−0.595429	−0.297714	−	0.954655i	$-0.596224\pi$
$827$	−17.1231	−0.595429	−0.297714	+	0.954655i	$0.596224\pi$
$828$	−1.00000	−0.0347524
$829$	32.2462	1.11996	0.559979	−	0.828507i	$-0.310809\pi$
$829$	32.2462	1.11996	0.559979	+	0.828507i	$0.310809\pi$
$830$	0	0
$831$	−20.2462	−0.702333
$832$	−2.00000	−0.0693375
$833$	−137.093	−4.74998
$834$	16.4924	0.571086
$835$	0	0
$836$	20.4924	0.708745
$837$	0	0
$838$	1.61553	0.0558075
$839$	−26.2462	−0.906120	−0.453060	−	0.891480i	$-0.649668\pi$
$839$	−26.2462	−0.906120	−0.453060	+	0.891480i	$0.649668\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−13.3693	−0.460737
$843$	−12.8769	−0.443504
$844$	16.4924	0.567693
$845$	0	0
$846$	−8.00000	−0.275046
$847$	78.1080	2.68382
$848$	4.24621	0.145815
$849$	2.24621	0.0770898
$850$	0	0
$851$	−7.12311	−0.244177
$852$	6.24621	0.213992
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	−4.49242	−0.153728
$855$	0	0
$856$	11.3693	0.388595
$857$	−24.7386	−0.845056	−0.422528	−	0.906350i	$-0.638857\pi$
$857$	−24.7386	−0.845056	−0.422528	+	0.906350i	$0.638857\pi$
$858$	10.2462	0.349800
$859$	48.4924	1.65454	0.827270	−	0.561804i	$-0.189893\pi$
$859$	48.4924	1.65454	0.827270	+	0.561804i	$0.189893\pi$
$860$	0	0
$861$	10.2462	0.349190
$862$	−26.2462	−0.893950
$863$	−48.9848	−1.66746	−0.833732	−	0.552170i	$-0.813800\pi$
$863$	−48.9848	−1.66746	−0.833732	+	0.552170i	$0.813800\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	10.4924	0.356547
$867$	33.7386	1.14582
$868$	0	0
$869$	−26.2462	−0.890342
$870$	0	0
$871$	−16.0000	−0.542139
$872$	−11.1231	−0.376676
$873$	−0.246211	−0.00833299
$874$	4.00000	0.135302
$875$	0	0
$876$	−12.2462	−0.413761
$877$	−3.26137	−0.110129	−0.0550643	−	0.998483i	$-0.517536\pi$
$877$	−3.26137	−0.110129	−0.0550643	+	0.998483i	$0.517536\pi$
$878$	0	0
$879$	−11.7538	−0.396445
$880$	0	0
$881$	31.6155	1.06515	0.532577	−	0.846381i	$-0.321224\pi$
$881$	31.6155	1.06515	0.532577	+	0.846381i	$0.321224\pi$
$882$	−19.2462	−0.648054
$883$	20.9848	0.706196	0.353098	−	0.935586i	$-0.385128\pi$
$883$	20.9848	0.706196	0.353098	+	0.935586i	$0.385128\pi$
$884$	14.2462	0.479152
$885$	0	0
$886$	−24.4924	−0.822839
$887$	−52.4924	−1.76252	−0.881262	−	0.472629i	$-0.843305\pi$
$887$	−52.4924	−1.76252	−0.881262	+	0.472629i	$0.843305\pi$
$888$	−7.12311	−0.239036
$889$	113.970	3.82242
$890$	0	0
$891$	5.12311	0.171630
$892$	−6.24621	−0.209139
$893$	32.0000	1.07084
$894$	10.0000	0.334450
$895$	0	0
$896$	−5.12311	−0.171151
$897$	2.00000	0.0667781
$898$	10.4924	0.350137
$899$	0	0
$900$	0	0
$901$	−30.2462	−1.00765
$902$	−10.2462	−0.341162
$903$	0	0
$904$	−0.876894	−0.0291651
$905$	0	0
$906$	10.2462	0.340408
$907$	−37.7538	−1.25359	−0.626797	−	0.779183i	$-0.715634\pi$
$907$	−37.7538	−1.25359	−0.626797	+	0.779183i	$0.715634\pi$
$908$	−19.3693	−0.642793
$909$	−16.2462	−0.538853
$910$	0	0
$911$	−13.7538	−0.455683	−0.227842	−	0.973698i	$-0.573167\pi$
$911$	−13.7538	−0.455683	−0.227842	+	0.973698i	$0.573167\pi$
$912$	4.00000	0.132453
$913$	−58.2462	−1.92767
$914$	−32.7386	−1.08290
$915$	0	0
$916$	−1.36932	−0.0452435
$917$	−20.4924	−0.676719
$918$	7.12311	0.235098
$919$	10.8769	0.358796	0.179398	−	0.983777i	$-0.442585\pi$
$919$	10.8769	0.358796	0.179398	+	0.983777i	$0.442585\pi$
$920$	0	0
$921$	−0.492423	−0.0162259
$922$	−32.7386	−1.07819
$923$	−12.4924	−0.411193
$924$	26.2462	0.863437
$925$	0	0
$926$	−36.0000	−1.18303
$927$	−15.3693	−0.504795
$928$	−2.00000	−0.0656532
$929$	−30.9848	−1.01658	−0.508290	−	0.861186i	$-0.669722\pi$
$929$	−30.9848	−1.01658	−0.508290	+	0.861186i	$0.669722\pi$
$930$	0	0
$931$	76.9848	2.52308
$932$	20.7386	0.679317
$933$	26.7386	0.875384
$934$	−3.36932	−0.110247
$935$	0	0
$936$	2.00000	0.0653720
$937$	16.7386	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$937$	16.7386	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$938$	−40.9848	−1.33820
$939$	−10.4924	−0.342407
$940$	0	0
$941$	−6.49242	−0.211647	−0.105823	−	0.994385i	$-0.533748\pi$
$941$	−6.49242	−0.211647	−0.105823	+	0.994385i	$0.533748\pi$
$942$	0.876894	0.0285708
$943$	−2.00000	−0.0651290
$944$	−14.2462	−0.463675
$945$	0	0
$946$	0	0
$947$	−22.2462	−0.722905	−0.361452	−	0.932391i	$-0.617719\pi$
$947$	−22.2462	−0.722905	−0.361452	+	0.932391i	$0.617719\pi$
$948$	−5.12311	−0.166391
$949$	24.4924	0.795058
$950$	0	0
$951$	−20.7386	−0.672496
$952$	36.4924	1.18273
$953$	33.8617	1.09689	0.548445	−	0.836187i	$-0.315220\pi$
$953$	33.8617	1.09689	0.548445	+	0.836187i	$0.315220\pi$
$954$	−4.24621	−0.137476
$955$	0	0
$956$	−1.75379	−0.0567216
$957$	10.2462	0.331213
$958$	20.4924	0.662080
$959$	97.9697	3.16361
$960$	0	0
$961$	−31.0000	−1.00000
$962$	14.2462	0.459316
$963$	−11.3693	−0.366371
$964$	−2.49242	−0.0802755
$965$	0	0
$966$	5.12311	0.164833
$967$	4.98485	0.160302	0.0801509	−	0.996783i	$-0.474460\pi$
$967$	4.98485	0.160302	0.0801509	+	0.996783i	$0.474460\pi$
$968$	−15.2462	−0.490032
$969$	−28.4924	−0.915308
$970$	0	0
$971$	34.8769	1.11925	0.559626	−	0.828745i	$-0.310945\pi$
$971$	34.8769	1.11925	0.559626	+	0.828745i	$0.310945\pi$
$972$	1.00000	0.0320750
$973$	−84.4924	−2.70870
$974$	40.4924	1.29746
$975$	0	0
$976$	0.876894	0.0280687
$977$	−21.8617	−0.699419	−0.349710	−	0.936858i	$-0.613720\pi$
$977$	−21.8617	−0.699419	−0.349710	+	0.936858i	$0.613720\pi$
$978$	−12.0000	−0.383718
$979$	16.0000	0.511362
$980$	0	0
$981$	11.1231	0.355133
$982$	12.0000	0.382935
$983$	26.2462	0.837124	0.418562	−	0.908188i	$-0.362534\pi$
$983$	26.2462	0.837124	0.418562	+	0.908188i	$0.362534\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	14.2462	0.453692
$987$	40.9848	1.30456
$988$	−8.00000	−0.254514
$989$	0	0
$990$	0	0
$991$	−37.4773	−1.19050	−0.595252	−	0.803539i	$-0.702948\pi$
$991$	−37.4773	−1.19050	−0.595252	+	0.803539i	$0.702948\pi$
$992$	0	0
$993$	−32.4924	−1.03112
$994$	−32.0000	−1.01498
$995$	0	0
$996$	−11.3693	−0.360251
$997$	−20.2462	−0.641204	−0.320602	−	0.947214i	$-0.603885\pi$
$997$	−20.2462	−0.641204	−0.320602	+	0.947214i	$0.603885\pi$
$998$	28.9848	0.917499
$999$	7.12311	0.225365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bi.1.2		2
5.2	odd	4		3450.2.d.v.2899.2		4
5.3	odd	4		3450.2.d.v.2899.3		4
5.4	even	2		690.2.a.l.1.1	✓	2
15.14	odd	2		2070.2.a.t.1.1		2
20.19	odd	2		5520.2.a.bs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.l.1.1	✓	2	5.4	even	2
2070.2.a.t.1.1		2	15.14	odd	2
3450.2.a.bi.1.2		2	1.1	even	1	trivial
3450.2.d.v.2899.2		4	5.2	odd	4
3450.2.d.v.2899.3		4	5.3	odd	4
5520.2.a.bs.1.2		2	20.19	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.12311 q^{11} +1.00000 q^{12} -2.00000 q^{13} -5.12311 q^{14} +1.00000 q^{16} -7.12311 q^{17} -1.00000 q^{18} +4.00000 q^{19} +5.12311 q^{21} -5.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{32} +5.12311 q^{33} +7.12311 q^{34} +1.00000 q^{36} +7.12311 q^{37} -4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{41} -5.12311 q^{42} +5.12311 q^{44} +1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +19.2462 q^{49} -7.12311 q^{51} -2.00000 q^{52} +4.24621 q^{53} -1.00000 q^{54} -5.12311 q^{56} +4.00000 q^{57} -2.00000 q^{58} -14.2462 q^{59} +0.876894 q^{61} +5.12311 q^{63} +1.00000 q^{64} -5.12311 q^{66} +8.00000 q^{67} -7.12311 q^{68} -1.00000 q^{69} +6.24621 q^{71} -1.00000 q^{72} -12.2462 q^{73} -7.12311 q^{74} +4.00000 q^{76} +26.2462 q^{77} +2.00000 q^{78} -5.12311 q^{79} +1.00000 q^{81} -2.00000 q^{82} -11.3693 q^{83} +5.12311 q^{84} +2.00000 q^{87} -5.12311 q^{88} +3.12311 q^{89} -10.2462 q^{91} -1.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} -0.246211 q^{97} -19.2462 q^{98} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} + 2 q^{28} + 4 q^{29} - 2 q^{32} + 2 q^{33} + 6 q^{34} + 2 q^{36} + 6 q^{37} - 8 q^{38} - 4 q^{39} + 4 q^{41} - 2 q^{42} + 2 q^{44} + 2 q^{46} + 16 q^{47} + 2 q^{48} + 22 q^{49} - 6 q^{51} - 4 q^{52} - 8 q^{53} - 2 q^{54} - 2 q^{56} + 8 q^{57} - 4 q^{58} - 12 q^{59} + 10 q^{61} + 2 q^{63} + 2 q^{64} - 2 q^{66} + 16 q^{67} - 6 q^{68} - 2 q^{69} - 4 q^{71} - 2 q^{72} - 8 q^{73} - 6 q^{74} + 8 q^{76} + 36 q^{77} + 4 q^{78} - 2 q^{79} + 2 q^{81} - 4 q^{82} + 2 q^{83} + 2 q^{84} + 4 q^{87} - 2 q^{88} - 2 q^{89} - 4 q^{91} - 2 q^{92} - 16 q^{94} - 2 q^{96} + 16 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^11 + 2 * q^12 - 4 * q^13 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 2 * q^18 + 8 * q^19 + 2 * q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 + 4 * q^26 + 2 * q^27 + 2 * q^28 + 4 * q^29 - 2 * q^32 + 2 * q^33 + 6 * q^34 + 2 * q^36 + 6 * q^37 - 8 * q^38 - 4 * q^39 + 4 * q^41 - 2 * q^42 + 2 * q^44 + 2 * q^46 + 16 * q^47 + 2 * q^48 + 22 * q^49 - 6 * q^51 - 4 * q^52 - 8 * q^53 - 2 * q^54 - 2 * q^56 + 8 * q^57 - 4 * q^58 - 12 * q^59 + 10 * q^61 + 2 * q^63 + 2 * q^64 - 2 * q^66 + 16 * q^67 - 6 * q^68 - 2 * q^69 - 4 * q^71 - 2 * q^72 - 8 * q^73 - 6 * q^74 + 8 * q^76 + 36 * q^77 + 4 * q^78 - 2 * q^79 + 2 * q^81 - 4 * q^82 + 2 * q^83 + 2 * q^84 + 4 * q^87 - 2 * q^88 - 2 * q^89 - 4 * q^91 - 2 * q^92 - 16 * q^94 - 2 * q^96 + 16 * q^97 - 22 * q^98 + 2 * q^99

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