LMFDB - Embedded newform 2070.3.c.b.91.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.c (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:	$56.4034147226$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.2
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -3.95881i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -3.95881i q^{7} -2.82843 q^{8} +3.16228i q^{10} -5.67111i q^{11} +11.1376 q^{13} +5.59860i q^{14} +4.00000 q^{16} +2.12953i q^{17} +20.9865i q^{19} -4.47214i q^{20} +8.02016i q^{22} +(10.6801 + 20.3700i) q^{23} -5.00000 q^{25} -15.7510 q^{26} -7.91762i q^{28} -36.0592 q^{29} +39.7941 q^{31} -5.65685 q^{32} -3.01161i q^{34} -8.85217 q^{35} +50.2613i q^{37} -29.6794i q^{38} +6.32456i q^{40} -41.6563 q^{41} +41.3234i q^{43} -11.3422i q^{44} +(-15.1039 - 28.8075i) q^{46} -60.1764 q^{47} +33.3278 q^{49} +7.07107 q^{50} +22.2753 q^{52} -3.29435i q^{53} -12.6810 q^{55} +11.1972i q^{56} +50.9954 q^{58} +83.2918 q^{59} +37.8857i q^{61} -56.2773 q^{62} +8.00000 q^{64} -24.9045i q^{65} -61.1024i q^{67} +4.25906i q^{68} +12.5189 q^{70} +0.572102 q^{71} +40.8795 q^{73} -71.0803i q^{74} +41.9730i q^{76} -22.4509 q^{77} +86.8569i q^{79} -8.94427i q^{80} +58.9109 q^{82} -84.6886i q^{83} +4.76177 q^{85} -58.4401i q^{86} +16.0403i q^{88} +94.7533i q^{89} -44.0918i q^{91} +(21.3601 + 40.7400i) q^{92} +85.1023 q^{94} +46.9273 q^{95} -171.701i q^{97} -47.1327 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4}+O(q^{10}) $ 32 * q + 64 * q^4	$ 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$		−	3.95881i		−	0.565544i	−0.959187	−	0.282772i	$-0.908746\pi$
$7$		−	3.95881i		−	0.565544i	0.959187	−	0.282772i	$-0.0912540\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$		−	5.67111i		−	0.515556i	−0.966204	−	0.257778i	$-0.917010\pi$
$11$		−	5.67111i		−	0.515556i	0.966204	−	0.257778i	$-0.0829903\pi$
$12$		0			0
$13$	11.1376			0.856742			0.428371	−	0.903603i	$-0.359088\pi$
$13$	11.1376			0.856742			0.428371	+	0.903603i	$0.359088\pi$
$14$			5.59860i			0.399900i
$15$		0			0
$16$	4.00000			0.250000
$17$			2.12953i			0.125266i	0.998037	+	0.0626332i	$0.0199498\pi$
$17$			2.12953i			0.125266i	−0.998037	+	0.0626332i	$0.980050\pi$
$18$		0			0
$19$			20.9865i			1.10455i	0.833661	+	0.552277i	$0.186241\pi$
$19$			20.9865i			1.10455i	−0.833661	+	0.552277i	$0.813759\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$			8.02016i			0.364553i
$23$	10.6801	+	20.3700i	0.464350	+	0.885652i
$23$	10.6801	+	20.3700i	0.464350	+	0.885652i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−15.7510			−0.605808
$27$		0			0
$28$		−	7.91762i		−	0.282772i
$29$	−36.0592			−1.24342			−0.621710	−	0.783248i	$-0.713562\pi$
$29$	−36.0592			−1.24342			−0.621710	+	0.783248i	$0.713562\pi$
$30$		0			0
$31$	39.7941			1.28368			0.641840	−	0.766838i	$-0.278171\pi$
$31$	39.7941			1.28368			0.641840	+	0.766838i	$0.278171\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$		−	3.01161i		−	0.0885767i
$35$	−8.85217			−0.252919
$36$		0			0
$37$			50.2613i			1.35841i	0.733946	+	0.679207i	$0.237676\pi$
$37$			50.2613i			1.35841i	−0.733946	+	0.679207i	$0.762324\pi$
$38$		−	29.6794i		−	0.781037i
$39$		0			0
$40$			6.32456i			0.158114i
$41$	−41.6563			−1.01601			−0.508003	−	0.861355i	$-0.669616\pi$
$41$	−41.6563			−1.01601			−0.508003	+	0.861355i	$0.669616\pi$
$42$		0			0
$43$			41.3234i			0.961009i	0.876992	+	0.480505i	$0.159547\pi$
$43$			41.3234i			0.961009i	−0.876992	+	0.480505i	$0.840453\pi$
$44$		−	11.3422i		−	0.257778i
$45$		0			0
$46$	−15.1039	−	28.8075i	−0.328345	−	0.626250i
$47$	−60.1764			−1.28035			−0.640175	−	0.768229i	$-0.721138\pi$
$47$	−60.1764			−1.28035			−0.640175	+	0.768229i	$0.721138\pi$
$48$		0			0
$49$	33.3278			0.680160
$50$	7.07107			0.141421
$51$		0			0
$52$	22.2753			0.428371
$53$		−	3.29435i		−	0.0621575i	−0.999517	−	0.0310787i	$-0.990106\pi$
$53$		−	3.29435i		−	0.0621575i	0.999517	−	0.0310787i	$-0.00989427\pi$
$54$		0			0
$55$	−12.6810			−0.230563
$56$			11.1972i			0.199950i
$57$		0			0
$58$	50.9954			0.879231
$59$	83.2918			1.41173			0.705863	−	0.708349i	$-0.250560\pi$
$59$	83.2918			1.41173			0.705863	+	0.708349i	$0.250560\pi$
$60$		0			0
$61$			37.8857i			0.621077i	0.950561	+	0.310538i	$0.100509\pi$
$61$			37.8857i			0.621077i	−0.950561	+	0.310538i	$0.899491\pi$
$62$	−56.2773			−0.907699
$63$		0			0
$64$	8.00000			0.125000
$65$		−	24.9045i		−	0.383147i
$66$		0			0
$67$		−	61.1024i		−	0.911975i	−0.889986	−	0.455988i	$-0.849286\pi$
$67$		−	61.1024i		−	0.911975i	0.889986	−	0.455988i	$-0.150714\pi$
$68$			4.25906i			0.0626332i
$69$		0			0
$70$	12.5189			0.178841
$71$	0.572102			0.00805777			0.00402888	−	0.999992i	$-0.498718\pi$
$71$	0.572102			0.00805777			0.00402888	+	0.999992i	$0.498718\pi$
$72$		0			0
$73$	40.8795			0.559994			0.279997	−	0.960001i	$-0.409667\pi$
$73$	40.8795			0.559994			0.279997	+	0.960001i	$0.409667\pi$
$74$		−	71.0803i		−	0.960544i
$75$		0			0
$76$			41.9730i			0.552277i
$77$	−22.4509			−0.291570
$78$		0			0
$79$			86.8569i			1.09945i	0.835344	+	0.549727i	$0.185268\pi$
$79$			86.8569i			1.09945i	−0.835344	+	0.549727i	$0.814732\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$	58.9109			0.718425
$83$		−	84.6886i		−	1.02034i	−0.860072	−	0.510172i	$-0.829582\pi$
$83$		−	84.6886i		−	1.02034i	0.860072	−	0.510172i	$-0.170418\pi$
$84$		0			0
$85$	4.76177			0.0560208
$86$		−	58.4401i		−	0.679536i
$87$		0			0
$88$			16.0403i			0.182276i
$89$			94.7533i			1.06464i	0.846542	+	0.532322i	$0.178680\pi$
$89$			94.7533i			1.06464i	−0.846542	+	0.532322i	$0.821320\pi$
$90$		0			0
$91$		−	44.0918i		−	0.484525i
$92$	21.3601	+	40.7400i	0.232175	+	0.442826i
$93$		0			0
$94$	85.1023			0.905344
$95$	46.9273			0.493971
$96$		0			0
$97$		−	171.701i		−	1.77011i	−0.465485	−	0.885056i	$-0.654120\pi$
$97$		−	171.701i		−	1.77011i	0.465485	−	0.885056i	$-0.345880\pi$
$98$	−47.1327			−0.480946
$99$		0			0
$100$	−10.0000			−0.100000
$101$	48.8836			0.483997			0.241998	−	0.970277i	$-0.422197\pi$
$101$	48.8836			0.483997			0.241998	+	0.970277i	$0.422197\pi$
$102$		0			0
$103$			3.88018i			0.0376716i	0.999823	+	0.0188358i	$0.00599598\pi$
$103$			3.88018i			0.0376716i	−0.999823	+	0.0188358i	$0.994004\pi$
$104$	−31.5020			−0.302904
$105$		0			0
$106$			4.65891i			0.0439520i
$107$			121.583i			1.13629i	0.822927	+	0.568147i	$0.192339\pi$
$107$			121.583i			1.13629i	−0.822927	+	0.568147i	$0.807661\pi$
$108$		0			0
$109$			140.035i			1.28473i	0.766401	+	0.642363i	$0.222046\pi$
$109$			140.035i			1.28473i	−0.766401	+	0.642363i	$0.777954\pi$
$110$	17.9336			0.163033
$111$		0			0
$112$		−	15.8352i		−	0.141386i
$113$			3.73201i			0.0330267i	0.999864	+	0.0165133i	$0.00525660\pi$
$113$			3.73201i			0.0330267i	−0.999864	+	0.0165133i	$0.994743\pi$
$114$		0			0
$115$	45.5487	−	23.8813i	0.396075	−	0.207664i
$116$	−72.1184			−0.621710
$117$		0			0
$118$	−117.792			−0.998240
$119$	8.43040			0.0708437
$120$		0			0
$121$	88.8385			0.734202
$122$		−	53.5784i		−	0.439167i
$123$		0			0
$124$	79.5882			0.641840
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	189.032			1.48844			0.744221	−	0.667933i	$-0.232821\pi$
$127$	189.032			1.48844			0.744221	+	0.667933i	$0.232821\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$			35.2203i			0.270926i
$131$	102.753			0.784374			0.392187	−	0.919885i	$-0.371719\pi$
$131$	102.753			0.784374			0.392187	+	0.919885i	$0.371719\pi$
$132$		0			0
$133$	83.0816			0.624674
$134$			86.4118i			0.644864i
$135$		0			0
$136$		−	6.02322i		−	0.0442884i
$137$			53.0102i			0.386936i	0.981107	+	0.193468i	$0.0619736\pi$
$137$			53.0102i			0.386936i	−0.981107	+	0.193468i	$0.938026\pi$
$138$		0			0
$139$	−230.317			−1.65696			−0.828478	−	0.560022i	$-0.810793\pi$
$139$	−230.317			−1.65696			−0.828478	+	0.560022i	$0.810793\pi$
$140$	−17.7043			−0.126460
$141$		0			0
$142$	−0.809074			−0.00569770
$143$		−	63.1628i		−	0.441698i
$144$		0			0
$145$			80.6308i			0.556074i
$146$	−57.8124			−0.395975
$147$		0			0
$148$			100.523i			0.679207i
$149$			34.0099i			0.228254i	0.993466	+	0.114127i	$0.0364071\pi$
$149$			34.0099i			0.228254i	−0.993466	+	0.114127i	$0.963593\pi$
$150$		0			0
$151$	−273.426			−1.81077			−0.905383	−	0.424596i	$-0.860416\pi$
$151$	−273.426			−1.81077			−0.905383	+	0.424596i	$0.860416\pi$
$152$		−	59.3588i		−	0.390519i
$153$		0			0
$154$	31.7503			0.206171
$155$		−	88.9823i		−	0.574079i
$156$		0			0
$157$		−	302.701i		−	1.92803i	−0.265841	−	0.964017i	$-0.585650\pi$
$157$		−	302.701i		−	1.92803i	0.265841	−	0.964017i	$-0.414350\pi$
$158$		−	122.834i		−	0.777431i
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	80.6409	−	42.2803i	0.500875	−	0.262611i
$162$		0			0
$163$	−101.769			−0.624352			−0.312176	−	0.950024i	$-0.601058\pi$
$163$	−101.769			−0.624352			−0.312176	+	0.950024i	$0.601058\pi$
$164$	−83.3126			−0.508003
$165$		0			0
$166$			119.768i			0.721493i
$167$	124.606			0.746143			0.373071	−	0.927803i	$-0.378305\pi$
$167$	124.606			0.746143			0.373071	+	0.927803i	$0.378305\pi$
$168$		0			0
$169$	−44.9529			−0.265994
$170$	−6.73416			−0.0396127
$171$		0			0
$172$			82.6468i			0.480505i
$173$	278.331			1.60885			0.804425	−	0.594054i	$-0.202474\pi$
$173$	278.331			1.60885			0.804425	+	0.594054i	$0.202474\pi$
$174$		0			0
$175$			19.7940i			0.113109i
$176$		−	22.6844i		−	0.128889i
$177$		0			0
$178$		−	134.001i		−	0.752817i
$179$	22.7044			0.126840			0.0634202	−	0.997987i	$-0.479799\pi$
$179$	22.7044			0.126840			0.0634202	+	0.997987i	$0.479799\pi$
$180$		0			0
$181$			151.385i			0.836379i	0.908360	+	0.418189i	$0.137335\pi$
$181$			151.385i			0.836379i	−0.908360	+	0.418189i	$0.862665\pi$
$182$			62.3552i			0.342611i
$183$		0			0
$184$	−30.2078	−	57.6150i	−0.164173	−	0.313125i
$185$	112.388			0.607502
$186$		0			0
$187$	12.0768			0.0645818
$188$	−120.353			−0.640175
$189$		0			0
$190$	−66.3652			−0.349290
$191$			68.1039i			0.356565i	0.983979	+	0.178282i	$0.0570541\pi$
$191$			68.1039i			0.356565i	−0.983979	+	0.178282i	$0.942946\pi$
$192$		0			0
$193$	243.269			1.26046			0.630230	−	0.776408i	$-0.282961\pi$
$193$	243.269			1.26046			0.630230	+	0.776408i	$0.282961\pi$
$194$			242.822i			1.25166i
$195$		0			0
$196$	66.6556			0.340080
$197$	131.358			0.666790			0.333395	−	0.942787i	$-0.391806\pi$
$197$	131.358			0.666790			0.333395	+	0.942787i	$0.391806\pi$
$198$		0			0
$199$		−	313.700i		−	1.57638i	−0.615432	−	0.788190i	$-0.711018\pi$
$199$		−	313.700i		−	1.57638i	0.615432	−	0.788190i	$-0.288982\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	−69.1319			−0.342237
$203$			142.751i			0.703209i
$204$		0			0
$205$			93.1463i			0.454372i
$206$		−	5.48740i		−	0.0266379i
$207$		0			0
$208$	44.5506			0.214185
$209$	119.017			0.569459
$210$		0			0
$211$	75.8630			0.359540			0.179770	−	0.983709i	$-0.442465\pi$
$211$	75.8630			0.359540			0.179770	+	0.983709i	$0.442465\pi$
$212$		−	6.58869i		−	0.0310787i
$213$		0			0
$214$		−	171.945i		−	0.803481i
$215$	92.4019			0.429777
$216$		0			0
$217$		−	157.537i		−	0.725978i
$218$		−	198.040i		−	0.908438i
$219$		0			0
$220$	−25.3620			−0.115282
$221$			23.7179i			0.107321i
$222$		0			0
$223$	−165.794			−0.743472			−0.371736	−	0.928338i	$-0.621237\pi$
$223$	−165.794			−0.743472			−0.371736	+	0.928338i	$0.621237\pi$
$224$			22.3944i			0.0999750i
$225$		0			0
$226$		−	5.27786i		−	0.0233534i
$227$			405.393i			1.78587i	0.450183	+	0.892937i	$0.351359\pi$
$227$			405.393i			1.78587i	−0.450183	+	0.892937i	$0.648641\pi$
$228$		0			0
$229$			382.310i			1.66947i	0.550649	+	0.834737i	$0.314380\pi$
$229$			382.310i			1.66947i	−0.550649	+	0.834737i	$0.685620\pi$
$230$	−64.4156	+	33.7733i	−0.280068	+	0.146840i
$231$		0			0
$232$	101.991			0.439615
$233$	125.133			0.537050			0.268525	−	0.963273i	$-0.413464\pi$
$233$	125.133			0.537050			0.268525	+	0.963273i	$0.413464\pi$
$234$		0			0
$235$			134.559i			0.572590i
$236$	166.584			0.705863
$237$		0			0
$238$	−11.9224			−0.0500941
$239$	129.207			0.540616			0.270308	−	0.962774i	$-0.412874\pi$
$239$	129.207			0.540616			0.270308	+	0.962774i	$0.412874\pi$
$240$		0			0
$241$			313.373i			1.30030i	0.759805	+	0.650151i	$0.225295\pi$
$241$			313.373i			1.30030i	−0.759805	+	0.650151i	$0.774705\pi$
$242$	−125.637			−0.519160
$243$		0			0
$244$			75.7713i			0.310538i
$245$		−	74.5233i		−	0.304177i
$246$		0			0
$247$			233.740i			0.946317i
$248$	−112.555			−0.453850
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$			152.125i			0.606075i	0.952979	+	0.303037i	$0.0980007\pi$
$251$			152.125i			0.606075i	−0.952979	+	0.303037i	$0.901999\pi$
$252$		0			0
$253$	115.520	−	60.5678i	0.456603	−	0.239398i
$254$	−267.332			−1.05249
$255$		0			0
$256$	16.0000			0.0625000
$257$	−19.5904			−0.0762272			−0.0381136	−	0.999273i	$-0.512135\pi$
$257$	−19.5904			−0.0762272			−0.0381136	+	0.999273i	$0.512135\pi$
$258$		0			0
$259$	198.975			0.768244
$260$		−	49.8090i		−	0.191573i
$261$		0			0
$262$	−145.315			−0.554636
$263$		−	127.348i		−	0.484212i	−0.970250	−	0.242106i	$-0.922162\pi$
$263$		−	127.348i		−	0.484212i	0.970250	−	0.242106i	$-0.0778381\pi$
$264$		0			0
$265$	−7.36638			−0.0277977
$266$	−117.495			−0.441711
$267$		0			0
$268$		−	122.205i		−	0.455988i
$269$	−142.959			−0.531448			−0.265724	−	0.964049i	$-0.585611\pi$
$269$	−142.959			−0.531448			−0.265724	+	0.964049i	$0.585611\pi$
$270$		0			0
$271$	−63.7008			−0.235058			−0.117529	−	0.993069i	$-0.537497\pi$
$271$	−63.7008			−0.235058			−0.117529	+	0.993069i	$0.537497\pi$
$272$			8.51812i			0.0313166i
$273$		0			0
$274$		−	74.9678i		−	0.273605i
$275$			28.3556i			0.103111i
$276$		0			0
$277$	448.470			1.61903			0.809513	−	0.587101i	$-0.199731\pi$
$277$	448.470			1.61903			0.809513	+	0.587101i	$0.199731\pi$
$278$	325.717			1.17164
$279$		0			0
$280$	25.0377			0.0894204
$281$			154.698i			0.550525i	0.961369	+	0.275263i	$0.0887648\pi$
$281$			154.698i			0.550525i	−0.961369	+	0.275263i	$0.911235\pi$
$282$		0			0
$283$			19.1545i			0.0676837i	0.999427	+	0.0338418i	$0.0107742\pi$
$283$			19.1545i			0.0676837i	−0.999427	+	0.0338418i	$0.989226\pi$
$284$	1.14420			0.00402888
$285$		0			0
$286$			89.3257i			0.312328i
$287$			164.909i			0.574597i
$288$		0			0
$289$	284.465			0.984308
$290$		−	114.029i		−	0.393204i
$291$		0			0
$292$	81.7591			0.279997
$293$		−	269.403i		−	0.919463i	−0.888058	−	0.459732i	$-0.847946\pi$
$293$		−	269.403i		−	0.919463i	0.888058	−	0.459732i	$-0.152054\pi$
$294$		0			0
$295$		−	186.246i		−	0.631343i
$296$		−	142.161i		−	0.480272i
$297$		0			0
$298$		−	48.0972i		−	0.161400i
$299$	118.951	+	226.874i	0.397828	+	0.758775i
$300$		0			0
$301$	163.592			0.543493
$302$	386.682			1.28040
$303$		0			0
$304$			83.9461i			0.276138i
$305$	84.7149			0.277754
$306$		0			0
$307$	331.722			1.08053			0.540263	−	0.841496i	$-0.318325\pi$
$307$	331.722			1.08053			0.540263	+	0.841496i	$0.318325\pi$
$308$	−44.9017			−0.145785
$309$		0			0
$310$			125.840i			0.405935i
$311$	397.584			1.27840			0.639202	−	0.769039i	$-0.279265\pi$
$311$	397.584			1.27840			0.639202	+	0.769039i	$0.279265\pi$
$312$		0			0
$313$			124.944i			0.399181i	0.979879	+	0.199590i	$0.0639611\pi$
$313$			124.944i			0.399181i	−0.979879	+	0.199590i	$0.936039\pi$
$314$			428.084i			1.36333i
$315$		0			0
$316$			173.714i			0.549727i
$317$	−272.823			−0.860642			−0.430321	−	0.902676i	$-0.641600\pi$
$317$	−272.823			−0.860642			−0.430321	+	0.902676i	$0.641600\pi$
$318$		0			0
$319$			204.496i			0.641052i
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	−114.043	+	59.7934i	−0.354172	+	0.185694i
$323$	−44.6914			−0.138363
$324$		0			0
$325$	−55.6882			−0.171348
$326$	143.924			0.441483
$327$		0			0
$328$	117.822			0.359213
$329$			238.227i			0.724094i
$330$		0			0
$331$	158.093			0.477621			0.238811	−	0.971066i	$-0.423243\pi$
$331$	158.093			0.477621			0.238811	+	0.971066i	$0.423243\pi$
$332$		−	169.377i		−	0.510172i
$333$		0			0
$334$	−176.219			−0.527603
$335$	−136.629			−0.407848
$336$		0			0
$337$			187.915i			0.557610i	0.960348	+	0.278805i	$0.0899383\pi$
$337$			187.915i			0.557610i	−0.960348	+	0.278805i	$0.910062\pi$
$338$	63.5730			0.188086
$339$		0			0
$340$	9.52354			0.0280104
$341$		−	225.677i		−	0.661809i
$342$		0			0
$343$		−	325.920i		−	0.950205i
$344$		−	116.880i		−	0.339768i
$345$		0			0
$346$	−393.620			−1.13763
$347$	313.436			0.903274			0.451637	−	0.892202i	$-0.350840\pi$
$347$	313.436			0.903274			0.451637	+	0.892202i	$0.350840\pi$
$348$		0			0
$349$	363.470			1.04146			0.520730	−	0.853721i	$-0.325660\pi$
$349$	363.470			1.04146			0.520730	+	0.853721i	$0.325660\pi$
$350$		−	27.9930i		−	0.0799800i
$351$		0			0
$352$			32.0807i			0.0911382i
$353$	524.725			1.48647			0.743236	−	0.669029i	$-0.233290\pi$
$353$	524.725			1.48647			0.743236	+	0.669029i	$0.233290\pi$
$354$		0			0
$355$		−	1.27926i		−	0.00360354i
$356$			189.507i			0.532322i
$357$		0			0
$358$	−32.1089			−0.0896897
$359$			311.701i			0.868249i	0.900853	+	0.434124i	$0.142942\pi$
$359$			311.701i			0.868249i	−0.900853	+	0.434124i	$0.857058\pi$
$360$		0			0
$361$	−79.4338			−0.220038
$362$		−	214.090i		−	0.591409i
$363$		0			0
$364$		−	88.1836i		−	0.242263i
$365$		−	91.4094i		−	0.250437i
$366$		0			0
$367$		−	468.490i		−	1.27654i	−0.769813	−	0.638269i	$-0.779651\pi$
$367$		−	468.490i		−	1.27654i	0.769813	−	0.638269i	$-0.220349\pi$
$368$	42.7202	+	81.4799i	0.116088	+	0.221413i
$369$		0			0
$370$	−158.940			−0.429568
$371$	−13.0417			−0.0351528
$372$		0			0
$373$		−	7.45302i		−	0.0199813i	−0.999950	−	0.00999064i	$-0.996820\pi$
$373$		−	7.45302i		−	0.0199813i	0.999950	−	0.00999064i	$-0.00318017\pi$
$374$	−17.0792			−0.0456662
$375$		0			0
$376$	170.205			0.452672
$377$	−401.614			−1.06529
$378$		0			0
$379$			444.034i			1.17159i	0.810458	+	0.585796i	$0.199218\pi$
$379$			444.034i			1.17159i	−0.810458	+	0.585796i	$0.800782\pi$
$380$	93.8545			0.246986
$381$		0			0
$382$		−	96.3134i		−	0.252129i
$383$			315.599i			0.824017i	0.911180	+	0.412009i	$0.135173\pi$
$383$			315.599i			0.824017i	−0.911180	+	0.412009i	$0.864827\pi$
$384$		0			0
$385$			50.2016i			0.130394i
$386$	−344.034			−0.891280
$387$		0			0
$388$		−	343.402i		−	0.885056i
$389$			126.333i			0.324763i	0.986728	+	0.162381i	$0.0519175\pi$
$389$			126.333i			0.324763i	−0.986728	+	0.162381i	$0.948083\pi$
$390$		0			0
$391$	−43.3785	+	22.7435i	−0.110942	+	0.0581675i
$392$	−94.2653			−0.240473
$393$		0			0
$394$	−185.768			−0.471492
$395$	194.218			0.491691
$396$		0			0
$397$	−165.211			−0.416148			−0.208074	−	0.978113i	$-0.566719\pi$
$397$	−165.211			−0.416148			−0.208074	+	0.978113i	$0.566719\pi$
$398$			443.638i			1.11467i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$		−	298.998i		−	0.745630i	−0.927906	−	0.372815i	$-0.878393\pi$
$401$		−	298.998i		−	0.745630i	0.927906	−	0.372815i	$-0.121607\pi$
$402$		0			0
$403$	443.212			1.09978
$404$	97.7673			0.241998
$405$		0			0
$406$		−	201.881i		−	0.497244i
$407$	285.038			0.700338
$408$		0			0
$409$	−55.5554			−0.135832			−0.0679162	−	0.997691i	$-0.521635\pi$
$409$	−55.5554			−0.135832			−0.0679162	+	0.997691i	$0.521635\pi$
$410$		−	131.729i		−	0.321290i
$411$		0			0
$412$			7.76036i			0.0188358i
$413$		−	329.736i		−	0.798393i
$414$		0			0
$415$	−189.369			−0.456312
$416$	−63.0040			−0.151452
$417$		0			0
$418$	−168.315			−0.402668
$419$		−	422.645i		−	1.00870i	−0.863500	−	0.504349i	$-0.831732\pi$
$419$		−	422.645i		−	1.00870i	0.863500	−	0.504349i	$-0.168268\pi$
$420$		0			0
$421$		−	106.651i		−	0.253328i	−0.991946	−	0.126664i	$-0.959573\pi$
$421$		−	106.651i		−	0.253328i	0.991946	−	0.126664i	$-0.0404269\pi$
$422$	−107.287			−0.254233
$423$		0			0
$424$			9.31782i			0.0219760i
$425$		−	10.6476i		−	0.0250533i
$426$		0			0
$427$	149.982			0.351246
$428$			243.167i			0.568147i
$429$		0			0
$430$	−130.676			−0.303898
$431$		−	305.614i		−	0.709080i	−0.935041	−	0.354540i	$-0.884637\pi$
$431$		−	305.614i		−	0.709080i	0.935041	−	0.354540i	$-0.115363\pi$
$432$		0			0
$433$		−	285.890i		−	0.660254i	−0.943936	−	0.330127i	$-0.892908\pi$
$433$		−	285.890i		−	0.660254i	0.943936	−	0.330127i	$-0.107092\pi$
$434$			222.791i			0.513344i
$435$		0			0
$436$			280.070i			0.642363i
$437$	−427.495	+	224.137i	−0.978249	+	0.512900i
$438$		0			0
$439$	−461.780			−1.05189			−0.525945	−	0.850519i	$-0.676288\pi$
$439$	−461.780			−1.05189			−0.525945	+	0.850519i	$0.676288\pi$
$440$	35.8673			0.0815165
$441$		0			0
$442$		−	33.5422i		−	0.0758874i
$443$	65.0709			0.146887			0.0734434	−	0.997299i	$-0.476601\pi$
$443$	65.0709			0.146887			0.0734434	+	0.997299i	$0.476601\pi$
$444$		0			0
$445$	211.875			0.476123
$446$	234.469			0.525714
$447$		0			0
$448$		−	31.6705i		−	0.0706930i
$449$	−547.308			−1.21895			−0.609475	−	0.792806i	$-0.708620\pi$
$449$	−547.308			−1.21895			−0.609475	+	0.792806i	$0.708620\pi$
$450$		0			0
$451$			236.237i			0.523808i
$452$			7.46403i			0.0165133i
$453$		0			0
$454$		−	573.313i		−	1.26280i
$455$	−98.5923			−0.216686
$456$		0			0
$457$			334.369i			0.731660i	0.930682	+	0.365830i	$0.119215\pi$
$457$			334.369i			0.731660i	−0.930682	+	0.365830i	$0.880785\pi$
$458$		−	540.667i		−	1.18050i
$459$		0			0
$460$	91.0973	−	47.7627i	0.198038	−	0.103832i
$461$	384.546			0.834156			0.417078	−	0.908871i	$-0.363054\pi$
$461$	384.546			0.834156			0.417078	+	0.908871i	$0.363054\pi$
$462$		0			0
$463$	741.501			1.60151			0.800757	−	0.598990i	$-0.204431\pi$
$463$	741.501			1.60151			0.800757	+	0.598990i	$0.204431\pi$
$464$	−144.237			−0.310855
$465$		0			0
$466$	−176.964			−0.379752
$467$			183.855i			0.393695i	0.980434	+	0.196847i	$0.0630704\pi$
$467$			183.855i			0.393695i	−0.980434	+	0.196847i	$0.936930\pi$
$468$		0			0
$469$	−241.893			−0.515763
$470$		−	190.295i		−	0.404882i
$471$		0			0
$472$	−235.585			−0.499120
$473$	234.350			0.495454
$474$		0			0
$475$		−	104.933i		−	0.220911i
$476$	16.8608			0.0354218
$477$		0			0
$478$	−182.727			−0.382273
$479$			643.670i			1.34378i	0.740652	+	0.671889i	$0.234517\pi$
$479$			643.670i			1.34378i	−0.740652	+	0.671889i	$0.765483\pi$
$480$		0			0
$481$			559.793i			1.16381i
$482$		−	443.176i		−	0.919453i
$483$		0			0
$484$	177.677			0.367101
$485$	−383.935			−0.791618
$486$		0			0
$487$	−404.712			−0.831031			−0.415516	−	0.909586i	$-0.636399\pi$
$487$	−404.712			−0.831031			−0.415516	+	0.909586i	$0.636399\pi$
$488$		−	107.157i		−	0.219584i
$489$		0			0
$490$			105.392i			0.215085i
$491$	−173.587			−0.353537			−0.176769	−	0.984252i	$-0.556564\pi$
$491$	−173.587			−0.353537			−0.176769	+	0.984252i	$0.556564\pi$
$492$		0			0
$493$		−	76.7891i		−	0.155759i
$494$		−	330.559i		−	0.669147i
$495$		0			0
$496$	159.176			0.320920
$497$		−	2.26484i		−	0.00455703i
$498$		0			0
$499$	6.44061			0.0129070			0.00645352	−	0.999979i	$-0.497946\pi$
$499$	6.44061			0.0129070			0.00645352	+	0.999979i	$0.497946\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$		−	215.137i		−	0.428559i
$503$		−	617.629i		−	1.22789i	−0.789348	−	0.613945i	$-0.789581\pi$
$503$		−	617.629i		−	1.22789i	0.789348	−	0.613945i	$-0.210419\pi$
$504$		0			0
$505$		−	109.307i		−	0.216450i
$506$	−163.371	+	85.6558i	−0.322867	+	0.169280i
$507$		0			0
$508$	378.064			0.744221
$509$	−200.449			−0.393809			−0.196904	−	0.980423i	$-0.563089\pi$
$509$	−200.449			−0.393809			−0.196904	+	0.980423i	$0.563089\pi$
$510$		0			0
$511$		−	161.834i		−	0.316701i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	27.7050			0.0539008
$515$	8.67634			0.0168473
$516$		0			0
$517$			341.267i			0.660091i
$518$	−281.393			−0.543230
$519$		0			0
$520$			70.4406i			0.135463i
$521$			201.610i			0.386968i	0.981103	+	0.193484i	$0.0619787\pi$
$521$			201.610i			0.386968i	−0.981103	+	0.193484i	$0.938021\pi$
$522$		0			0
$523$		−	678.779i		−	1.29786i	−0.760850	−	0.648928i	$-0.775218\pi$
$523$		−	678.779i		−	1.29786i	0.760850	−	0.648928i	$-0.224782\pi$
$524$	205.506			0.392187
$525$		0			0
$526$			180.097i			0.342389i
$527$			84.7427i			0.160802i
$528$		0			0
$529$	−300.873	+	435.105i	−0.568757	+	0.822505i
$530$	10.4176			0.0196559
$531$		0			0
$532$	166.163			0.312337
$533$	−463.953			−0.870455
$534$		0			0
$535$	271.869			0.508166
$536$			172.824i			0.322432i
$537$		0			0
$538$	202.175			0.375790
$539$		−	189.006i		−	0.350660i
$540$		0			0
$541$	465.332			0.860133			0.430067	−	0.902797i	$-0.358490\pi$
$541$	465.332			0.860133			0.430067	+	0.902797i	$0.358490\pi$
$542$	90.0865			0.166211
$543$		0			0
$544$		−	12.0464i		−	0.0221442i
$545$	313.128			0.574547
$546$		0			0
$547$	−309.522			−0.565853			−0.282926	−	0.959142i	$-0.591305\pi$
$547$	−309.522			−0.565853			−0.282926	+	0.959142i	$0.591305\pi$
$548$			106.020i			0.193468i
$549$		0			0
$550$		−	40.1008i		−	0.0729106i
$551$		−	756.756i		−	1.37342i
$552$		0			0
$553$	343.850			0.621790
$554$	−634.233			−1.14482
$555$		0			0
$556$	−460.634			−0.828478
$557$			826.642i			1.48410i	0.670346	+	0.742049i	$0.266146\pi$
$557$			826.642i			1.48410i	−0.670346	+	0.742049i	$0.733854\pi$
$558$		0			0
$559$			460.245i			0.823337i
$560$	−35.4087			−0.0632298
$561$		0			0
$562$		−	218.775i		−	0.389280i
$563$		−	1018.64i		−	1.80931i	−0.426140	−	0.904657i	$-0.640127\pi$
$563$		−	1018.64i		−	1.80931i	0.426140	−	0.904657i	$-0.359873\pi$
$564$		0			0
$565$	8.34503			0.0147700
$566$		−	27.0885i		−	0.0478596i
$567$		0			0
$568$	−1.61815			−0.00284885
$569$		−	766.176i		−	1.34653i	−0.739401	−	0.673266i	$-0.764891\pi$
$569$		−	766.176i		−	1.34653i	0.739401	−	0.673266i	$-0.235109\pi$
$570$		0			0
$571$			119.265i			0.208870i	0.994532	+	0.104435i	$0.0333035\pi$
$571$			119.265i			0.208870i	−0.994532	+	0.104435i	$0.966697\pi$
$572$		−	126.326i		−	0.220849i
$573$		0			0
$574$		−	233.217i		−	0.406301i
$575$	−53.4003	−	101.850i	−0.0928701	−	0.177130i
$576$		0			0
$577$	−73.2216			−0.126901			−0.0634503	−	0.997985i	$-0.520210\pi$
$577$	−73.2216			−0.126901			−0.0634503	+	0.997985i	$0.520210\pi$
$578$	−402.294			−0.696011
$579$		0			0
$580$			161.262i			0.278037i
$581$	−335.266			−0.577050
$582$		0			0
$583$	−18.6826			−0.0320456
$584$	−115.625			−0.197988
$585$		0			0
$586$			380.993i			0.650159i
$587$	−829.222			−1.41264			−0.706322	−	0.707891i	$-0.749647\pi$
$587$	−829.222			−1.41264			−0.706322	+	0.707891i	$0.749647\pi$
$588$		0			0
$589$			835.139i			1.41789i
$590$			263.392i			0.446427i
$591$		0			0
$592$			201.045i			0.339604i
$593$	−82.6719			−0.139413			−0.0697065	−	0.997568i	$-0.522206\pi$
$593$	−82.6719			−0.139413			−0.0697065	+	0.997568i	$0.522206\pi$
$594$		0			0
$595$		−	18.8509i		−	0.0316823i
$596$			68.0198i			0.114127i
$597$		0			0
$598$	−168.222	−	320.848i	−0.281307	−	0.536535i
$599$	−390.672			−0.652208			−0.326104	−	0.945334i	$-0.605736\pi$
$599$	−390.672			−0.652208			−0.326104	+	0.945334i	$0.605736\pi$
$600$		0			0
$601$	−726.400			−1.20865			−0.604326	−	0.796737i	$-0.706558\pi$
$601$	−726.400			−1.20865			−0.604326	+	0.796737i	$0.706558\pi$
$602$	−231.353			−0.384308
$603$		0			0
$604$	−546.851			−0.905383
$605$		−	198.649i		−	0.328345i
$606$		0			0
$607$	−289.612			−0.477120			−0.238560	−	0.971128i	$-0.576675\pi$
$607$	−289.612			−0.477120			−0.238560	+	0.971128i	$0.576675\pi$
$608$		−	118.718i		−	0.195259i
$609$		0			0
$610$	−119.805			−0.196402
$611$	−670.224			−1.09693
$612$		0			0
$613$		−	457.354i		−	0.746091i	−0.927813	−	0.373045i	$-0.878314\pi$
$613$		−	457.354i		−	0.746091i	0.927813	−	0.373045i	$-0.121686\pi$
$614$	−469.125			−0.764048
$615$		0			0
$616$	63.5006			0.103085
$617$			934.894i			1.51523i	0.652705	+	0.757613i	$0.273634\pi$
$617$			934.894i			1.51523i	−0.652705	+	0.757613i	$0.726366\pi$
$618$		0			0
$619$			79.2136i			0.127970i	0.997951	+	0.0639851i	$0.0203810\pi$
$619$			79.2136i			0.127970i	−0.997951	+	0.0639851i	$0.979619\pi$
$620$		−	177.965i		−	0.287040i
$621$		0			0
$622$	−562.268			−0.903968
$623$	375.110			0.602103
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	176.697i		−	0.282263i
$627$		0			0
$628$		−	605.403i		−	0.964017i
$629$	−107.033			−0.170164
$630$		0			0
$631$		−	405.115i		−	0.642021i	−0.947076	−	0.321011i	$-0.895977\pi$
$631$		−	405.115i		−	0.642021i	0.947076	−	0.321011i	$-0.104023\pi$
$632$		−	245.668i		−	0.388716i
$633$		0			0
$634$	385.831			0.608566
$635$		−	422.689i		−	0.665652i
$636$		0			0
$637$	371.193			0.582721
$638$		−	289.200i		−	0.453292i
$639$		0			0
$640$			25.2982i			0.0395285i
$641$			984.727i			1.53623i	0.640309	+	0.768117i	$0.278806\pi$
$641$			984.727i			1.53623i	−0.640309	+	0.768117i	$0.721194\pi$
$642$		0			0
$643$		−	330.197i		−	0.513526i	−0.966474	−	0.256763i	$-0.917344\pi$
$643$		−	330.197i		−	0.513526i	0.966474	−	0.256763i	$-0.0826560\pi$
$644$	161.282	−	84.5606i	0.250438	−	0.131305i
$645$		0			0
$646$	63.2032			0.0978377
$647$	−948.869			−1.46657			−0.733284	−	0.679923i	$-0.762013\pi$
$647$	−948.869			−1.46657			−0.733284	+	0.679923i	$0.762013\pi$
$648$		0			0
$649$		−	472.357i		−	0.727823i
$650$	78.7550			0.121162
$651$		0			0
$652$	−203.539			−0.312176
$653$	378.498			0.579630			0.289815	−	0.957083i	$-0.406406\pi$
$653$	378.498			0.579630			0.289815	+	0.957083i	$0.406406\pi$
$654$		0			0
$655$		−	229.763i		−	0.350783i
$656$	−166.625			−0.254002
$657$		0			0
$658$		−	336.904i		−	0.512012i
$659$		−	470.040i		−	0.713262i	−0.934245	−	0.356631i	$-0.883925\pi$
$659$		−	470.040i		−	0.713262i	0.934245	−	0.356631i	$-0.116075\pi$
$660$		0			0
$661$			244.609i			0.370058i	0.982733	+	0.185029i	$0.0592380\pi$
$661$			244.609i			0.370058i	−0.982733	+	0.185029i	$0.940762\pi$
$662$	−223.577			−0.337729
$663$		0			0
$664$			239.536i			0.360746i
$665$		−	185.776i		−	0.279363i
$666$		0			0
$667$	−385.114	−	734.525i	−0.577383	−	1.10124i
$668$	249.212			0.373071
$669$		0			0
$670$	193.223			0.288392
$671$	214.854			0.320199
$672$		0			0
$673$	297.873			0.442605			0.221302	−	0.975205i	$-0.428969\pi$
$673$	297.873			0.442605			0.221302	+	0.975205i	$0.428969\pi$
$674$		−	265.751i		−	0.394290i
$675$		0			0
$676$	−89.9059			−0.132997
$677$			306.899i			0.453322i	0.973974	+	0.226661i	$0.0727809\pi$
$677$			306.899i			0.453322i	−0.973974	+	0.226661i	$0.927219\pi$
$678$		0			0
$679$	−679.731			−1.00108
$680$	−13.4683			−0.0198064
$681$		0			0
$682$			319.155i			0.467969i
$683$	−270.473			−0.396007			−0.198003	−	0.980201i	$-0.563446\pi$
$683$	−270.473			−0.396007			−0.198003	+	0.980201i	$0.563446\pi$
$684$		0			0
$685$	118.534			0.173043
$686$			460.921i			0.671896i
$687$		0			0
$688$			165.294i			0.240252i
$689$		−	36.6913i		−	0.0532529i
$690$		0			0
$691$	−994.734			−1.43956			−0.719779	−	0.694204i	$-0.755757\pi$
$691$	−994.734			−1.43956			−0.719779	+	0.694204i	$0.755757\pi$
$692$	556.662			0.804425
$693$		0			0
$694$	−443.266			−0.638711
$695$			515.004i			0.741013i
$696$		0			0
$697$		−	88.7082i		−	0.127272i
$698$	−514.024			−0.736424
$699$		0			0
$700$			39.5881i			0.0565544i
$701$		−	615.121i		−	0.877490i	−0.898612	−	0.438745i	$-0.855423\pi$
$701$		−	615.121i		−	0.877490i	0.898612	−	0.438745i	$-0.144577\pi$
$702$		0			0
$703$	−1054.81			−1.50044
$704$		−	45.3689i		−	0.0644444i
$705$		0			0
$706$	−742.073			−1.05109
$707$		−	193.521i		−	0.273721i
$708$		0			0
$709$		−	768.270i		−	1.08360i	−0.840509	−	0.541798i	$-0.817744\pi$
$709$		−	768.270i		−	1.08360i	0.840509	−	0.541798i	$-0.182256\pi$
$710$			1.80914i			0.00254809i
$711$		0			0
$712$		−	268.003i		−	0.376408i
$713$	425.003	+	810.605i	0.596078	+	1.13689i
$714$		0			0
$715$	−141.236			−0.197533
$716$	45.4088			0.0634202
$717$		0			0
$718$		−	440.812i		−	0.613944i
$719$	613.243			0.852910			0.426455	−	0.904509i	$-0.359762\pi$
$719$	613.243			0.852910			0.426455	+	0.904509i	$0.359762\pi$
$720$		0			0
$721$	15.3609			0.0213050
$722$	112.336			0.155590
$723$		0			0
$724$			302.769i			0.418189i
$725$	180.296			0.248684
$726$		0			0
$727$			1148.29i			1.57949i	0.613436	+	0.789744i	$0.289787\pi$
$727$			1148.29i			1.57949i	−0.613436	+	0.789744i	$0.710213\pi$
$728$			124.710i			0.171306i
$729$		0			0
$730$			129.272i			0.177086i
$731$	−87.9994			−0.120382
$732$		0			0
$733$		−	570.538i		−	0.778361i	−0.921162	−	0.389180i	$-0.872758\pi$
$733$		−	570.538i		−	0.778361i	0.921162	−	0.389180i	$-0.127242\pi$
$734$			662.544i			0.902649i
$735$		0			0
$736$	−60.4155	−	115.230i	−0.0820863	−	0.156563i
$737$	−346.518			−0.470174
$738$		0			0
$739$	−1062.30			−1.43749			−0.718744	−	0.695275i	$-0.755283\pi$
$739$	−1062.30			−1.43749			−0.718744	+	0.695275i	$0.755283\pi$
$740$	224.776			0.303751
$741$		0			0
$742$	18.4437			0.0248568
$743$		−	166.131i		−	0.223595i	−0.993731	−	0.111797i	$-0.964339\pi$
$743$		−	166.131i		−	0.223595i	0.993731	−	0.111797i	$-0.0356608\pi$
$744$		0			0
$745$	76.0484			0.102078
$746$			10.5402i			0.0141289i
$747$		0			0
$748$	24.1536			0.0322909
$749$	481.325			0.642624
$750$		0			0
$751$		−	1210.26i		−	1.61153i	−0.592236	−	0.805765i	$-0.701755\pi$
$751$		−	1210.26i		−	1.61153i	0.592236	−	0.805765i	$-0.298245\pi$
$752$	−240.706			−0.320087
$753$		0			0
$754$	567.968			0.753274
$755$			611.398i			0.809799i
$756$		0			0
$757$		−	500.893i		−	0.661681i	−0.943687	−	0.330841i	$-0.892668\pi$
$757$		−	500.893i		−	0.661681i	0.943687	−	0.330841i	$-0.107332\pi$
$758$		−	627.958i		−	0.828441i
$759$		0			0
$760$	−132.730			−0.174645
$761$	−788.375			−1.03597			−0.517987	−	0.855389i	$-0.673318\pi$
$761$	−788.375			−1.03597			−0.517987	+	0.855389i	$0.673318\pi$
$762$		0			0
$763$	554.372			0.726569
$764$			136.208i			0.178282i
$765$		0			0
$766$		−	446.324i		−	0.582668i
$767$	927.674			1.20948
$768$		0			0
$769$			115.079i			0.149647i	0.997197	+	0.0748237i	$0.0238394\pi$
$769$			115.079i			0.149647i	−0.997197	+	0.0748237i	$0.976161\pi$
$770$		−	70.9958i		−	0.0922024i
$771$		0			0
$772$	486.538			0.630230
$773$		−	1396.31i		−	1.80635i	−0.429272	−	0.903175i	$-0.641230\pi$
$773$		−	1396.31i		−	1.80635i	0.429272	−	0.903175i	$-0.358770\pi$
$774$		0			0
$775$	−198.970			−0.256736
$776$			485.643i			0.625829i
$777$		0			0
$778$		−	178.661i		−	0.229642i
$779$		−	874.220i		−	1.12223i
$780$		0			0
$781$		−	3.24445i		−	0.00415423i
$782$	61.3464	−	32.1642i	0.0784481	−	0.0411306i
$783$		0			0
$784$	133.311			0.170040
$785$	−676.861			−0.862243
$786$		0			0
$787$		−	608.869i		−	0.773659i	−0.922151	−	0.386829i	$-0.873570\pi$
$787$		−	608.869i		−	0.773659i	0.922151	−	0.386829i	$-0.126430\pi$
$788$	262.715			0.333395
$789$		0			0
$790$	−274.666			−0.347678
$791$	14.7743			0.0186780
$792$		0			0
$793$			421.957i			0.532102i
$794$	233.643			0.294261
$795$		0			0
$796$		−	627.399i		−	0.788190i
$797$			651.157i			0.817009i	0.912756	+	0.408505i	$0.133950\pi$
$797$			651.157i			0.817009i	−0.912756	+	0.408505i	$0.866050\pi$
$798$		0			0
$799$		−	128.147i		−	0.160385i
$800$	28.2843			0.0353553
$801$		0			0
$802$			422.847i			0.527240i
$803$		−	231.832i		−	0.288708i
$804$		0			0
$805$	−94.5417	−	180.319i	−0.117443	−	0.223998i
$806$	−626.797			−0.777664
$807$		0			0
$808$	−138.264			−0.171119
$809$	521.791			0.644983			0.322492	−	0.946572i	$-0.395480\pi$
$809$	521.791			0.644983			0.322492	+	0.946572i	$0.395480\pi$
$810$		0			0
$811$	−1004.88			−1.23907			−0.619533	−	0.784971i	$-0.712678\pi$
$811$	−1004.88			−1.23907			−0.619533	+	0.784971i	$0.712678\pi$
$812$			285.503i			0.351604i
$813$		0			0
$814$	−403.104			−0.495214
$815$			227.563i			0.279219i
$816$		0			0
$817$	−867.234			−1.06149
$818$	78.5673			0.0960480
$819$		0			0
$820$			186.293i			0.227186i
$821$	752.571			0.916652			0.458326	−	0.888784i	$-0.348449\pi$
$821$	752.571			0.916652			0.458326	+	0.888784i	$0.348449\pi$
$822$		0			0
$823$	−1119.05			−1.35972			−0.679858	−	0.733343i	$-0.737959\pi$
$823$	−1119.05			−1.35972			−0.679858	+	0.733343i	$0.737959\pi$
$824$		−	10.9748i		−	0.0133189i
$825$		0			0
$826$			466.318i			0.564549i
$827$		−	305.110i		−	0.368936i	−0.982839	−	0.184468i	$-0.940944\pi$
$827$		−	305.110i		−	0.368936i	0.982839	−	0.184468i	$-0.0590562\pi$
$828$		0			0
$829$	−1139.25			−1.37424			−0.687121	−	0.726543i	$-0.741126\pi$
$829$	−1139.25			−1.37424			−0.687121	+	0.726543i	$0.741126\pi$
$830$	267.809			0.322661
$831$		0			0
$832$	89.1011			0.107093
$833$			70.9726i			0.0852012i
$834$		0			0
$835$		−	278.627i		−	0.333685i
$836$	238.034			0.284729
$837$		0			0
$838$			597.710i			0.713258i
$839$			811.707i			0.967470i	0.875215	+	0.483735i	$0.160720\pi$
$839$			811.707i			0.967470i	−0.875215	+	0.483735i	$0.839280\pi$
$840$		0			0
$841$	459.264			0.546093
$842$			150.827i			0.179130i
$843$		0			0
$844$	151.726			0.179770
$845$			100.518i			0.118956i
$846$		0			0
$847$		−	351.695i		−	0.415224i
$848$		−	13.1774i		−	0.0155394i
$849$		0			0
$850$			15.0580i			0.0177153i
$851$	−1023.82	+	536.794i	−1.20308	+	0.630781i
$852$		0			0
$853$	−336.775			−0.394812			−0.197406	−	0.980322i	$-0.563252\pi$
$853$	−336.775			−0.394812			−0.197406	+	0.980322i	$0.563252\pi$
$854$	−212.107			−0.248369
$855$		0			0
$856$		−	343.890i		−	0.401740i
$857$	764.331			0.891869			0.445934	−	0.895066i	$-0.352871\pi$
$857$	764.331			0.891869			0.445934	+	0.895066i	$0.352871\pi$
$858$		0			0
$859$	−368.659			−0.429172			−0.214586	−	0.976705i	$-0.568840\pi$
$859$	−368.659			−0.429172			−0.214586	+	0.976705i	$0.568840\pi$
$860$	184.804			0.214888
$861$		0			0
$862$			432.203i			0.501395i
$863$	731.744			0.847907			0.423954	−	0.905684i	$-0.360642\pi$
$863$	731.744			0.847907			0.423954	+	0.905684i	$0.360642\pi$
$864$		0			0
$865$		−	622.367i		−	0.719500i
$866$			404.310i			0.466870i
$867$		0			0
$868$		−	315.074i		−	0.362989i
$869$	492.575			0.566830
$870$		0			0
$871$		−	680.536i		−	0.781327i
$872$		−	396.079i		−	0.454219i
$873$		0			0
$874$	604.569	−	316.978i	0.691727	−	0.362675i
$875$	44.2608			0.0505838
$876$		0			0
$877$	−53.7425			−0.0612799			−0.0306400	−	0.999530i	$-0.509755\pi$
$877$	−53.7425			−0.0612799			−0.0306400	+	0.999530i	$0.509755\pi$
$878$	653.055			0.743799
$879$		0			0
$880$	−50.7240			−0.0576409
$881$			860.235i			0.976430i	0.872723	+	0.488215i	$0.162352\pi$
$881$			860.235i			0.976430i	−0.872723	+	0.488215i	$0.837648\pi$
$882$		0			0
$883$	−445.282			−0.504283			−0.252142	−	0.967690i	$-0.581135\pi$
$883$	−445.282			−0.504283			−0.252142	+	0.967690i	$0.581135\pi$
$884$			47.4359i			0.0536605i
$885$		0			0
$886$	−92.0241			−0.103865
$887$	−121.682			−0.137184			−0.0685921	−	0.997645i	$-0.521851\pi$
$887$	−121.682			−0.137184			−0.0685921	+	0.997645i	$0.521851\pi$
$888$		0			0
$889$		−	748.343i		−	0.841780i
$890$	−299.636			−0.336670
$891$		0			0
$892$	−331.589			−0.371736
$893$		−	1262.89i		−	1.41421i
$894$		0			0
$895$		−	50.7686i		−	0.0567247i
$896$			44.7888i			0.0499875i
$897$		0			0
$898$	774.011			0.861927
$899$	−1434.94			−1.59615
$900$		0			0
$901$	7.01541			0.00778625
$902$		−	334.090i		−	0.370388i
$903$		0			0
$904$		−	10.5557i		−	0.0116767i
$905$	338.506			0.374040
$906$		0			0
$907$			914.181i			1.00792i	0.863728	+	0.503959i	$0.168124\pi$
$907$			914.181i			1.00792i	−0.863728	+	0.503959i	$0.831876\pi$
$908$			810.786i			0.892937i
$909$		0			0
$910$	139.431			0.153220
$911$		−	936.862i		−	1.02839i	−0.857674	−	0.514195i	$-0.828091\pi$
$911$		−	936.862i		−	1.02839i	0.857674	−	0.514195i	$-0.171909\pi$
$912$		0			0
$913$	−480.278			−0.526044
$914$		−	472.869i		−	0.517362i
$915$		0			0
$916$			764.619i			0.834737i
$917$		−	406.780i		−	0.443598i
$918$		0			0
$919$			1272.45i			1.38460i	0.721610	+	0.692300i	$0.243403\pi$
$919$			1272.45i			1.38460i	−0.721610	+	0.692300i	$0.756597\pi$
$920$	−128.831	+	67.5466i	−0.140034	+	0.0734202i
$921$		0			0
$922$	−543.830			−0.589838
$923$	6.37186			0.00690343
$924$		0			0
$925$		−	251.307i		−	0.271683i
$926$	−1048.64			−1.13244
$927$		0			0
$928$	203.981			0.219808
$929$	1343.27			1.44593			0.722967	−	0.690882i	$-0.242778\pi$
$929$	1343.27			1.44593			0.722967	+	0.690882i	$0.242778\pi$
$930$		0			0
$931$			699.435i			0.751273i
$932$	250.265			0.268525
$933$		0			0
$934$		−	260.011i		−	0.278384i
$935$		−	27.0045i		−	0.0288819i
$936$		0			0
$937$		−	850.505i		−	0.907690i	−0.891081	−	0.453845i	$-0.850052\pi$
$937$		−	850.505i		−	0.907690i	0.891081	−	0.453845i	$-0.149948\pi$
$938$	342.088			0.364699
$939$		0			0
$940$			269.117i			0.286295i
$941$			1778.92i			1.89045i	0.326417	+	0.945226i	$0.394159\pi$
$941$			1778.92i			1.89045i	−0.326417	+	0.945226i	$0.605841\pi$
$942$		0			0
$943$	−444.892	−	848.538i	−0.471783	−	0.899828i
$944$	333.167			0.352931
$945$		0			0
$946$	−331.420			−0.350339
$947$	−121.252			−0.128038			−0.0640190	−	0.997949i	$-0.520392\pi$
$947$	−121.252			−0.128038			−0.0640190	+	0.997949i	$0.520392\pi$
$948$		0			0
$949$	455.302			0.479770
$950$			148.397i			0.156207i
$951$		0			0
$952$	−23.8448			−0.0250470
$953$		−	1183.70i		−	1.24207i	−0.783781	−	0.621037i	$-0.786712\pi$
$953$		−	1183.70i		−	1.24207i	0.783781	−	0.621037i	$-0.213288\pi$
$954$		0			0
$955$	152.285			0.159461
$956$	258.414			0.270308
$957$		0			0
$958$		−	910.286i		−	0.950195i
$959$	209.857			0.218829
$960$		0			0
$961$	622.570			0.647835
$962$		−	791.667i		−	0.822938i
$963$		0			0
$964$			626.746i			0.650151i
$965$		−	543.966i		−	0.563695i
$966$		0			0
$967$	1135.29			1.17403			0.587015	−	0.809576i	$-0.300303\pi$
$967$	1135.29			1.17403			0.587015	+	0.809576i	$0.300303\pi$
$968$	−251.273			−0.259580
$969$		0			0
$970$	542.966			0.559758
$971$			1446.26i			1.48945i	0.667369	+	0.744727i	$0.267421\pi$
$971$			1446.26i			1.48945i	−0.667369	+	0.744727i	$0.732579\pi$
$972$		0			0
$973$			911.780i			0.937082i
$974$	572.349			0.587628
$975$		0			0
$976$			151.543i			0.155269i
$977$		−	77.0844i		−	0.0788991i	−0.999222	−	0.0394496i	$-0.987440\pi$
$977$		−	77.0844i		−	0.0788991i	0.999222	−	0.0394496i	$-0.0125604\pi$
$978$		0			0
$979$	537.356			0.548883
$980$		−	149.047i		−	0.152088i
$981$		0			0
$982$	245.489			0.249989
$983$			496.933i			0.505527i	0.967528	+	0.252763i	$0.0813394\pi$
$983$			496.933i			0.505527i	−0.967528	+	0.252763i	$0.918661\pi$
$984$		0			0
$985$		−	293.725i		−	0.298198i
$986$			108.596i			0.110138i
$987$		0			0
$988$			467.481i			0.473158i
$989$	−841.757	+	441.336i	−0.851120	+	0.446245i
$990$		0			0
$991$	290.841			0.293482			0.146741	−	0.989175i	$-0.453122\pi$
$991$	290.841			0.293482			0.146741	+	0.989175i	$0.453122\pi$
$992$	−225.109			−0.226925
$993$		0			0
$994$			3.20297i			0.00322230i
$995$	−701.454			−0.704978
$996$		0			0
$997$	−1034.44			−1.03756			−0.518778	−	0.854909i	$-0.673613\pi$
$997$	−1034.44			−1.03756			−0.518778	+	0.854909i	$0.673613\pi$
$998$	−9.10840			−0.00912665
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.2		32
3.2	odd	2		690.3.c.a.91.21	yes	32
23.22	odd	2	inner	2070.3.c.b.91.15		32
69.68	even	2		690.3.c.a.91.20	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.20	✓	32	69.68	even	2
690.3.c.a.91.21	yes	32	3.2	odd	2
2070.3.c.b.91.2		32	1.1	even	1	trivial
2070.3.c.b.91.15		32	23.22	odd	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -3.95881i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -3.95881i q^{7} -2.82843 q^{8} +3.16228i q^{10} -5.67111i q^{11} +11.1376 q^{13} +5.59860i q^{14} +4.00000 q^{16} +2.12953i q^{17} +20.9865i q^{19} -4.47214i q^{20} +8.02016i q^{22} +(10.6801 + 20.3700i) q^{23} -5.00000 q^{25} -15.7510 q^{26} -7.91762i q^{28} -36.0592 q^{29} +39.7941 q^{31} -5.65685 q^{32} -3.01161i q^{34} -8.85217 q^{35} +50.2613i q^{37} -29.6794i q^{38} +6.32456i q^{40} -41.6563 q^{41} +41.3234i q^{43} -11.3422i q^{44} +(-15.1039 - 28.8075i) q^{46} -60.1764 q^{47} +33.3278 q^{49} +7.07107 q^{50} +22.2753 q^{52} -3.29435i q^{53} -12.6810 q^{55} +11.1972i q^{56} +50.9954 q^{58} +83.2918 q^{59} +37.8857i q^{61} -56.2773 q^{62} +8.00000 q^{64} -24.9045i q^{65} -61.1024i q^{67} +4.25906i q^{68} +12.5189 q^{70} +0.572102 q^{71} +40.8795 q^{73} -71.0803i q^{74} +41.9730i q^{76} -22.4509 q^{77} +86.8569i q^{79} -8.94427i q^{80} +58.9109 q^{82} -84.6886i q^{83} +4.76177 q^{85} -58.4401i q^{86} +16.0403i q^{88} +94.7533i q^{89} -44.0918i q^{91} +(21.3601 + 40.7400i) q^{92} +85.1023 q^{94} +46.9273 q^{95} -171.701i q^{97} -47.1327 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 32 * q + 64 * q^4	\( 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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