LMFDB - Embedded newform 6030.2.d.l.2411.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (of order $2$, degree $1$, 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:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.12
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.l.2411.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.0653098i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.0653098i q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.77403 q^{11} -2.43181i q^{13} -0.0653098i q^{14} +1.00000 q^{16} +0.153991i q^{17} -7.53622 q^{19} +1.00000 q^{20} -1.77403 q^{22} -2.47214i q^{23} +1.00000 q^{25} -2.43181i q^{26} -0.0653098i q^{28} -4.70989i q^{29} -2.93050i q^{31} +1.00000 q^{32} +0.153991i q^{34} -0.0653098i q^{35} -7.68209 q^{37} -7.53622 q^{38} +1.00000 q^{40} -6.63164 q^{41} +4.71603i q^{43} -1.77403 q^{44} -2.47214i q^{46} -6.51453i q^{47} +6.99573 q^{49} +1.00000 q^{50} -2.43181i q^{52} -10.9762 q^{53} -1.77403 q^{55} -0.0653098i q^{56} -4.70989i q^{58} -0.249084i q^{59} -4.77423i q^{61} -2.93050i q^{62} +1.00000 q^{64} -2.43181i q^{65} +(-1.89948 - 7.96191i) q^{67} +0.153991i q^{68} -0.0653098i q^{70} +0.869907i q^{71} +6.50148 q^{73} -7.68209 q^{74} -7.53622 q^{76} +0.115862i q^{77} -6.75242i q^{79} +1.00000 q^{80} -6.63164 q^{82} +0.191151i q^{83} +0.153991i q^{85} +4.71603i q^{86} -1.77403 q^{88} +9.07101i q^{89} -0.158821 q^{91} -2.47214i q^{92} -6.51453i q^{94} -7.53622 q^{95} +5.89823i q^{97} +6.99573 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8} + 24 q^{10} + 12 q^{11} + 24 q^{16} + 4 q^{19} + 24 q^{20} + 12 q^{22} + 24 q^{25} + 24 q^{32} - 16 q^{37} + 4 q^{38} + 24 q^{40} + 8 q^{41} + 12 q^{44} - 20 q^{49} + 24 q^{50} + 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} - 16 q^{74} + 4 q^{76} + 24 q^{80} + 8 q^{82} + 12 q^{88} + 4 q^{95} - 20 q^{98}+O(q^{100}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8 + 24 * q^10 + 12 * q^11 + 24 * q^16 + 4 * q^19 + 24 * q^20 + 12 * q^22 + 24 * q^25 + 24 * q^32 - 16 * q^37 + 4 * q^38 + 24 * q^40 + 8 * q^41 + 12 * q^44 - 20 * q^49 + 24 * q^50 + 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 - 16 * q^74 + 4 * q^76 + 24 * q^80 + 8 * q^82 + 12 * q^88 + 4 * q^95 - 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	0.0653098i		−	0.0246848i	−0.999924	−	0.0123424i	$-0.996071\pi$
$7$		−	0.0653098i		−	0.0246848i	0.999924	−	0.0123424i	$-0.00392881\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	−1.77403			−0.534891			−0.267446	−	0.963573i	$-0.586180\pi$
$11$	−1.77403			−0.534891			−0.267446	+	0.963573i	$0.586180\pi$
$12$		0			0
$13$		−	2.43181i		−	0.674462i	−0.941422	−	0.337231i	$-0.890510\pi$
$13$		−	2.43181i		−	0.674462i	0.941422	−	0.337231i	$-0.109490\pi$
$14$		−	0.0653098i		−	0.0174548i
$15$		0			0
$16$	1.00000			0.250000
$17$			0.153991i			0.0373484i	0.999826	+	0.0186742i	$0.00594453\pi$
$17$			0.153991i			0.0373484i	−0.999826	+	0.0186742i	$0.994055\pi$
$18$		0			0
$19$	−7.53622			−1.72893			−0.864464	−	0.502695i	$-0.832342\pi$
$19$	−7.53622			−1.72893			−0.864464	+	0.502695i	$0.832342\pi$
$20$	1.00000			0.223607
$21$		0			0
$22$	−1.77403			−0.378225
$23$		−	2.47214i		−	0.515476i	−0.966215	−	0.257738i	$-0.917023\pi$
$23$		−	2.47214i		−	0.515476i	0.966215	−	0.257738i	$-0.0829771\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	2.43181i		−	0.476917i
$27$		0			0
$28$		−	0.0653098i		−	0.0123424i
$29$		−	4.70989i		−	0.874604i	−0.899315	−	0.437302i	$-0.855934\pi$
$29$		−	4.70989i		−	0.874604i	0.899315	−	0.437302i	$-0.144066\pi$
$30$		0			0
$31$		−	2.93050i		−	0.526334i	−0.964750	−	0.263167i	$-0.915233\pi$
$31$		−	2.93050i		−	0.526334i	0.964750	−	0.263167i	$-0.0847670\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			0.153991i			0.0264093i
$35$		−	0.0653098i		−	0.0110394i
$36$		0			0
$37$	−7.68209			−1.26293			−0.631464	−	0.775405i	$-0.717546\pi$
$37$	−7.68209			−1.26293			−0.631464	+	0.775405i	$0.717546\pi$
$38$	−7.53622			−1.22254
$39$		0			0
$40$	1.00000			0.158114
$41$	−6.63164			−1.03569			−0.517844	−	0.855475i	$-0.673265\pi$
$41$	−6.63164			−1.03569			−0.517844	+	0.855475i	$0.673265\pi$
$42$		0			0
$43$			4.71603i			0.719188i	0.933109	+	0.359594i	$0.117085\pi$
$43$			4.71603i			0.719188i	−0.933109	+	0.359594i	$0.882915\pi$
$44$	−1.77403			−0.267446
$45$		0			0
$46$		−	2.47214i		−	0.364496i
$47$		−	6.51453i		−	0.950242i	−0.879921	−	0.475121i	$-0.842404\pi$
$47$		−	6.51453i		−	0.950242i	0.879921	−	0.475121i	$-0.157596\pi$
$48$		0			0
$49$	6.99573			0.999391
$50$	1.00000			0.141421
$51$		0			0
$52$		−	2.43181i		−	0.337231i
$53$	−10.9762			−1.50770			−0.753849	−	0.657048i	$-0.771805\pi$
$53$	−10.9762			−1.50770			−0.753849	+	0.657048i	$0.771805\pi$
$54$		0			0
$55$	−1.77403			−0.239211
$56$		−	0.0653098i		−	0.00872739i
$57$		0			0
$58$		−	4.70989i		−	0.618439i
$59$		−	0.249084i		−	0.0324279i	−0.999869	−	0.0162140i	$-0.994839\pi$
$59$		−	0.249084i		−	0.0324279i	0.999869	−	0.0162140i	$-0.00516129\pi$
$60$		0			0
$61$		−	4.77423i		−	0.611277i	−0.952148	−	0.305639i	$-0.901130\pi$
$61$		−	4.77423i		−	0.611277i	0.952148	−	0.305639i	$-0.0988700\pi$
$62$		−	2.93050i		−	0.372174i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	2.43181i		−	0.301629i
$66$		0			0
$67$	−1.89948	−	7.96191i	−0.232058	−	0.972702i
$67$	−1.89948	−	7.96191i	−0.232058	−	0.972702i
$68$			0.153991i			0.0186742i
$69$		0			0
$70$		−	0.0653098i		−	0.00780602i
$71$			0.869907i			0.103239i	0.998667	+	0.0516195i	$0.0164383\pi$
$71$			0.869907i			0.103239i	−0.998667	+	0.0516195i	$0.983562\pi$
$72$		0			0
$73$	6.50148			0.760941			0.380470	−	0.924793i	$-0.375762\pi$
$73$	6.50148			0.760941			0.380470	+	0.924793i	$0.375762\pi$
$74$	−7.68209			−0.893025
$75$		0			0
$76$	−7.53622			−0.864464
$77$			0.115862i			0.0132037i
$78$		0			0
$79$		−	6.75242i		−	0.759707i	−0.925047	−	0.379853i	$-0.875974\pi$
$79$		−	6.75242i		−	0.759707i	0.925047	−	0.379853i	$-0.124026\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	−6.63164			−0.732342
$83$			0.191151i			0.0209815i	0.999945	+	0.0104908i	$0.00333937\pi$
$83$			0.191151i			0.0209815i	−0.999945	+	0.0104908i	$0.996661\pi$
$84$		0			0
$85$			0.153991i			0.0167027i
$86$			4.71603i			0.508543i
$87$		0			0
$88$	−1.77403			−0.189113
$89$			9.07101i			0.961525i	0.876851	+	0.480762i	$0.159640\pi$
$89$			9.07101i			0.961525i	−0.876851	+	0.480762i	$0.840360\pi$
$90$		0			0
$91$	−0.158821			−0.0166490
$92$		−	2.47214i		−	0.257738i
$93$		0			0
$94$		−	6.51453i		−	0.671922i
$95$	−7.53622			−0.773200
$96$		0			0
$97$			5.89823i			0.598874i	0.954116	+	0.299437i	$0.0967989\pi$
$97$			5.89823i			0.598874i	−0.954116	+	0.299437i	$0.903201\pi$
$98$	6.99573			0.706676
$99$		0			0
$100$	1.00000			0.100000
$101$	−9.76869			−0.972021			−0.486010	−	0.873953i	$-0.661548\pi$
$101$	−9.76869			−0.972021			−0.486010	+	0.873953i	$0.661548\pi$
$102$		0			0
$103$	0.522248			0.0514586			0.0257293	−	0.999669i	$-0.491809\pi$
$103$	0.522248			0.0514586			0.0257293	+	0.999669i	$0.491809\pi$
$104$		−	2.43181i		−	0.238458i
$105$		0			0
$106$	−10.9762			−1.06610
$107$		−	5.73358i		−	0.554286i	−0.960829	−	0.277143i	$-0.910612\pi$
$107$		−	5.73358i		−	0.554286i	0.960829	−	0.277143i	$-0.0893876\pi$
$108$		0			0
$109$			2.56974i			0.246137i	0.992398	+	0.123068i	$0.0392734\pi$
$109$			2.56974i			0.246137i	−0.992398	+	0.123068i	$0.960727\pi$
$110$	−1.77403			−0.169147
$111$		0			0
$112$		−	0.0653098i		−	0.00617120i
$113$	4.20520			0.395592			0.197796	−	0.980243i	$-0.436622\pi$
$113$	4.20520			0.395592			0.197796	+	0.980243i	$0.436622\pi$
$114$		0			0
$115$		−	2.47214i		−	0.230528i
$116$		−	4.70989i		−	0.437302i
$117$		0			0
$118$		−	0.249084i		−	0.0229300i
$119$	0.0100572			0.000921937
$120$		0			0
$121$	−7.85281			−0.713891
$122$		−	4.77423i		−	0.432238i
$123$		0			0
$124$		−	2.93050i		−	0.263167i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−2.02370			−0.179575			−0.0897874	−	0.995961i	$-0.528619\pi$
$127$	−2.02370			−0.179575			−0.0897874	+	0.995961i	$0.528619\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	2.43181i		−	0.213284i
$131$		−	12.4384i		−	1.08675i	−0.839489	−	0.543376i	$-0.817146\pi$
$131$		−	12.4384i		−	1.08675i	0.839489	−	0.543376i	$-0.182854\pi$
$132$		0			0
$133$			0.492189i			0.0426782i
$134$	−1.89948	−	7.96191i	−0.164090	−	0.687804i
$135$		0			0
$136$			0.153991i			0.0132047i
$137$	11.9283			1.01910			0.509552	−	0.860440i	$-0.329811\pi$
$137$	11.9283			1.01910			0.509552	+	0.860440i	$0.329811\pi$
$138$		0			0
$139$		−	22.5170i		−	1.90987i	−0.296815	−	0.954935i	$-0.595925\pi$
$139$		−	22.5170i		−	1.90987i	0.296815	−	0.954935i	$-0.404075\pi$
$140$		−	0.0653098i		−	0.00551969i
$141$		0			0
$142$			0.869907i			0.0730010i
$143$			4.31411i			0.360764i
$144$		0			0
$145$		−	4.70989i		−	0.391135i
$146$	6.50148			0.538066
$147$		0			0
$148$	−7.68209			−0.631464
$149$		−	4.83865i		−	0.396397i	−0.980162	−	0.198199i	$-0.936491\pi$
$149$		−	4.83865i		−	0.396397i	0.980162	−	0.198199i	$-0.0635091\pi$
$150$		0			0
$151$	4.83447			0.393423			0.196712	−	0.980461i	$-0.436974\pi$
$151$	4.83447			0.393423			0.196712	+	0.980461i	$0.436974\pi$
$152$	−7.53622			−0.611268
$153$		0			0
$154$			0.115862i			0.00933641i
$155$		−	2.93050i		−	0.235384i
$156$		0			0
$157$	5.33801			0.426019			0.213010	−	0.977050i	$-0.431673\pi$
$157$	5.33801			0.426019			0.213010	+	0.977050i	$0.431673\pi$
$158$		−	6.75242i		−	0.537194i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−0.161455			−0.0127244
$162$		0			0
$163$	−8.18026			−0.640727			−0.320364	−	0.947295i	$-0.603805\pi$
$163$	−8.18026			−0.640727			−0.320364	+	0.947295i	$0.603805\pi$
$164$	−6.63164			−0.517844
$165$		0			0
$166$			0.191151i			0.0148362i
$167$			12.6623i			0.979837i	0.871768	+	0.489918i	$0.162973\pi$
$167$			12.6623i			0.979837i	−0.871768	+	0.489918i	$0.837027\pi$
$168$		0			0
$169$	7.08631			0.545101
$170$			0.153991i			0.0118106i
$171$		0			0
$172$			4.71603i			0.359594i
$173$			13.4777i			1.02469i	0.858780	+	0.512344i	$0.171223\pi$
$173$			13.4777i			1.02469i	−0.858780	+	0.512344i	$0.828777\pi$
$174$		0			0
$175$		−	0.0653098i		−	0.00493696i
$176$	−1.77403			−0.133723
$177$		0			0
$178$			9.07101i			0.679901i
$179$	−11.2585			−0.841497			−0.420749	−	0.907177i	$-0.638233\pi$
$179$	−11.2585			−0.841497			−0.420749	+	0.907177i	$0.638233\pi$
$180$		0			0
$181$	−7.85745			−0.584040			−0.292020	−	0.956412i	$-0.594327\pi$
$181$	−7.85745			−0.584040			−0.292020	+	0.956412i	$0.594327\pi$
$182$	−0.158821			−0.0117726
$183$		0			0
$184$		−	2.47214i		−	0.182248i
$185$	−7.68209			−0.564798
$186$		0			0
$187$		−	0.273186i		−	0.0199773i
$188$		−	6.51453i		−	0.475121i
$189$		0			0
$190$	−7.53622			−0.546735
$191$	−12.8617			−0.930641			−0.465320	−	0.885142i	$-0.654061\pi$
$191$	−12.8617			−0.930641			−0.465320	+	0.885142i	$0.654061\pi$
$192$		0			0
$193$	14.0408			1.01068			0.505341	−	0.862920i	$-0.331367\pi$
$193$	14.0408			1.01068			0.505341	+	0.862920i	$0.331367\pi$
$194$			5.89823i			0.423468i
$195$		0			0
$196$	6.99573			0.499695
$197$	0.308330			0.0219676			0.0109838	−	0.999940i	$-0.496504\pi$
$197$	0.308330			0.0219676			0.0109838	+	0.999940i	$0.496504\pi$
$198$		0			0
$199$	8.42981			0.597573			0.298787	−	0.954320i	$-0.403418\pi$
$199$	8.42981			0.597573			0.298787	+	0.954320i	$0.403418\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	−9.76869			−0.687323
$203$	−0.307602			−0.0215894
$204$		0			0
$205$	−6.63164			−0.463174
$206$	0.522248			0.0363867
$207$		0			0
$208$		−	2.43181i		−	0.168616i
$209$	13.3695			0.924788
$210$		0			0
$211$	17.9482			1.23561			0.617804	−	0.786332i	$-0.288023\pi$
$211$	17.9482			1.23561			0.617804	+	0.786332i	$0.288023\pi$
$212$	−10.9762			−0.753849
$213$		0			0
$214$		−	5.73358i		−	0.391939i
$215$			4.71603i			0.321631i
$216$		0			0
$217$	−0.191391			−0.0129924
$218$			2.56974i			0.174045i
$219$		0			0
$220$	−1.77403			−0.119605
$221$	0.374478			0.0251901
$222$		0			0
$223$	−12.3469			−0.826808			−0.413404	−	0.910548i	$-0.635660\pi$
$223$	−12.3469			−0.826808			−0.413404	+	0.910548i	$0.635660\pi$
$224$		−	0.0653098i		−	0.00436370i
$225$		0			0
$226$	4.20520			0.279726
$227$		−	22.0078i		−	1.46071i	−0.683068	−	0.730355i	$-0.739355\pi$
$227$		−	22.0078i		−	1.46071i	0.683068	−	0.730355i	$-0.260645\pi$
$228$		0			0
$229$			22.4736i			1.48510i	0.669793	+	0.742548i	$0.266383\pi$
$229$			22.4736i			1.48510i	−0.669793	+	0.742548i	$0.733617\pi$
$230$		−	2.47214i		−	0.163008i
$231$		0			0
$232$		−	4.70989i		−	0.309219i
$233$	−13.0257			−0.853343			−0.426671	−	0.904407i	$-0.640314\pi$
$233$	−13.0257			−0.853343			−0.426671	+	0.904407i	$0.640314\pi$
$234$		0			0
$235$		−	6.51453i		−	0.424961i
$236$		−	0.249084i		−	0.0162140i
$237$		0			0
$238$	0.0100572			0.000651908
$239$	−10.3177			−0.667394			−0.333697	−	0.942680i	$-0.608296\pi$
$239$	−10.3177			−0.667394			−0.333697	+	0.942680i	$0.608296\pi$
$240$		0			0
$241$	24.7261			1.59275			0.796375	−	0.604803i	$-0.206748\pi$
$241$	24.7261			1.59275			0.796375	+	0.604803i	$0.206748\pi$
$242$	−7.85281			−0.504797
$243$		0			0
$244$		−	4.77423i		−	0.305639i
$245$	6.99573			0.446941
$246$		0			0
$247$			18.3266i			1.16610i
$248$		−	2.93050i		−	0.186087i
$249$		0			0
$250$	1.00000			0.0632456
$251$	−4.18360			−0.264066			−0.132033	−	0.991245i	$-0.542151\pi$
$251$	−4.18360			−0.264066			−0.132033	+	0.991245i	$0.542151\pi$
$252$		0			0
$253$			4.38565i			0.275724i
$254$	−2.02370			−0.126979
$255$		0			0
$256$	1.00000			0.0625000
$257$			25.4086i			1.58495i	0.609907	+	0.792473i	$0.291207\pi$
$257$			25.4086i			1.58495i	−0.609907	+	0.792473i	$0.708793\pi$
$258$		0			0
$259$			0.501716i			0.0311751i
$260$		−	2.43181i		−	0.150814i
$261$		0			0
$262$		−	12.4384i		−	0.768450i
$263$			7.95225i			0.490356i	0.969478	+	0.245178i	$0.0788465\pi$
$263$			7.95225i			0.490356i	−0.969478	+	0.245178i	$0.921154\pi$
$264$		0			0
$265$	−10.9762			−0.674263
$266$			0.492189i			0.0301781i
$267$		0			0
$268$	−1.89948	−	7.96191i	−0.116029	−	0.486351i
$269$			10.1919i			0.621409i	0.950507	+	0.310704i	$0.100565\pi$
$269$			10.1919i			0.621409i	−0.950507	+	0.310704i	$0.899435\pi$
$270$		0			0
$271$		−	25.0474i		−	1.52152i	−0.649032	−	0.760761i	$-0.724826\pi$
$271$		−	25.0474i		−	1.52152i	0.649032	−	0.760761i	$-0.275174\pi$
$272$			0.153991i			0.00933710i
$273$		0			0
$274$	11.9283			0.720615
$275$	−1.77403			−0.106978
$276$		0			0
$277$	3.80033			0.228340			0.114170	−	0.993461i	$-0.463579\pi$
$277$	3.80033			0.228340			0.114170	+	0.993461i	$0.463579\pi$
$278$		−	22.5170i		−	1.35048i
$279$		0			0
$280$		−	0.0653098i		−	0.00390301i
$281$	−2.75854			−0.164561			−0.0822804	−	0.996609i	$-0.526220\pi$
$281$	−2.75854			−0.164561			−0.0822804	+	0.996609i	$0.526220\pi$
$282$		0			0
$283$	−5.49204			−0.326468			−0.163234	−	0.986587i	$-0.552193\pi$
$283$	−5.49204			−0.326468			−0.163234	+	0.986587i	$0.552193\pi$
$284$			0.869907i			0.0516195i
$285$		0			0
$286$			4.31411i			0.255099i
$287$			0.433111i			0.0255657i
$288$		0			0
$289$	16.9763			0.998605
$290$		−	4.70989i		−	0.276574i
$291$		0			0
$292$	6.50148			0.380470
$293$		−	24.8003i		−	1.44885i	−0.689355	−	0.724423i	$-0.742106\pi$
$293$		−	24.8003i		−	1.44885i	0.689355	−	0.724423i	$-0.257894\pi$
$294$		0			0
$295$		−	0.249084i		−	0.0145022i
$296$	−7.68209			−0.446512
$297$		0			0
$298$		−	4.83865i		−	0.280295i
$299$	−6.01176			−0.347669
$300$		0			0
$301$	0.308003			0.0177530
$302$	4.83447			0.278192
$303$		0			0
$304$	−7.53622			−0.432232
$305$		−	4.77423i		−	0.273372i
$306$		0			0
$307$	8.16194			0.465827			0.232913	−	0.972497i	$-0.425174\pi$
$307$	8.16194			0.465827			0.232913	+	0.972497i	$0.425174\pi$
$308$			0.115862i			0.00660184i
$309$		0			0
$310$		−	2.93050i		−	0.166441i
$311$	−30.1040			−1.70704			−0.853520	−	0.521060i	$-0.825537\pi$
$311$	−30.1040			−1.70704			−0.853520	+	0.521060i	$0.825537\pi$
$312$		0			0
$313$		−	24.0475i		−	1.35925i	−0.733561	−	0.679623i	$-0.762143\pi$
$313$		−	24.0475i		−	1.35925i	0.733561	−	0.679623i	$-0.237857\pi$
$314$	5.33801			0.301241
$315$		0			0
$316$		−	6.75242i		−	0.379853i
$317$		−	12.7637i		−	0.716879i	−0.933553	−	0.358440i	$-0.883309\pi$
$317$		−	12.7637i		−	0.716879i	0.933553	−	0.358440i	$-0.116691\pi$
$318$		0			0
$319$			8.35550i			0.467818i
$320$	1.00000			0.0559017
$321$		0			0
$322$	−0.161455			−0.00899752
$323$		−	1.16051i		−	0.0645727i
$324$		0			0
$325$		−	2.43181i		−	0.134892i
$326$	−8.18026			−0.453062
$327$		0			0
$328$	−6.63164			−0.366171
$329$	−0.425463			−0.0234565
$330$		0			0
$331$		−	22.8567i		−	1.25632i	−0.778085	−	0.628158i	$-0.783809\pi$
$331$		−	22.8567i		−	1.25632i	0.778085	−	0.628158i	$-0.216191\pi$
$332$			0.191151i			0.0104908i
$333$		0			0
$334$			12.6623i			0.692849i
$335$	−1.89948	−	7.96191i	−0.103779	−	0.435006i
$336$		0			0
$337$		−	4.35733i		−	0.237359i	−0.992933	−	0.118679i	$-0.962134\pi$
$337$		−	4.35733i		−	0.237359i	0.992933	−	0.118679i	$-0.0378661\pi$
$338$	7.08631			0.385444
$339$		0			0
$340$			0.153991i			0.00835136i
$341$			5.19881i			0.281531i
$342$		0			0
$343$		−	0.914059i		−	0.0493545i
$344$			4.71603i			0.254271i
$345$		0			0
$346$			13.4777i			0.724564i
$347$	12.2164			0.655808			0.327904	−	0.944711i	$-0.393658\pi$
$347$	12.2164			0.655808			0.327904	+	0.944711i	$0.393658\pi$
$348$		0			0
$349$	0.809753			0.0433451			0.0216726	−	0.999765i	$-0.493101\pi$
$349$	0.809753			0.0433451			0.0216726	+	0.999765i	$0.493101\pi$
$350$		−	0.0653098i		−	0.00349096i
$351$		0			0
$352$	−1.77403			−0.0945563
$353$	−21.5978			−1.14954			−0.574768	−	0.818316i	$-0.694908\pi$
$353$	−21.5978			−1.14954			−0.574768	+	0.818316i	$0.694908\pi$
$354$		0			0
$355$			0.869907i			0.0461699i
$356$			9.07101i			0.480762i
$357$		0			0
$358$	−11.2585			−0.595028
$359$			12.0551i			0.636245i	0.948050	+	0.318123i	$0.103052\pi$
$359$			12.0551i			0.636245i	−0.948050	+	0.318123i	$0.896948\pi$
$360$		0			0
$361$	37.7946			1.98919
$362$	−7.85745			−0.412979
$363$		0			0
$364$	−0.158821			−0.00832448
$365$	6.50148			0.340303
$366$		0			0
$367$		−	12.5880i		−	0.657089i	−0.944489	−	0.328544i	$-0.893442\pi$
$367$		−	12.5880i		−	0.657089i	0.944489	−	0.328544i	$-0.106558\pi$
$368$		−	2.47214i		−	0.128869i
$369$		0			0
$370$	−7.68209			−0.399373
$371$			0.716854i			0.0372172i
$372$		0			0
$373$		−	18.4622i		−	0.955934i	−0.878378	−	0.477967i	$-0.841374\pi$
$373$		−	18.4622i		−	0.955934i	0.878378	−	0.477967i	$-0.158626\pi$
$374$		−	0.273186i		−	0.0141261i
$375$		0			0
$376$		−	6.51453i		−	0.335961i
$377$	−11.4535			−0.589888
$378$		0			0
$379$		−	13.7123i		−	0.704353i	−0.935934	−	0.352177i	$-0.885442\pi$
$379$		−	13.7123i		−	0.704353i	0.935934	−	0.352177i	$-0.114558\pi$
$380$	−7.53622			−0.386600
$381$		0			0
$382$	−12.8617			−0.658062
$383$	−6.63995			−0.339286			−0.169643	−	0.985506i	$-0.554261\pi$
$383$	−6.63995			−0.339286			−0.169643	+	0.985506i	$0.554261\pi$
$384$		0			0
$385$			0.115862i			0.00590486i
$386$	14.0408			0.714660
$387$		0			0
$388$			5.89823i			0.299437i
$389$			32.0186i			1.62341i	0.584068	+	0.811704i	$0.301460\pi$
$389$			32.0186i			1.62341i	−0.584068	+	0.811704i	$0.698540\pi$
$390$		0			0
$391$	0.380688			0.0192522
$392$	6.99573			0.353338
$393$		0			0
$394$	0.308330			0.0155334
$395$		−	6.75242i		−	0.339751i
$396$		0			0
$397$	−28.9222			−1.45156			−0.725782	−	0.687925i	$-0.758522\pi$
$397$	−28.9222			−1.45156			−0.725782	+	0.687925i	$0.758522\pi$
$398$	8.42981			0.422548
$399$		0			0
$400$	1.00000			0.0500000
$401$	−5.21024			−0.260187			−0.130094	−	0.991502i	$-0.541528\pi$
$401$	−5.21024			−0.260187			−0.130094	+	0.991502i	$0.541528\pi$
$402$		0			0
$403$	−7.12642			−0.354992
$404$	−9.76869			−0.486010
$405$		0			0
$406$	−0.307602			−0.0152660
$407$	13.6283			0.675529
$408$		0			0
$409$			34.9742i			1.72936i	0.502321	+	0.864681i	$0.332480\pi$
$409$			34.9742i			1.72936i	−0.502321	+	0.864681i	$0.667520\pi$
$410$	−6.63164			−0.327513
$411$		0			0
$412$	0.522248			0.0257293
$413$	−0.0162676			−0.000800476
$414$		0			0
$415$			0.191151i			0.00938322i
$416$		−	2.43181i		−	0.119229i
$417$		0			0
$418$	13.3695			0.653924
$419$		−	3.86279i		−	0.188710i	−0.995539	−	0.0943549i	$-0.969921\pi$
$419$		−	3.86279i		−	0.188710i	0.995539	−	0.0943549i	$-0.0300788\pi$
$420$		0			0
$421$	15.9611			0.777898			0.388949	−	0.921259i	$-0.372838\pi$
$421$	15.9611			0.777898			0.388949	+	0.921259i	$0.372838\pi$
$422$	17.9482			0.873707
$423$		0			0
$424$	−10.9762			−0.533052
$425$			0.153991i			0.00746968i
$426$		0			0
$427$	−0.311804			−0.0150893
$428$		−	5.73358i		−	0.277143i
$429$		0			0
$430$			4.71603i			0.227427i
$431$			8.57010i			0.412807i	0.978467	+	0.206404i	$0.0661760\pi$
$431$			8.57010i			0.412807i	−0.978467	+	0.206404i	$0.933824\pi$
$432$		0			0
$433$		−	15.2995i		−	0.735247i	−0.929975	−	0.367623i	$-0.880172\pi$
$433$		−	15.2995i		−	0.735247i	0.929975	−	0.367623i	$-0.119828\pi$
$434$	−0.191391			−0.00918704
$435$		0			0
$436$			2.56974i			0.123068i
$437$			18.6306i			0.891221i
$438$		0			0
$439$	16.4912			0.787082			0.393541	−	0.919307i	$-0.371250\pi$
$439$	16.4912			0.787082			0.393541	+	0.919307i	$0.371250\pi$
$440$	−1.77403			−0.0845737
$441$		0			0
$442$	0.374478			0.0178121
$443$	−0.145588			−0.00691711			−0.00345855	−	0.999994i	$-0.501101\pi$
$443$	−0.145588			−0.00691711			−0.00345855	+	0.999994i	$0.501101\pi$
$444$		0			0
$445$			9.07101i			0.430007i
$446$	−12.3469			−0.584642
$447$		0			0
$448$		−	0.0653098i		−	0.00308560i
$449$			3.32880i			0.157096i	0.996910	+	0.0785480i	$0.0250284\pi$
$449$			3.32880i			0.157096i	−0.996910	+	0.0785480i	$0.974972\pi$
$450$		0			0
$451$	11.7647			0.553980
$452$	4.20520			0.197796
$453$		0			0
$454$		−	22.0078i		−	1.03288i
$455$	−0.158821			−0.00744564
$456$		0			0
$457$	−3.19043			−0.149242			−0.0746210	−	0.997212i	$-0.523775\pi$
$457$	−3.19043			−0.149242			−0.0746210	+	0.997212i	$0.523775\pi$
$458$			22.4736i			1.05012i
$459$		0			0
$460$		−	2.47214i		−	0.115264i
$461$			11.5897i			0.539787i	0.962890	+	0.269894i	$0.0869885\pi$
$461$			11.5897i			0.539787i	−0.962890	+	0.269894i	$0.913011\pi$
$462$		0			0
$463$			18.4892i			0.859265i	0.903004	+	0.429633i	$0.141357\pi$
$463$			18.4892i			0.859265i	−0.903004	+	0.429633i	$0.858643\pi$
$464$		−	4.70989i		−	0.218651i
$465$		0			0
$466$	−13.0257			−0.603404
$467$		−	1.63639i		−	0.0757231i	−0.999283	−	0.0378616i	$-0.987945\pi$
$467$		−	1.63639i		−	0.0757231i	0.999283	−	0.0378616i	$-0.0120546\pi$
$468$		0			0
$469$	−0.519991	+	0.124054i	−0.0240109	+	0.00572830i
$470$		−	6.51453i		−	0.300493i
$471$		0			0
$472$		−	0.249084i		−	0.0114650i
$473$		−	8.36639i		−	0.384687i
$474$		0			0
$475$	−7.53622			−0.345786
$476$	0.0100572			0.000460969
$477$		0			0
$478$	−10.3177			−0.471919
$479$		−	5.70312i		−	0.260582i	−0.991476	−	0.130291i	$-0.958409\pi$
$479$		−	5.70312i		−	0.260582i	0.991476	−	0.130291i	$-0.0415912\pi$
$480$		0			0
$481$			18.6814i			0.851797i
$482$	24.7261			1.12624
$483$		0			0
$484$	−7.85281			−0.356946
$485$			5.89823i			0.267825i
$486$		0			0
$487$			26.7105i			1.21037i	0.796085	+	0.605184i	$0.206901\pi$
$487$			26.7105i			1.21037i	−0.796085	+	0.605184i	$0.793099\pi$
$488$		−	4.77423i		−	0.216119i
$489$		0			0
$490$	6.99573			0.316035
$491$		−	1.60752i		−	0.0725464i	−0.999342	−	0.0362732i	$-0.988451\pi$
$491$		−	1.60752i		−	0.0725464i	0.999342	−	0.0362732i	$-0.0115487\pi$
$492$		0			0
$493$	0.725282			0.0326651
$494$			18.3266i			0.824555i
$495$		0			0
$496$		−	2.93050i		−	0.131583i
$497$	0.0568135			0.00254843
$498$		0			0
$499$			12.8436i			0.574957i	0.957787	+	0.287479i	$0.0928170\pi$
$499$			12.8436i			0.574957i	−0.957787	+	0.287479i	$0.907183\pi$
$500$	1.00000			0.0447214
$501$		0			0
$502$	−4.18360			−0.186723
$503$	21.0938			0.940528			0.470264	−	0.882526i	$-0.344159\pi$
$503$	21.0938			0.940528			0.470264	+	0.882526i	$0.344159\pi$
$504$		0			0
$505$	−9.76869			−0.434701
$506$			4.38565i			0.194966i
$507$		0			0
$508$	−2.02370			−0.0897874
$509$		−	37.2310i		−	1.65024i	−0.564960	−	0.825119i	$-0.691108\pi$
$509$		−	37.2310i		−	1.65024i	0.564960	−	0.825119i	$-0.308892\pi$
$510$		0			0
$511$		−	0.424610i		−	0.0187837i
$512$	1.00000			0.0441942
$513$		0			0
$514$			25.4086i			1.12073i
$515$	0.522248			0.0230130
$516$		0			0
$517$			11.5570i			0.508276i
$518$			0.501716i			0.0220441i
$519$		0			0
$520$		−	2.43181i		−	0.106642i
$521$	−24.2109			−1.06070			−0.530349	−	0.847779i	$-0.677939\pi$
$521$	−24.2109			−1.06070			−0.530349	+	0.847779i	$0.677939\pi$
$522$		0			0
$523$	3.30138			0.144359			0.0721796	−	0.997392i	$-0.477005\pi$
$523$	3.30138			0.144359			0.0721796	+	0.997392i	$0.477005\pi$
$524$		−	12.4384i		−	0.543376i
$525$		0			0
$526$			7.95225i			0.346734i
$527$	0.451272			0.0196577
$528$		0			0
$529$	16.8885			0.734285
$530$	−10.9762			−0.476776
$531$		0			0
$532$			0.492189i			0.0213391i
$533$			16.1269i			0.698532i
$534$		0			0
$535$		−	5.73358i		−	0.247884i
$536$	−1.89948	−	7.96191i	−0.0820449	−	0.343902i
$537$		0			0
$538$			10.1919i			0.439402i
$539$	−12.4107			−0.534565
$540$		0			0
$541$			3.24352i			0.139450i	0.997566	+	0.0697248i	$0.0222121\pi$
$541$			3.24352i			0.139450i	−0.997566	+	0.0697248i	$0.977788\pi$
$542$		−	25.0474i		−	1.07588i
$543$		0			0
$544$			0.153991i			0.00660233i
$545$			2.56974i			0.110076i
$546$		0			0
$547$		−	10.5386i		−	0.450596i	−0.974290	−	0.225298i	$-0.927664\pi$
$547$		−	10.5386i		−	0.450596i	0.974290	−	0.225298i	$-0.0723356\pi$
$548$	11.9283			0.509552
$549$		0			0
$550$	−1.77403			−0.0756450
$551$			35.4948i			1.51213i
$552$		0			0
$553$	−0.441000			−0.0187532
$554$	3.80033			0.161461
$555$		0			0
$556$		−	22.5170i		−	0.954935i
$557$		−	27.7024i		−	1.17379i	−0.809663	−	0.586895i	$-0.800350\pi$
$557$		−	27.7024i		−	1.17379i	0.809663	−	0.586895i	$-0.199650\pi$
$558$		0			0
$559$	11.4685			0.485065
$560$		−	0.0653098i		−	0.00275984i
$561$		0			0
$562$	−2.75854			−0.116362
$563$	0.777978			0.0327879			0.0163939	−	0.999866i	$-0.494781\pi$
$563$	0.777978			0.0327879			0.0163939	+	0.999866i	$0.494781\pi$
$564$		0			0
$565$	4.20520			0.176914
$566$	−5.49204			−0.230848
$567$		0			0
$568$			0.869907i			0.0365005i
$569$			23.4062i			0.981240i	0.871374	+	0.490620i	$0.163230\pi$
$569$			23.4062i			0.981240i	−0.871374	+	0.490620i	$0.836770\pi$
$570$		0			0
$571$	−24.0284			−1.00556			−0.502779	−	0.864415i	$-0.667689\pi$
$571$	−24.0284			−1.00556			−0.502779	+	0.864415i	$0.667689\pi$
$572$			4.31411i			0.180382i
$573$		0			0
$574$			0.433111i			0.0180777i
$575$		−	2.47214i		−	0.103095i
$576$		0			0
$577$		−	37.5440i		−	1.56298i	−0.623920	−	0.781488i	$-0.714461\pi$
$577$		−	37.5440i		−	1.56298i	0.623920	−	0.781488i	$-0.285539\pi$
$578$	16.9763			0.706120
$579$		0			0
$580$		−	4.70989i		−	0.195567i
$581$	0.0124840			0.000517924
$582$		0			0
$583$	19.4721			0.806454
$584$	6.50148			0.269033
$585$		0			0
$586$		−	24.8003i		−	1.02449i
$587$	−35.4221			−1.46203			−0.731013	−	0.682364i	$-0.760952\pi$
$587$	−35.4221			−1.46203			−0.731013	+	0.682364i	$0.760952\pi$
$588$		0			0
$589$			22.0849i			0.909993i
$590$		−	0.249084i		−	0.0102546i
$591$		0			0
$592$	−7.68209			−0.315732
$593$	11.7848			0.483943			0.241971	−	0.970283i	$-0.422206\pi$
$593$	11.7848			0.483943			0.241971	+	0.970283i	$0.422206\pi$
$594$		0			0
$595$	0.0100572			0.000412303
$596$		−	4.83865i		−	0.198199i
$597$		0			0
$598$	−6.01176			−0.245839
$599$	13.3290			0.544606			0.272303	−	0.962211i	$-0.412215\pi$
$599$	13.3290			0.544606			0.272303	+	0.962211i	$0.412215\pi$
$600$		0			0
$601$	6.97705			0.284600			0.142300	−	0.989824i	$-0.454550\pi$
$601$	6.97705			0.284600			0.142300	+	0.989824i	$0.454550\pi$
$602$	0.308003			0.0125533
$603$		0			0
$604$	4.83447			0.196712
$605$	−7.85281			−0.319262
$606$		0			0
$607$	8.90675			0.361514			0.180757	−	0.983528i	$-0.442145\pi$
$607$	8.90675			0.361514			0.180757	+	0.983528i	$0.442145\pi$
$608$	−7.53622			−0.305634
$609$		0			0
$610$		−	4.77423i		−	0.193303i
$611$	−15.8421			−0.640902
$612$		0			0
$613$	35.5233			1.43477			0.717386	−	0.696676i	$-0.245339\pi$
$613$	35.5233			1.43477			0.717386	+	0.696676i	$0.245339\pi$
$614$	8.16194			0.329389
$615$		0			0
$616$			0.115862i			0.00466820i
$617$		−	28.9605i		−	1.16590i	−0.812506	−	0.582952i	$-0.801897\pi$
$617$		−	28.9605i		−	1.16590i	0.812506	−	0.582952i	$-0.198103\pi$
$618$		0			0
$619$	24.8930			1.00053			0.500266	−	0.865872i	$-0.333235\pi$
$619$	24.8930			1.00053			0.500266	+	0.865872i	$0.333235\pi$
$620$		−	2.93050i		−	0.117692i
$621$		0			0
$622$	−30.1040			−1.20706
$623$	0.592426			0.0237350
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	24.0475i		−	0.961133i
$627$		0			0
$628$	5.33801			0.213010
$629$		−	1.18298i		−	0.0471683i
$630$		0			0
$631$			17.1639i			0.683282i	0.939830	+	0.341641i	$0.110983\pi$
$631$			17.1639i			0.683282i	−0.939830	+	0.341641i	$0.889017\pi$
$632$		−	6.75242i		−	0.268597i
$633$		0			0
$634$		−	12.7637i		−	0.506910i
$635$	−2.02370			−0.0803083
$636$		0			0
$637$		−	17.0123i		−	0.674051i
$638$			8.35550i			0.330797i
$639$		0			0
$640$	1.00000			0.0395285
$641$	−19.7499			−0.780075			−0.390037	−	0.920799i	$-0.627538\pi$
$641$	−19.7499			−0.780075			−0.390037	+	0.920799i	$0.627538\pi$
$642$		0			0
$643$	7.23094			0.285160			0.142580	−	0.989783i	$-0.454460\pi$
$643$	7.23094			0.285160			0.142580	+	0.989783i	$0.454460\pi$
$644$	−0.161455			−0.00636221
$645$		0			0
$646$		−	1.16051i		−	0.0456598i
$647$	−13.9332			−0.547772			−0.273886	−	0.961762i	$-0.588309\pi$
$647$	−13.9332			−0.547772			−0.273886	+	0.961762i	$0.588309\pi$
$648$		0			0
$649$			0.441883i			0.0173454i
$650$		−	2.43181i		−	0.0953834i
$651$		0			0
$652$	−8.18026			−0.320364
$653$	−28.3837			−1.11074			−0.555371	−	0.831603i	$-0.687424\pi$
$653$	−28.3837			−1.11074			−0.555371	+	0.831603i	$0.687424\pi$
$654$		0			0
$655$		−	12.4384i		−	0.486010i
$656$	−6.63164			−0.258922
$657$		0			0
$658$	−0.425463			−0.0165863
$659$			48.6904i			1.89671i	0.317212	+	0.948355i	$0.397253\pi$
$659$			48.6904i			1.89671i	−0.317212	+	0.948355i	$0.602747\pi$
$660$		0			0
$661$		−	45.8500i		−	1.78336i	−0.452667	−	0.891680i	$-0.649527\pi$
$661$		−	45.8500i		−	1.78336i	0.452667	−	0.891680i	$-0.350473\pi$
$662$		−	22.8567i		−	0.888350i
$663$		0			0
$664$			0.191151i			0.00741808i
$665$			0.492189i			0.0190863i
$666$		0			0
$667$	−11.6435			−0.450837
$668$			12.6623i			0.489918i
$669$		0			0
$670$	−1.89948	−	7.96191i	−0.0733832	−	0.307595i
$671$			8.46964i			0.326967i
$672$		0			0
$673$		−	29.1290i		−	1.12284i	−0.827531	−	0.561420i	$-0.810255\pi$
$673$		−	29.1290i		−	1.12284i	0.827531	−	0.561420i	$-0.189745\pi$
$674$		−	4.35733i		−	0.167838i
$675$		0			0
$676$	7.08631			0.272550
$677$	−16.2820			−0.625767			−0.312883	−	0.949792i	$-0.601295\pi$
$677$	−16.2820			−0.625767			−0.312883	+	0.949792i	$0.601295\pi$
$678$		0			0
$679$	0.385212			0.0147831
$680$			0.153991i			0.00590530i
$681$		0			0
$682$			5.19881i			0.199073i
$683$	−13.7529			−0.526240			−0.263120	−	0.964763i	$-0.584752\pi$
$683$	−13.7529			−0.526240			−0.263120	+	0.964763i	$0.584752\pi$
$684$		0			0
$685$	11.9283			0.455757
$686$		−	0.914059i		−	0.0348989i
$687$		0			0
$688$			4.71603i			0.179797i
$689$			26.6920i			1.01689i
$690$		0			0
$691$	−6.53606			−0.248644			−0.124322	−	0.992242i	$-0.539676\pi$
$691$	−6.53606			−0.248644			−0.124322	+	0.992242i	$0.539676\pi$
$692$			13.4777i			0.512344i
$693$		0			0
$694$	12.2164			0.463727
$695$		−	22.5170i		−	0.854120i
$696$		0			0
$697$		−	1.02121i		−	0.0386813i
$698$	0.809753			0.0306496
$699$		0			0
$700$		−	0.0653098i		−	0.00246848i
$701$	34.0117			1.28460			0.642302	−	0.766452i	$-0.277980\pi$
$701$	34.0117			1.28460			0.642302	+	0.766452i	$0.277980\pi$
$702$		0			0
$703$	57.8939			2.18351
$704$	−1.77403			−0.0668614
$705$		0			0
$706$	−21.5978			−0.812845
$707$			0.637991i			0.0239941i
$708$		0			0
$709$	15.6106			0.586269			0.293135	−	0.956071i	$-0.405302\pi$
$709$	15.6106			0.586269			0.293135	+	0.956071i	$0.405302\pi$
$710$			0.869907i			0.0326470i
$711$		0			0
$712$			9.07101i			0.339950i
$713$	−7.24460			−0.271312
$714$		0			0
$715$			4.31411i			0.161339i
$716$	−11.2585			−0.420749
$717$		0			0
$718$			12.0551i			0.449893i
$719$		−	13.3039i		−	0.496153i	−0.968740	−	0.248077i	$-0.920201\pi$
$719$		−	13.3039i		−	0.496153i	0.968740	−	0.248077i	$-0.0797985\pi$
$720$		0			0
$721$		−	0.0341079i		−	0.00127025i
$722$	37.7946			1.40657
$723$		0			0
$724$	−7.85745			−0.292020
$725$		−	4.70989i		−	0.174921i
$726$		0			0
$727$		−	18.0075i		−	0.667863i	−0.942597	−	0.333931i	$-0.891625\pi$
$727$		−	18.0075i		−	0.667863i	0.942597	−	0.333931i	$-0.108375\pi$
$728$	−0.158821			−0.00588630
$729$		0			0
$730$	6.50148			0.240631
$731$	−0.726228			−0.0268605
$732$		0			0
$733$			35.6889i			1.31820i	0.752056	+	0.659100i	$0.229062\pi$
$733$			35.6889i			1.31820i	−0.752056	+	0.659100i	$0.770938\pi$
$734$		−	12.5880i		−	0.464632i
$735$		0			0
$736$		−	2.47214i		−	0.0911241i
$737$	3.36973	+	14.1247i	0.124126	+	0.520290i
$738$		0			0
$739$		−	11.1681i		−	0.410826i	−0.978675	−	0.205413i	$-0.934146\pi$
$739$		−	11.1681i		−	0.410826i	0.978675	−	0.205413i	$-0.0658538\pi$
$740$	−7.68209			−0.282399
$741$		0			0
$742$			0.716854i			0.0263165i
$743$			35.7844i			1.31280i	0.754413	+	0.656400i	$0.227922\pi$
$743$			35.7844i			1.31280i	−0.754413	+	0.656400i	$0.772078\pi$
$744$		0			0
$745$		−	4.83865i		−	0.177274i
$746$		−	18.4622i		−	0.675947i
$747$		0			0
$748$		−	0.273186i		−	0.00998866i
$749$	−0.374459			−0.0136824
$750$		0			0
$751$	33.1894			1.21110			0.605549	−	0.795808i	$-0.292954\pi$
$751$	33.1894			1.21110			0.605549	+	0.795808i	$0.292954\pi$
$752$		−	6.51453i		−	0.237560i
$753$		0			0
$754$	−11.4535			−0.417114
$755$	4.83447			0.175944
$756$		0			0
$757$			38.7915i			1.40990i	0.709257	+	0.704950i	$0.249031\pi$
$757$			38.7915i			1.40990i	−0.709257	+	0.704950i	$0.750969\pi$
$758$		−	13.7123i		−	0.498053i
$759$		0			0
$760$	−7.53622			−0.273368
$761$			11.9373i			0.432726i	0.976313	+	0.216363i	$0.0694195\pi$
$761$			11.9373i			0.432726i	−0.976313	+	0.216363i	$0.930580\pi$
$762$		0			0
$763$	0.167829			0.00607583
$764$	−12.8617			−0.465320
$765$		0			0
$766$	−6.63995			−0.239911
$767$	−0.605724			−0.0218714
$768$		0			0
$769$			33.0492i			1.19179i	0.803064	+	0.595893i	$0.203202\pi$
$769$			33.0492i			1.19179i	−0.803064	+	0.595893i	$0.796798\pi$
$770$			0.115862i			0.00417537i
$771$		0			0
$772$	14.0408			0.505341
$773$			26.5278i			0.954138i	0.878866	+	0.477069i	$0.158301\pi$
$773$			26.5278i			0.954138i	−0.878866	+	0.477069i	$0.841699\pi$
$774$		0			0
$775$		−	2.93050i		−	0.105267i
$776$			5.89823i			0.211734i
$777$		0			0
$778$			32.0186i			1.14792i
$779$	49.9775			1.79063
$780$		0			0
$781$		−	1.54324i		−	0.0552216i
$782$	0.380688			0.0136134
$783$		0			0
$784$	6.99573			0.249848
$785$	5.33801			0.190522
$786$		0			0
$787$			29.9813i			1.06872i	0.845257	+	0.534359i	$0.179447\pi$
$787$			29.9813i			1.06872i	−0.845257	+	0.534359i	$0.820553\pi$
$788$	0.308330			0.0109838
$789$		0			0
$790$		−	6.75242i		−	0.240240i
$791$		−	0.274641i		−	0.00976510i
$792$		0			0
$793$	−11.6100			−0.412284
$794$	−28.9222			−1.02641
$795$		0			0
$796$	8.42981			0.298787
$797$			31.7795i			1.12569i	0.826564	+	0.562843i	$0.190293\pi$
$797$			31.7795i			1.12569i	−0.826564	+	0.562843i	$0.809707\pi$
$798$		0			0
$799$	1.00318			0.0354900
$800$	1.00000			0.0353553
$801$		0			0
$802$	−5.21024			−0.183980
$803$	−11.5338			−0.407020
$804$		0			0
$805$	−0.161455			−0.00569053
$806$	−7.12642			−0.251018
$807$		0			0
$808$	−9.76869			−0.343661
$809$	25.8994			0.910574			0.455287	−	0.890345i	$-0.349537\pi$
$809$	25.8994			0.910574			0.455287	+	0.890345i	$0.349537\pi$
$810$		0			0
$811$			10.4680i			0.367580i	0.982966	+	0.183790i	$0.0588366\pi$
$811$			10.4680i			0.367580i	−0.982966	+	0.183790i	$0.941163\pi$
$812$	−0.307602			−0.0107947
$813$		0			0
$814$	13.6283			0.477671
$815$	−8.18026			−0.286542
$816$		0			0
$817$		−	35.5410i		−	1.24342i
$818$			34.9742i			1.22284i
$819$		0			0
$820$	−6.63164			−0.231587
$821$		−	11.3668i		−	0.396705i	−0.980131	−	0.198352i	$-0.936441\pi$
$821$		−	11.3668i		−	0.396705i	0.980131	−	0.198352i	$-0.0635591\pi$
$822$		0			0
$823$	−3.75400			−0.130856			−0.0654282	−	0.997857i	$-0.520841\pi$
$823$	−3.75400			−0.130856			−0.0654282	+	0.997857i	$0.520841\pi$
$824$	0.522248			0.0181934
$825$		0			0
$826$	−0.0162676			−0.000566022
$827$		−	44.5824i		−	1.55028i	−0.631789	−	0.775141i	$-0.717679\pi$
$827$		−	44.5824i		−	1.55028i	0.631789	−	0.775141i	$-0.282321\pi$
$828$		0			0
$829$	8.18144			0.284153			0.142077	−	0.989856i	$-0.454622\pi$
$829$	8.18144			0.284153			0.142077	+	0.989856i	$0.454622\pi$
$830$			0.191151i			0.00663494i
$831$		0			0
$832$		−	2.43181i		−	0.0843078i
$833$			1.07728i			0.0373256i
$834$		0			0
$835$			12.6623i			0.438196i
$836$	13.3695			0.462394
$837$		0			0
$838$		−	3.86279i		−	0.133438i
$839$			47.6723i			1.64583i	0.568163	+	0.822916i	$0.307654\pi$
$839$			47.6723i			1.64583i	−0.568163	+	0.822916i	$0.692346\pi$
$840$		0			0
$841$	6.81696			0.235067
$842$	15.9611			0.550057
$843$		0			0
$844$	17.9482			0.617804
$845$	7.08631			0.243776
$846$		0			0
$847$			0.512865i			0.0176223i
$848$	−10.9762			−0.376924
$849$		0			0
$850$			0.153991i			0.00528186i
$851$			18.9912i			0.651009i
$852$		0			0
$853$	45.8566			1.57010			0.785050	−	0.619433i	$-0.212637\pi$
$853$	45.8566			1.57010			0.785050	+	0.619433i	$0.212637\pi$
$854$	−0.311804			−0.0106697
$855$		0			0
$856$		−	5.73358i		−	0.195970i
$857$	22.7619			0.777532			0.388766	−	0.921336i	$-0.372901\pi$
$857$	22.7619			0.777532			0.388766	+	0.921336i	$0.372901\pi$
$858$		0			0
$859$	30.7398			1.04883			0.524414	−	0.851463i	$-0.324284\pi$
$859$	30.7398			1.04883			0.524414	+	0.851463i	$0.324284\pi$
$860$			4.71603i			0.160815i
$861$		0			0
$862$			8.57010i			0.291899i
$863$		−	17.9794i		−	0.612027i	−0.952027	−	0.306013i	$-0.901005\pi$
$863$		−	17.9794i		−	0.612027i	0.952027	−	0.306013i	$-0.0989952\pi$
$864$		0			0
$865$			13.4777i			0.458255i
$866$		−	15.2995i		−	0.519898i
$867$		0			0
$868$	−0.191391			−0.00649622
$869$			11.9790i			0.406361i
$870$		0			0
$871$	−19.3618	+	4.61916i	−0.656051	+	0.156514i
$872$			2.56974i			0.0870225i
$873$		0			0
$874$			18.6306i			0.630188i
$875$		−	0.0653098i		−	0.00220787i
$876$		0			0
$877$	−6.36926			−0.215075			−0.107537	−	0.994201i	$-0.534297\pi$
$877$	−6.36926			−0.215075			−0.107537	+	0.994201i	$0.534297\pi$
$878$	16.4912			0.556551
$879$		0			0
$880$	−1.77403			−0.0598027
$881$			5.97053i			0.201152i	0.994929	+	0.100576i	$0.0320686\pi$
$881$			5.97053i			0.201152i	−0.994929	+	0.100576i	$0.967931\pi$
$882$		0			0
$883$		−	4.36007i		−	0.146728i	−0.997305	−	0.0733641i	$-0.976626\pi$
$883$		−	4.36007i		−	0.146728i	0.997305	−	0.0733641i	$-0.0233735\pi$
$884$	0.374478			0.0125950
$885$		0			0
$886$	−0.145588			−0.00489113
$887$			5.17391i			0.173723i	0.996220	+	0.0868615i	$0.0276838\pi$
$887$			5.17391i			0.173723i	−0.996220	+	0.0868615i	$0.972316\pi$
$888$		0			0
$889$			0.132168i			0.00443277i
$890$			9.07101i			0.304061i
$891$		0			0
$892$	−12.3469			−0.413404
$893$			49.0949i			1.64290i
$894$		0			0
$895$	−11.2585			−0.376329
$896$		−	0.0653098i		−	0.00218185i
$897$		0			0
$898$			3.32880i			0.111084i
$899$	−13.8023			−0.460334
$900$		0			0
$901$		−	1.69024i		−	0.0563101i
$902$	11.7647			0.391723
$903$		0			0
$904$	4.20520			0.139863
$905$	−7.85745			−0.261191
$906$		0			0
$907$	−26.4919			−0.879650			−0.439825	−	0.898083i	$-0.644960\pi$
$907$	−26.4919			−0.879650			−0.439825	+	0.898083i	$0.644960\pi$
$908$		−	22.0078i		−	0.730355i
$909$		0			0
$910$	−0.158821			−0.00526486
$911$		−	54.2582i		−	1.79765i	−0.438303	−	0.898827i	$-0.644420\pi$
$911$		−	54.2582i		−	1.79765i	0.438303	−	0.898827i	$-0.355580\pi$
$912$		0			0
$913$		−	0.339108i		−	0.0112228i
$914$	−3.19043			−0.105530
$915$		0			0
$916$			22.4736i			0.742548i
$917$	−0.812353			−0.0268263
$918$		0			0
$919$			39.2581i			1.29500i	0.762064	+	0.647502i	$0.224186\pi$
$919$			39.2581i			1.29500i	−0.762064	+	0.647502i	$0.775814\pi$
$920$		−	2.47214i		−	0.0815039i
$921$		0			0
$922$			11.5897i			0.381687i
$923$	2.11545			0.0696308
$924$		0			0
$925$	−7.68209			−0.252586
$926$			18.4892i			0.607592i
$927$		0			0
$928$		−	4.70989i		−	0.154610i
$929$	34.2131			1.12249			0.561247	−	0.827648i	$-0.310322\pi$
$929$	34.2131			1.12249			0.561247	+	0.827648i	$0.310322\pi$
$930$		0			0
$931$	−52.7214			−1.72787
$932$	−13.0257			−0.426671
$933$		0			0
$934$		−	1.63639i		−	0.0535443i
$935$		−	0.273186i		−	0.00893413i
$936$		0			0
$937$		−	12.8823i		−	0.420846i	−0.977610	−	0.210423i	$-0.932516\pi$
$937$		−	12.8823i		−	0.420846i	0.977610	−	0.210423i	$-0.0674841\pi$
$938$	−0.519991	+	0.124054i	−0.0169783	+	0.00405052i
$939$		0			0
$940$		−	6.51453i		−	0.212481i
$941$	10.9590			0.357255			0.178627	−	0.983917i	$-0.442834\pi$
$941$	10.9590			0.357255			0.178627	+	0.983917i	$0.442834\pi$
$942$		0			0
$943$			16.3943i			0.533872i
$944$		−	0.249084i		−	0.00810698i
$945$		0			0
$946$		−	8.36639i		−	0.272015i
$947$		−	25.6954i		−	0.834989i	−0.908679	−	0.417495i	$-0.862908\pi$
$947$		−	25.6954i		−	0.834989i	0.908679	−	0.417495i	$-0.137092\pi$
$948$		0			0
$949$		−	15.8104i		−	0.513226i
$950$	−7.53622			−0.244507
$951$		0			0
$952$	0.0100572			0.000325954
$953$		−	29.3797i		−	0.951702i	−0.879526	−	0.475851i	$-0.842140\pi$
$953$		−	29.3797i		−	0.951702i	0.879526	−	0.475851i	$-0.157860\pi$
$954$		0			0
$955$	−12.8617			−0.416195
$956$	−10.3177			−0.333697
$957$		0			0
$958$		−	5.70312i		−	0.184259i
$959$		−	0.779035i		−	0.0251564i
$960$		0			0
$961$	22.4122			0.722973
$962$			18.6814i			0.602312i
$963$		0			0
$964$	24.7261			0.796375
$965$	14.0408			0.451990
$966$		0			0
$967$	−58.1886			−1.87122			−0.935609	−	0.353038i	$-0.885149\pi$
$967$	−58.1886			−1.87122			−0.935609	+	0.353038i	$0.885149\pi$
$968$	−7.85281			−0.252399
$969$		0			0
$970$			5.89823i			0.189381i
$971$		−	47.8260i		−	1.53481i	−0.641164	−	0.767404i	$-0.721548\pi$
$971$		−	47.8260i		−	1.53481i	0.641164	−	0.767404i	$-0.278452\pi$
$972$		0			0
$973$	−1.47058			−0.0471447
$974$			26.7105i			0.855860i
$975$		0			0
$976$		−	4.77423i		−	0.152819i
$977$		−	14.9856i		−	0.479430i	−0.970843	−	0.239715i	$-0.922946\pi$
$977$		−	14.9856i		−	0.479430i	0.970843	−	0.239715i	$-0.0770540\pi$
$978$		0			0
$979$		−	16.0923i		−	0.514311i
$980$	6.99573			0.223471
$981$		0			0
$982$		−	1.60752i		−	0.0512981i
$983$	−2.79283			−0.0890774			−0.0445387	−	0.999008i	$-0.514182\pi$
$983$	−2.79283			−0.0890774			−0.0445387	+	0.999008i	$0.514182\pi$
$984$		0			0
$985$	0.308330			0.00982421
$986$	0.725282			0.0230977
$987$		0			0
$988$			18.3266i			0.583048i
$989$	11.6587			0.370724
$990$		0			0
$991$		−	15.8286i		−	0.502811i	−0.967882	−	0.251405i	$-0.919107\pi$
$991$		−	15.8286i		−	0.502811i	0.967882	−	0.251405i	$-0.0808927\pi$
$992$		−	2.93050i		−	0.0930436i
$993$		0			0
$994$	0.0568135			0.00180201
$995$	8.42981			0.267243
$996$		0			0
$997$	−39.0143			−1.23559			−0.617797	−	0.786338i	$-0.711975\pi$
$997$	−39.0143			−1.23559			−0.617797	+	0.786338i	$0.711975\pi$
$998$			12.8436i			0.406556i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.l.2411.12	yes	24
3.2	odd	2		6030.2.d.k.2411.12	✓	24
67.66	odd	2		6030.2.d.k.2411.13	yes	24
201.200	even	2	inner	6030.2.d.l.2411.13	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.12	✓	24	3.2	odd	2
6030.2.d.k.2411.13	yes	24	67.66	odd	2
6030.2.d.l.2411.12	yes	24	1.1	even	1	trivial
6030.2.d.l.2411.13	yes	24	201.200	even	2	inner

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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.0653098i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.0653098i q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.77403 q^{11} -2.43181i q^{13} -0.0653098i q^{14} +1.00000 q^{16} +0.153991i q^{17} -7.53622 q^{19} +1.00000 q^{20} -1.77403 q^{22} -2.47214i q^{23} +1.00000 q^{25} -2.43181i q^{26} -0.0653098i q^{28} -4.70989i q^{29} -2.93050i q^{31} +1.00000 q^{32} +0.153991i q^{34} -0.0653098i q^{35} -7.68209 q^{37} -7.53622 q^{38} +1.00000 q^{40} -6.63164 q^{41} +4.71603i q^{43} -1.77403 q^{44} -2.47214i q^{46} -6.51453i q^{47} +6.99573 q^{49} +1.00000 q^{50} -2.43181i q^{52} -10.9762 q^{53} -1.77403 q^{55} -0.0653098i q^{56} -4.70989i q^{58} -0.249084i q^{59} -4.77423i q^{61} -2.93050i q^{62} +1.00000 q^{64} -2.43181i q^{65} +(-1.89948 - 7.96191i) q^{67} +0.153991i q^{68} -0.0653098i q^{70} +0.869907i q^{71} +6.50148 q^{73} -7.68209 q^{74} -7.53622 q^{76} +0.115862i q^{77} -6.75242i q^{79} +1.00000 q^{80} -6.63164 q^{82} +0.191151i q^{83} +0.153991i q^{85} +4.71603i q^{86} -1.77403 q^{88} +9.07101i q^{89} -0.158821 q^{91} -2.47214i q^{92} -6.51453i q^{94} -7.53622 q^{95} +5.89823i q^{97} +6.99573 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8} + 24 q^{10} + 12 q^{11} + 24 q^{16} + 4 q^{19} + 24 q^{20} + 12 q^{22} + 24 q^{25} + 24 q^{32} - 16 q^{37} + 4 q^{38} + 24 q^{40} + 8 q^{41} + 12 q^{44} - 20 q^{49} + 24 q^{50} + 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} - 16 q^{74} + 4 q^{76} + 24 q^{80} + 8 q^{82} + 12 q^{88} + 4 q^{95} - 20 q^{98}+O(q^{100}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8 + 24 * q^10 + 12 * q^11 + 24 * q^16 + 4 * q^19 + 24 * q^20 + 12 * q^22 + 24 * q^25 + 24 * q^32 - 16 * q^37 + 4 * q^38 + 24 * q^40 + 8 * q^41 + 12 * q^44 - 20 * q^49 + 24 * q^50 + 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 - 16 * q^74 + 4 * q^76 + 24 * q^80 + 8 * q^82 + 12 * q^88 + 4 * q^95 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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