LMFDB - Embedded newform 1840.2.i.b.1471.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.i (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8x^{14} + 33x^{12} - 98x^{10} + 272x^{8} - 882x^{6} + 2673x^{4} - 5832x^{2} + 6561 $ x^16 - 8x^14 + 33x^12 - 98x^10 + 272x^8 - 882x^6 + 2673x^4 - 5832*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.1
Root			$0.739948 - 1.56604i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.b.1471.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.13208i q^{3} -1.00000i q^{5} -1.01363 q^{7} -6.80991 q^{9} +O(q^{10})$	$q-3.13208i q^{3} -1.00000i q^{5} -1.01363 q^{7} -6.80991 q^{9} +1.65218 q^{11} -4.63515 q^{13} -3.13208 q^{15} +6.97256i q^{17} -4.61197 q^{19} +3.17476i q^{21} +(4.40042 + 1.90690i) q^{23} -1.00000 q^{25} +11.9329i q^{27} +0.837348 q^{29} +4.14570i q^{31} -5.17476i q^{33} +1.01363i q^{35} -3.47250i q^{37} +14.5176i q^{39} -7.51217 q^{41} +8.42577 q^{43} +6.80991i q^{45} +0.172285i q^{47} -5.97256 q^{49} +21.8386 q^{51} +2.29774i q^{53} -1.65218i q^{55} +14.4451i q^{57} +1.89814i q^{59} -12.4451i q^{61} +6.90271 q^{63} +4.63515i q^{65} -10.2376 q^{67} +(5.97256 - 13.7825i) q^{69} +0.841342i q^{71} -13.3100 q^{73} +3.13208i q^{75} -1.67470 q^{77} -4.61197 q^{79} +16.9451 q^{81} -11.8898 q^{83} +6.97256 q^{85} -2.62264i q^{87} -8.11975i q^{89} +4.69831 q^{91} +12.9847 q^{93} +4.61197i q^{95} +14.7946i q^{97} -11.2512 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100}) $ 16 * q - 16 * q^9 - 8 * q^13 - 16 * q^25 + 36 * q^29 - 44 * q^41 + 20 * q^49 - 20 * q^69 - 48 * q^73 - 72 * q^77 + 40 * q^81 - 4 * q^85 + 88 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	3.13208i		−	1.80831i	−0.427209	−	0.904153i	$-0.640503\pi$
$3$		−	3.13208i		−	1.80831i	0.427209	−	0.904153i	$-0.359497\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−1.01363			−0.383115			−0.191558	−	0.981481i	$-0.561354\pi$
$7$	−1.01363			−0.383115			−0.191558	+	0.981481i	$0.561354\pi$
$8$		0			0
$9$	−6.80991			−2.26997
$10$		0			0
$11$	1.65218			0.498151			0.249076	−	0.968484i	$-0.419873\pi$
$11$	1.65218			0.498151			0.249076	+	0.968484i	$0.419873\pi$
$12$		0			0
$13$	−4.63515			−1.28556			−0.642779	−	0.766051i	$-0.722219\pi$
$13$	−4.63515			−1.28556			−0.642779	+	0.766051i	$0.722219\pi$
$14$		0			0
$15$	−3.13208			−0.808699
$16$		0			0
$17$			6.97256i			1.69109i	0.533901	+	0.845547i	$0.320726\pi$
$17$			6.97256i			1.69109i	−0.533901	+	0.845547i	$0.679274\pi$
$18$		0			0
$19$	−4.61197			−1.05806			−0.529030	−	0.848603i	$-0.677444\pi$
$19$	−4.61197			−1.05806			−0.529030	+	0.848603i	$0.677444\pi$
$20$		0			0
$21$			3.17476i			0.692789i
$22$		0			0
$23$	4.40042	+	1.90690i	0.917552	+	0.397616i
$23$	4.40042	+	1.90690i	0.917552	+	0.397616i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			11.9329i			2.29649i
$28$		0			0
$29$	0.837348			0.155492			0.0777458	−	0.996973i	$-0.475228\pi$
$29$	0.837348			0.155492			0.0777458	+	0.996973i	$0.475228\pi$
$30$		0			0
$31$			4.14570i			0.744591i	0.928114	+	0.372295i	$0.121429\pi$
$31$			4.14570i			0.744591i	−0.928114	+	0.372295i	$0.878571\pi$
$32$		0			0
$33$		−	5.17476i		−	0.900810i
$34$		0			0
$35$			1.01363i			0.171334i
$36$		0			0
$37$		−	3.47250i		−	0.570875i	−0.958397	−	0.285437i	$-0.907861\pi$
$37$		−	3.47250i		−	0.570875i	0.958397	−	0.285437i	$-0.0921389\pi$
$38$		0			0
$39$			14.5176i			2.32468i
$40$		0			0
$41$	−7.51217			−1.17320			−0.586602	−	0.809875i	$-0.699535\pi$
$41$	−7.51217			−1.17320			−0.586602	+	0.809875i	$0.699535\pi$
$42$		0			0
$43$	8.42577			1.28492			0.642459	−	0.766320i	$-0.277914\pi$
$43$	8.42577			1.28492			0.642459	+	0.766320i	$0.277914\pi$
$44$		0			0
$45$			6.80991i			1.01516i
$46$		0			0
$47$			0.172285i			0.0251304i	0.999921	+	0.0125652i	$0.00399973\pi$
$47$			0.172285i			0.0251304i	−0.999921	+	0.0125652i	$0.996000\pi$
$48$		0			0
$49$	−5.97256			−0.853223
$50$		0			0
$51$	21.8386			3.05802
$52$		0			0
$53$			2.29774i			0.315619i	0.987470	+	0.157809i	$0.0504431\pi$
$53$			2.29774i			0.315619i	−0.987470	+	0.157809i	$0.949557\pi$
$54$		0			0
$55$		−	1.65218i		−	0.222780i
$56$		0			0
$57$			14.4451i			1.91329i
$58$		0			0
$59$			1.89814i			0.247117i	0.992337	+	0.123558i	$0.0394306\pi$
$59$			1.89814i			0.247117i	−0.992337	+	0.123558i	$0.960569\pi$
$60$		0			0
$61$		−	12.4451i		−	1.59343i	−0.604358	−	0.796713i	$-0.706570\pi$
$61$		−	12.4451i		−	1.59343i	0.604358	−	0.796713i	$-0.293430\pi$
$62$		0			0
$63$	6.90271			0.869660
$64$		0			0
$65$			4.63515i			0.574919i
$66$		0			0
$67$	−10.2376			−1.25072			−0.625359	−	0.780337i	$-0.715048\pi$
$67$	−10.2376			−1.25072			−0.625359	+	0.780337i	$0.715048\pi$
$68$		0			0
$69$	5.97256	−	13.7825i	0.719012	−	1.65921i
$70$		0			0
$71$			0.841342i			0.0998489i	0.998753	+	0.0499245i	$0.0158981\pi$
$71$			0.841342i			0.0998489i	−0.998753	+	0.0499245i	$0.984102\pi$
$72$		0			0
$73$	−13.3100			−1.55781			−0.778907	−	0.627139i	$-0.784226\pi$
$73$	−13.3100			−1.55781			−0.778907	+	0.627139i	$0.784226\pi$
$74$		0			0
$75$			3.13208i			0.361661i
$76$		0			0
$77$	−1.67470			−0.190849
$78$		0			0
$79$	−4.61197			−0.518888			−0.259444	−	0.965758i	$-0.583539\pi$
$79$	−4.61197			−0.518888			−0.259444	+	0.965758i	$0.583539\pi$
$80$		0			0
$81$	16.9451			1.88279
$82$		0			0
$83$	−11.8898			−1.30507			−0.652535	−	0.757759i	$-0.726294\pi$
$83$	−11.8898			−1.30507			−0.652535	+	0.757759i	$0.726294\pi$
$84$		0			0
$85$	6.97256			0.756280
$86$		0			0
$87$		−	2.62264i		−	0.281176i
$88$		0			0
$89$		−	8.11975i		−	0.860692i	−0.902664	−	0.430346i	$-0.858392\pi$
$89$		−	8.11975i		−	0.860692i	0.902664	−	0.430346i	$-0.141608\pi$
$90$		0			0
$91$	4.69831			0.492517
$92$		0			0
$93$	12.9847			1.34645
$94$		0			0
$95$			4.61197i			0.473178i
$96$		0			0
$97$			14.7946i			1.50216i	0.660210	+	0.751081i	$0.270467\pi$
$97$			14.7946i			1.50216i	−0.660210	+	0.751081i	$0.729533\pi$
$98$		0			0
$99$	−11.2512			−1.13079
$100$		0			0
$101$	2.29774			0.228633			0.114317	−	0.993444i	$-0.463532\pi$
$101$	2.29774			0.228633			0.114317	+	0.993444i	$0.463532\pi$
$102$		0			0
$103$	15.0650			1.48440			0.742199	−	0.670179i	$-0.233783\pi$
$103$	15.0650			1.48440			0.742199	+	0.670179i	$0.233783\pi$
$104$		0			0
$105$	3.17476			0.309825
$106$		0			0
$107$	8.55489			0.827032			0.413516	−	0.910497i	$-0.364301\pi$
$107$	8.55489			0.827032			0.413516	+	0.910497i	$0.364301\pi$
$108$		0			0
$109$		−	5.50006i		−	0.526810i	−0.964685	−	0.263405i	$-0.915154\pi$
$109$		−	5.50006i		−	0.526810i	0.964685	−	0.263405i	$-0.0848456\pi$
$110$		0			0
$111$	−10.8761			−1.03232
$112$		0			0
$113$			9.82201i			0.923977i	0.886886	+	0.461989i	$0.152864\pi$
$113$			9.82201i			0.923977i	−0.886886	+	0.461989i	$0.847136\pi$
$114$		0			0
$115$	1.90690	−	4.40042i	0.177819	−	0.410342i
$116$		0			0
$117$	31.5649			2.91818
$118$		0			0
$119$		−	7.06758i		−	0.647884i
$120$		0			0
$121$	−8.27030			−0.751845
$122$		0			0
$123$			23.5287i			2.12151i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	16.5929i		−	1.47238i	−0.676773	−	0.736192i	$-0.736622\pi$
$127$		−	16.5929i		−	1.47238i	0.676773	−	0.736192i	$-0.263378\pi$
$128$		0			0
$129$		−	26.3902i		−	2.32353i
$130$		0			0
$131$		−	10.6253i		−	0.928338i	−0.885747	−	0.464169i	$-0.846353\pi$
$131$		−	10.6253i		−	0.928338i	0.885747	−	0.464169i	$-0.153647\pi$
$132$		0			0
$133$	4.67482			0.405358
$134$		0			0
$135$	11.9329			1.02702
$136$		0			0
$137$			3.27030i			0.279400i	0.990194	+	0.139700i	$0.0446139\pi$
$137$			3.27030i			0.279400i	−0.990194	+	0.139700i	$0.955386\pi$
$138$		0			0
$139$			16.7604i			1.42159i	0.703397	+	0.710797i	$0.251666\pi$
$139$			16.7604i			1.42159i	−0.703397	+	0.710797i	$0.748334\pi$
$140$		0			0
$141$	0.539610			0.0454434
$142$		0			0
$143$	−7.65811			−0.640403
$144$		0			0
$145$		−	0.837348i		−	0.0695380i
$146$		0			0
$147$			18.7065i			1.54289i
$148$		0			0
$149$			1.32518i			0.108563i	0.998526	+	0.0542814i	$0.0172868\pi$
$149$			1.32518i			0.108563i	−0.998526	+	0.0542814i	$0.982713\pi$
$150$		0			0
$151$		−	6.57080i		−	0.534725i	−0.963596	−	0.267362i	$-0.913848\pi$
$151$		−	6.57080i		−	0.534725i	0.963596	−	0.267362i	$-0.0861520\pi$
$152$		0			0
$153$		−	47.4825i		−	3.83873i
$154$		0			0
$155$	4.14570			0.332991
$156$		0			0
$157$			18.4968i			1.47621i	0.674687	+	0.738104i	$0.264279\pi$
$157$			18.4968i			1.47621i	−0.674687	+	0.738104i	$0.735721\pi$
$158$		0			0
$159$	7.19669			0.570735
$160$		0			0
$161$	−4.46039	−	1.93289i	−0.351528	−	0.152333i
$162$		0			0
$163$			20.3892i			1.59701i	0.601991	+	0.798503i	$0.294374\pi$
$163$			20.3892i			1.59701i	−0.601991	+	0.798503i	$0.705626\pi$
$164$		0			0
$165$	−5.17476			−0.402854
$166$		0			0
$167$			18.5518i			1.43558i	0.696261	+	0.717789i	$0.254846\pi$
$167$			18.5518i			1.43558i	−0.696261	+	0.717789i	$0.745154\pi$
$168$		0			0
$169$	8.48460			0.652662
$170$		0			0
$171$	31.4071			2.40176
$172$		0			0
$173$	−8.34952			−0.634802			−0.317401	−	0.948291i	$-0.602810\pi$
$173$	−8.34952			−0.634802			−0.317401	+	0.948291i	$0.602810\pi$
$174$		0			0
$175$	1.01363			0.0766230
$176$		0			0
$177$	5.94512			0.446863
$178$		0			0
$179$		−	23.5592i		−	1.76090i	−0.474142	−	0.880449i	$-0.657242\pi$
$179$		−	23.5592i		−	1.76090i	0.474142	−	0.880449i	$-0.342758\pi$
$180$		0			0
$181$		−	1.84958i		−	0.137478i	−0.997635	−	0.0687391i	$-0.978102\pi$
$181$		−	1.84958i		−	0.137478i	0.997635	−	0.0687391i	$-0.0218976\pi$
$182$		0			0
$183$	−38.9789			−2.88140
$184$		0			0
$185$	−3.47250			−0.255303
$186$		0			0
$187$			11.5199i			0.842421i
$188$		0			0
$189$		−	12.0955i		−	0.879821i
$190$		0			0
$191$	−8.81837			−0.638075			−0.319037	−	0.947742i	$-0.603360\pi$
$191$	−8.81837			−0.638075			−0.319037	+	0.947742i	$0.603360\pi$
$192$		0			0
$193$	3.36485			0.242207			0.121104	−	0.992640i	$-0.461357\pi$
$193$	3.36485			0.242207			0.121104	+	0.992640i	$0.461357\pi$
$194$		0			0
$195$	14.5176			1.03963
$196$		0			0
$197$	23.2000			1.65293			0.826464	−	0.562989i	$-0.190349\pi$
$197$	23.2000			1.65293			0.826464	+	0.562989i	$0.190349\pi$
$198$		0			0
$199$	21.3772			1.51539			0.757694	−	0.652610i	$-0.226326\pi$
$199$	21.3772			1.51539			0.757694	+	0.652610i	$0.226326\pi$
$200$		0			0
$201$			32.0649i			2.26168i
$202$		0			0
$203$	−0.848759			−0.0595712
$204$		0			0
$205$			7.51217i			0.524673i
$206$		0			0
$207$	−29.9665	−	12.9858i	−2.08281	−	0.902577i
$208$		0			0
$209$	−7.61982			−0.527074
$210$		0			0
$211$			5.86631i			0.403853i	0.979401	+	0.201927i	$0.0647203\pi$
$211$			5.86631i			0.403853i	−0.979401	+	0.201927i	$0.935280\pi$
$212$		0			0
$213$	2.63515			0.180557
$214$		0			0
$215$		−	8.42577i		−	0.574633i
$216$		0			0
$217$		−	4.20220i		−	0.285264i
$218$		0			0
$219$			41.6879i			2.81700i
$220$		0			0
$221$		−	32.3189i		−	2.17400i
$222$		0			0
$223$		−	14.8826i		−	0.996613i	−0.867001	−	0.498307i	$-0.833955\pi$
$223$		−	14.8826i		−	0.996613i	0.867001	−	0.498307i	$-0.166045\pi$
$224$		0			0
$225$	6.80991			0.453994
$226$		0			0
$227$	−19.7633			−1.31174			−0.655869	−	0.754875i	$-0.727697\pi$
$227$	−19.7633			−1.31174			−0.655869	+	0.754875i	$0.727697\pi$
$228$		0			0
$229$			11.7154i			0.774172i	0.922044	+	0.387086i	$0.126518\pi$
$229$			11.7154i			0.774172i	−0.922044	+	0.387086i	$0.873482\pi$
$230$		0			0
$231$			5.24528i			0.345114i
$232$		0			0
$233$	−11.6595			−0.763838			−0.381919	−	0.924196i	$-0.624737\pi$
$233$	−11.6595			−0.763838			−0.381919	+	0.924196i	$0.624737\pi$
$234$		0			0
$235$	0.172285			0.0112386
$236$		0			0
$237$			14.4451i			0.938307i
$238$		0			0
$239$		−	21.7474i		−	1.40672i	−0.710833	−	0.703361i	$-0.751682\pi$
$239$		−	21.7474i		−	1.40672i	0.710833	−	0.703361i	$-0.248318\pi$
$240$		0			0
$241$			9.59560i			0.618107i	0.951045	+	0.309054i	$0.100012\pi$
$241$			9.59560i			0.618107i	−0.951045	+	0.309054i	$0.899988\pi$
$242$		0			0
$243$		−	17.2746i		−	1.10817i
$244$		0			0
$245$			5.97256i			0.381573i
$246$		0			0
$247$	21.3772			1.36020
$248$		0			0
$249$			37.2396i			2.35997i
$250$		0			0
$251$	−2.20965			−0.139472			−0.0697358	−	0.997565i	$-0.522216\pi$
$251$	−2.20965			−0.139472			−0.0697358	+	0.997565i	$0.522216\pi$
$252$		0			0
$253$	7.27030	+	3.15055i	0.457080	+	0.198073i
$254$		0			0
$255$		−	21.8386i		−	1.36759i
$256$		0			0
$257$	−15.5801			−0.971863			−0.485931	−	0.873997i	$-0.661519\pi$
$257$	−15.5801			−0.971863			−0.485931	+	0.873997i	$0.661519\pi$
$258$		0			0
$259$			3.51982i			0.218711i
$260$		0			0
$261$	−5.70226			−0.352961
$262$		0			0
$263$	−29.4609			−1.81664			−0.908320	−	0.418276i	$-0.862634\pi$
$263$	−29.4609			−1.81664			−0.908320	+	0.418276i	$0.862634\pi$
$264$		0			0
$265$	2.29774			0.141149
$266$		0			0
$267$	−25.4317			−1.55639
$268$		0			0
$269$	−30.7276			−1.87349			−0.936747	−	0.350007i	$-0.886179\pi$
$269$	−30.7276			−1.87349			−0.936747	+	0.350007i	$0.886179\pi$
$270$		0			0
$271$			6.98124i			0.424080i	0.977261	+	0.212040i	$0.0680107\pi$
$271$			6.98124i			0.424080i	−0.977261	+	0.212040i	$0.931989\pi$
$272$		0			0
$273$		−	14.7155i		−	0.890621i
$274$		0			0
$275$	−1.65218			−0.0996303
$276$		0			0
$277$	−24.9540			−1.49934			−0.749670	−	0.661812i	$-0.769788\pi$
$277$	−24.9540			−1.49934			−0.749670	+	0.661812i	$0.769788\pi$
$278$		0			0
$279$		−	28.2319i		−	1.69020i
$280$		0			0
$281$			12.0649i			0.719730i	0.933004	+	0.359865i	$0.117177\pi$
$281$			12.0649i			0.719730i	−0.933004	+	0.359865i	$0.882823\pi$
$282$		0			0
$283$	−1.22652			−0.0729091			−0.0364546	−	0.999335i	$-0.511606\pi$
$283$	−1.22652			−0.0729091			−0.0364546	+	0.999335i	$0.511606\pi$
$284$		0			0
$285$	14.4451			0.855651
$286$		0			0
$287$	7.61454			0.449472
$288$		0			0
$289$	−31.6166			−1.85980
$290$		0			0
$291$	46.3377			2.71637
$292$		0			0
$293$			15.6474i			0.914130i	0.889433	+	0.457065i	$0.151099\pi$
$293$			15.6474i			0.914130i	−0.889433	+	0.457065i	$0.848901\pi$
$294$		0			0
$295$	1.89814			0.110514
$296$		0			0
$297$			19.7154i			1.14400i
$298$		0			0
$299$	−20.3966	−	8.83877i	−1.17957	−	0.511159i
$300$		0			0
$301$	−8.54059			−0.492272
$302$		0			0
$303$		−	7.19669i		−	0.413439i
$304$		0			0
$305$	−12.4451			−0.712602
$306$		0			0
$307$			8.21288i			0.468734i	0.972148	+	0.234367i	$0.0753017\pi$
$307$			8.21288i			0.468734i	−0.972148	+	0.234367i	$0.924698\pi$
$308$		0			0
$309$		−	47.1848i		−	2.68425i
$310$		0			0
$311$		−	17.9083i		−	1.01549i	−0.861508	−	0.507745i	$-0.830479\pi$
$311$		−	17.9083i		−	1.01549i	0.861508	−	0.507745i	$-0.169521\pi$
$312$		0			0
$313$			17.9969i			1.01724i	0.860990	+	0.508622i	$0.169845\pi$
$313$			17.9969i			1.01724i	−0.860990	+	0.508622i	$0.830155\pi$
$314$		0			0
$315$		−	6.90271i		−	0.388924i
$316$		0			0
$317$	−9.59560			−0.538943			−0.269471	−	0.963008i	$-0.586849\pi$
$317$	−9.59560			−0.538943			−0.269471	+	0.963008i	$0.586849\pi$
$318$		0			0
$319$	1.38345			0.0774584
$320$		0			0
$321$		−	26.7946i		−	1.49553i
$322$		0			0
$323$		−	32.1573i		−	1.78928i
$324$		0			0
$325$	4.63515			0.257112
$326$		0			0
$327$	−17.2266			−0.952634
$328$		0			0
$329$		−	0.174633i		−	0.00962782i
$330$		0			0
$331$		−	18.9700i		−	1.04269i	−0.853347	−	0.521343i	$-0.825431\pi$
$331$		−	18.9700i		−	1.04269i	0.853347	−	0.521343i	$-0.174569\pi$
$332$		0			0
$333$			23.6474i			1.29587i
$334$		0			0
$335$			10.2376i			0.559338i
$336$		0			0
$337$		−	32.0649i		−	1.74668i	−0.487107	−	0.873342i	$-0.661948\pi$
$337$		−	32.0649i		−	1.74668i	0.487107	−	0.873342i	$-0.338052\pi$
$338$		0			0
$339$	30.7633			1.67083
$340$		0			0
$341$			6.84945i			0.370919i
$342$		0			0
$343$	13.1493			0.709998
$344$		0			0
$345$	−13.7825	−	5.97256i	−0.742023	−	0.321552i
$346$		0			0
$347$		−	12.6932i		−	0.681405i	−0.940171	−	0.340703i	$-0.889335\pi$
$347$		−	12.6932i		−	0.681405i	0.940171	−	0.340703i	$-0.110665\pi$
$348$		0			0
$349$	−0.813134			−0.0435261			−0.0217630	−	0.999763i	$-0.506928\pi$
$349$	−0.813134			−0.0435261			−0.0217630	+	0.999763i	$0.506928\pi$
$350$		0			0
$351$		−	55.3109i		−	2.95228i
$352$		0			0
$353$	25.3342			1.34840			0.674201	−	0.738548i	$-0.264488\pi$
$353$	25.3342			1.34840			0.674201	+	0.738548i	$0.264488\pi$
$354$		0			0
$355$	0.841342			0.0446538
$356$		0			0
$357$	−22.1362			−1.17157
$358$		0			0
$359$	−7.16619			−0.378217			−0.189108	−	0.981956i	$-0.560560\pi$
$359$	−7.16619			−0.378217			−0.189108	+	0.981956i	$0.560560\pi$
$360$		0			0
$361$	2.27030			0.119489
$362$		0			0
$363$			25.9032i			1.35957i
$364$		0			0
$365$			13.3100i			0.696676i
$366$		0			0
$367$	−24.8490			−1.29711			−0.648553	−	0.761170i	$-0.724625\pi$
$367$	−24.8490			−1.29711			−0.648553	+	0.761170i	$0.724625\pi$
$368$		0			0
$369$	51.1572			2.66314
$370$		0			0
$371$		−	2.32905i		−	0.120918i
$372$		0			0
$373$			6.89011i			0.356757i	0.983962	+	0.178378i	$0.0570851\pi$
$373$			6.89011i			0.356757i	−0.983962	+	0.178378i	$0.942915\pi$
$374$		0			0
$375$	3.13208			0.161740
$376$		0			0
$377$	−3.88123			−0.199894
$378$		0			0
$379$	−17.0747			−0.877070			−0.438535	−	0.898714i	$-0.644503\pi$
$379$	−17.0747			−0.877070			−0.438535	+	0.898714i	$0.644503\pi$
$380$		0			0
$381$	−51.9703			−2.66252
$382$		0			0
$383$	18.0981			0.924768			0.462384	−	0.886680i	$-0.346994\pi$
$383$	18.0981			0.924768			0.462384	+	0.886680i	$0.346994\pi$
$384$		0			0
$385$			1.67470i			0.0853504i
$386$		0			0
$387$	−57.3787			−2.91673
$388$		0			0
$389$			14.6748i			0.744043i	0.928224	+	0.372022i	$0.121335\pi$
$389$			14.6748i			0.744043i	−0.928224	+	0.372022i	$0.878665\pi$
$390$		0			0
$391$	−13.2960	+	30.6822i	−0.672407	+	1.55167i
$392$		0			0
$393$	−33.2793			−1.67872
$394$		0			0
$395$			4.61197i			0.232054i
$396$		0			0
$397$	−15.6837			−0.787142			−0.393571	−	0.919294i	$-0.628761\pi$
$397$	−15.6837			−0.787142			−0.393571	+	0.919294i	$0.628761\pi$
$398$		0			0
$399$		−	14.6419i		−	0.733012i
$400$		0			0
$401$			20.3189i			1.01468i	0.861747	+	0.507338i	$0.169370\pi$
$401$			20.3189i			1.01468i	−0.861747	+	0.507338i	$0.830630\pi$
$402$		0			0
$403$		−	19.2160i		−	0.957215i
$404$		0			0
$405$		−	16.9451i		−	0.842010i
$406$		0			0
$407$		−	5.73719i		−	0.284382i
$408$		0			0
$409$	−27.7275			−1.37103			−0.685517	−	0.728056i	$-0.740424\pi$
$409$	−27.7275			−1.37103			−0.685517	+	0.728056i	$0.740424\pi$
$410$		0			0
$411$	10.2428			0.505241
$412$		0			0
$413$		−	1.92401i		−	0.0946741i
$414$		0			0
$415$			11.8898i			0.583645i
$416$		0			0
$417$	52.4947			2.57068
$418$		0			0
$419$	20.0137			0.977735			0.488867	−	0.872358i	$-0.337410\pi$
$419$	20.0137			0.977735			0.488867	+	0.872358i	$0.337410\pi$
$420$		0			0
$421$		−	8.42084i		−	0.410407i	−0.978719	−	0.205203i	$-0.934214\pi$
$421$		−	8.42084i		−	0.410407i	0.978719	−	0.205203i	$-0.0657856\pi$
$422$		0			0
$423$		−	1.17325i		−	0.0570452i
$424$		0			0
$425$		−	6.97256i		−	0.338219i
$426$		0			0
$427$			12.6146i			0.610466i
$428$		0			0
$429$			23.9858i			1.15804i
$430$		0			0
$431$	34.3111			1.65271			0.826353	−	0.563153i	$-0.190412\pi$
$431$	34.3111			1.65271			0.826353	+	0.563153i	$0.190412\pi$
$432$		0			0
$433$		−	7.67160i		−	0.368673i	−0.982863	−	0.184337i	$-0.940986\pi$
$433$		−	7.67160i		−	0.368673i	0.982863	−	0.184337i	$-0.0590137\pi$
$434$		0			0
$435$	−2.62264			−0.125746
$436$		0			0
$437$	−20.2946	−	8.79457i	−0.970824	−	0.420702i
$438$		0			0
$439$		−	14.3353i		−	0.684184i	−0.939666	−	0.342092i	$-0.888864\pi$
$439$		−	14.3353i		−	0.684184i	0.939666	−	0.342092i	$-0.111136\pi$
$440$		0			0
$441$	40.6726			1.93679
$442$		0			0
$443$		−	9.13800i		−	0.434160i	−0.976154	−	0.217080i	$-0.930347\pi$
$443$		−	9.13800i		−	0.434160i	0.976154	−	0.217080i	$-0.0696532\pi$
$444$		0			0
$445$	−8.11975			−0.384913
$446$		0			0
$447$	4.15056			0.196315
$448$		0			0
$449$	4.37708			0.206567			0.103284	−	0.994652i	$-0.467065\pi$
$449$	4.37708			0.206567			0.103284	+	0.994652i	$0.467065\pi$
$450$		0			0
$451$	−12.4115			−0.584433
$452$		0			0
$453$	−20.5803			−0.966945
$454$		0			0
$455$		−	4.69831i		−	0.220260i
$456$		0			0
$457$		−	0.267071i		−	0.0124931i	−0.999980	−	0.00624653i	$-0.998012\pi$
$457$		−	0.267071i		−	0.0124931i	0.999980	−	0.00624653i	$-0.00198834\pi$
$458$		0			0
$459$	−83.2030			−3.88358
$460$		0			0
$461$	−15.7550			−0.733785			−0.366892	−	0.930263i	$-0.619578\pi$
$461$	−15.7550			−0.733785			−0.366892	+	0.930263i	$0.619578\pi$
$462$		0			0
$463$		−	5.49649i		−	0.255443i	−0.991810	−	0.127722i	$-0.959234\pi$
$463$		−	5.49649i		−	0.255443i	0.991810	−	0.127722i	$-0.0407664\pi$
$464$		0			0
$465$		−	12.9847i		−	0.602149i
$466$		0			0
$467$	17.4648			0.808173			0.404087	−	0.914721i	$-0.367589\pi$
$467$	17.4648			0.808173			0.404087	+	0.914721i	$0.367589\pi$
$468$		0			0
$469$	10.3771			0.479169
$470$		0			0
$471$	57.9335			2.66944
$472$		0			0
$473$	13.9209			0.640084
$474$		0			0
$475$	4.61197			0.211612
$476$		0			0
$477$		−	15.6474i		−	0.716444i
$478$		0			0
$479$	−30.8394			−1.40909			−0.704543	−	0.709661i	$-0.748848\pi$
$479$	−30.8394			−1.40909			−0.704543	+	0.709661i	$0.748848\pi$
$480$		0			0
$481$			16.0955i			0.733893i
$482$		0			0
$483$	−6.05395	+	13.9703i	−0.275464	+	0.635670i
$484$		0			0
$485$	14.7946			0.671787
$486$		0			0
$487$			22.0109i			0.997408i	0.866772	+	0.498704i	$0.166191\pi$
$487$			22.0109i			0.997408i	−0.866772	+	0.498704i	$0.833809\pi$
$488$		0			0
$489$	63.8606			2.88787
$490$		0			0
$491$		−	34.8027i		−	1.57062i	−0.619101	−	0.785311i	$-0.712503\pi$
$491$		−	34.8027i		−	1.57062i	0.619101	−	0.785311i	$-0.287497\pi$
$492$		0			0
$493$			5.83846i			0.262951i
$494$		0			0
$495$			11.2512i			0.505704i
$496$		0			0
$497$		−	0.852807i		−	0.0382536i
$498$		0			0
$499$			39.8761i			1.78510i	0.450951	+	0.892549i	$0.351085\pi$
$499$			39.8761i			1.78510i	−0.450951	+	0.892549i	$0.648915\pi$
$500$		0			0
$501$	58.1055			2.59596
$502$		0			0
$503$	10.3031			0.459393			0.229697	−	0.973262i	$-0.426227\pi$
$503$	10.3031			0.459393			0.229697	+	0.973262i	$0.426227\pi$
$504$		0			0
$505$		−	2.29774i		−	0.102248i
$506$		0			0
$507$		−	26.5744i		−	1.18021i
$508$		0			0
$509$	0.515396			0.0228445			0.0114223	−	0.999935i	$-0.496364\pi$
$509$	0.515396			0.0228445			0.0114223	+	0.999935i	$0.496364\pi$
$510$		0			0
$511$	13.4913			0.596822
$512$		0			0
$513$		−	55.0343i		−	2.42983i
$514$		0			0
$515$		−	15.0650i		−	0.663843i
$516$		0			0
$517$			0.284646i			0.0125187i
$518$		0			0
$519$			26.1513i			1.14792i
$520$		0			0
$521$		−	25.9451i		−	1.13668i	−0.822795	−	0.568338i	$-0.807587\pi$
$521$		−	25.9451i		−	1.13668i	0.822795	−	0.568338i	$-0.192413\pi$
$522$		0			0
$523$	42.3008			1.84968			0.924842	−	0.380353i	$-0.124197\pi$
$523$	42.3008			1.84968			0.924842	+	0.380353i	$0.124197\pi$
$524$		0			0
$525$		−	3.17476i		−	0.138558i
$526$		0			0
$527$	−28.9062			−1.25917
$528$		0			0
$529$	15.7275	+	16.7823i	0.683803	+	0.729667i
$530$		0			0
$531$		−	12.9262i		−	0.560947i
$532$		0			0
$533$	34.8200			1.50822
$534$		0			0
$535$		−	8.55489i		−	0.369860i
$536$		0			0
$537$	−73.7892			−3.18424
$538$		0			0
$539$	−9.86775			−0.425034
$540$		0			0
$541$	7.40526			0.318377			0.159188	−	0.987248i	$-0.449112\pi$
$541$	7.40526			0.318377			0.159188	+	0.987248i	$0.449112\pi$
$542$		0			0
$543$	−5.79303			−0.248603
$544$		0			0
$545$	−5.50006			−0.235597
$546$		0			0
$547$		−	27.6617i		−	1.18273i	−0.806404	−	0.591365i	$-0.798589\pi$
$547$		−	27.6617i		−	1.18273i	0.806404	−	0.591365i	$-0.201411\pi$
$548$		0			0
$549$			84.7497i			3.61703i
$550$		0			0
$551$	−3.86183			−0.164519
$552$		0			0
$553$	4.67482			0.198794
$554$		0			0
$555$			10.8761i			0.461666i
$556$		0			0
$557$		−	35.0923i		−	1.48691i	−0.668787	−	0.743454i	$-0.733186\pi$
$557$		−	35.0923i		−	1.48691i	0.668787	−	0.743454i	$-0.266814\pi$
$558$		0			0
$559$	−39.0547			−1.65184
$560$		0			0
$561$	36.0813			1.52335
$562$		0			0
$563$	−8.44855			−0.356064			−0.178032	−	0.984025i	$-0.556973\pi$
$563$	−8.44855			−0.356064			−0.178032	+	0.984025i	$0.556973\pi$
$564$		0			0
$565$	9.82201			0.413215
$566$		0			0
$567$	−17.1760			−0.721326
$568$		0			0
$569$			10.0000i			0.419222i	0.977785	+	0.209611i	$0.0672197\pi$
$569$			10.0000i			0.419222i	−0.977785	+	0.209611i	$0.932780\pi$
$570$		0			0
$571$	19.2234			0.804473			0.402237	−	0.915536i	$-0.368233\pi$
$571$	19.2234			0.804473			0.402237	+	0.915536i	$0.368233\pi$
$572$		0			0
$573$			27.6198i			1.15383i
$574$		0			0
$575$	−4.40042	−	1.90690i	−0.183510	−	0.0795232i
$576$		0			0
$577$	33.4701			1.39338			0.696690	−	0.717372i	$-0.254655\pi$
$577$	33.4701			1.39338			0.696690	+	0.717372i	$0.254655\pi$
$578$		0			0
$579$		−	10.5390i		−	0.437985i
$580$		0			0
$581$	12.0518			0.499992
$582$		0			0
$583$			3.79628i			0.157226i
$584$		0			0
$585$		−	31.5649i		−	1.30505i
$586$		0			0
$587$		−	3.88222i		−	0.160237i	−0.996785	−	0.0801183i	$-0.974470\pi$
$587$		−	3.88222i		−	0.160237i	0.996785	−	0.0801183i	$-0.0255298\pi$
$588$		0			0
$589$		−	19.1199i		−	0.787821i
$590$		0			0
$591$		−	72.6641i		−	2.98900i
$592$		0			0
$593$	−34.2704			−1.40732			−0.703659	−	0.710538i	$-0.748452\pi$
$593$	−34.2704			−1.40732			−0.703659	+	0.710538i	$0.748452\pi$
$594$		0			0
$595$	−7.06758			−0.289742
$596$		0			0
$597$		−	66.9550i		−	2.74029i
$598$		0			0
$599$			11.2512i			0.459712i	0.973225	+	0.229856i	$0.0738255\pi$
$599$			11.2512i			0.459712i	−0.973225	+	0.229856i	$0.926175\pi$
$600$		0			0
$601$	−8.51204			−0.347214			−0.173607	−	0.984815i	$-0.555542\pi$
$601$	−8.51204			−0.347214			−0.173607	+	0.984815i	$0.555542\pi$
$602$		0			0
$603$	69.7169			2.83909
$604$		0			0
$605$			8.27030i			0.336235i
$606$		0			0
$607$			22.1143i			0.897594i	0.893634	+	0.448797i	$0.148147\pi$
$607$			22.1143i			0.897594i	−0.893634	+	0.448797i	$0.851853\pi$
$608$		0			0
$609$			2.65838i			0.107723i
$610$		0			0
$611$		−	0.798567i		−	0.0323066i
$612$		0			0
$613$			32.7801i			1.32398i	0.749515	+	0.661988i	$0.230287\pi$
$613$			32.7801i			1.32398i	−0.749515	+	0.661988i	$0.769713\pi$
$614$		0			0
$615$	23.5287			0.948769
$616$		0			0
$617$		−	8.44196i		−	0.339860i	−0.985456	−	0.169930i	$-0.945646\pi$
$617$		−	8.44196i		−	0.339860i	0.985456	−	0.169930i	$-0.0543542\pi$
$618$		0			0
$619$	−40.2663			−1.61844			−0.809219	−	0.587507i	$-0.800110\pi$
$619$	−40.2663			−1.61844			−0.809219	+	0.587507i	$0.800110\pi$
$620$		0			0
$621$	−22.7549	+	52.5099i	−0.913123	+	2.10715i
$622$		0			0
$623$			8.23040i			0.329744i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			23.8658i			0.953110i
$628$		0			0
$629$	24.2122			0.965403
$630$		0			0
$631$	−30.2566			−1.20449			−0.602247	−	0.798310i	$-0.705728\pi$
$631$	−30.2566			−1.20449			−0.602247	+	0.798310i	$0.705728\pi$
$632$		0			0
$633$	18.3737			0.730290
$634$		0			0
$635$	−16.5929			−0.658470
$636$		0			0
$637$	27.6837			1.09687
$638$		0			0
$639$		−	5.72946i		−	0.226654i
$640$		0			0
$641$		−	13.3352i		−	0.526708i	−0.964699	−	0.263354i	$-0.915171\pi$
$641$		−	13.3352i		−	0.526708i	0.964699	−	0.263354i	$-0.0848286\pi$
$642$		0			0
$643$	7.48096			0.295020			0.147510	−	0.989061i	$-0.452874\pi$
$643$	7.48096			0.295020			0.147510	+	0.989061i	$0.452874\pi$
$644$		0			0
$645$	−26.3902			−1.03911
$646$		0			0
$647$		−	35.3854i		−	1.39114i	−0.718457	−	0.695572i	$-0.755151\pi$
$647$		−	35.3854i		−	1.39114i	0.718457	−	0.695572i	$-0.244849\pi$
$648$		0			0
$649$			3.13607i			0.123102i
$650$		0			0
$651$	−13.1616			−0.515844
$652$		0			0
$653$	19.6353			0.768388			0.384194	−	0.923252i	$-0.374479\pi$
$653$	19.6353			0.768388			0.384194	+	0.923252i	$0.374479\pi$
$654$		0			0
$655$	−10.6253			−0.415165
$656$		0			0
$657$	90.6397			3.53619
$658$		0			0
$659$	18.8885			0.735792			0.367896	−	0.929867i	$-0.380078\pi$
$659$	18.8885			0.735792			0.367896	+	0.929867i	$0.380078\pi$
$660$		0			0
$661$		−	50.0504i		−	1.94673i	−0.229251	−	0.973367i	$-0.573628\pi$
$661$		−	50.0504i		−	1.94673i	0.229251	−	0.973367i	$-0.426372\pi$
$662$		0			0
$663$	−101.225			−3.93126
$664$		0			0
$665$		−	4.67482i		−	0.181282i
$666$		0			0
$667$	3.68469	+	1.59674i	0.142672	+	0.0618260i
$668$		0			0
$669$	−46.6135			−1.80218
$670$		0			0
$671$		−	20.5615i		−	0.793767i
$672$		0			0
$673$	22.2243			0.856684			0.428342	−	0.903617i	$-0.359098\pi$
$673$	22.2243			0.856684			0.428342	+	0.903617i	$0.359098\pi$
$674$		0			0
$675$		−	11.9329i		−	0.459298i
$676$		0			0
$677$		−	46.1880i		−	1.77515i	−0.460665	−	0.887574i	$-0.652389\pi$
$677$		−	46.1880i		−	1.77515i	0.460665	−	0.887574i	$-0.347611\pi$
$678$		0			0
$679$		−	14.9962i		−	0.575501i
$680$		0			0
$681$			61.9002i			2.37202i
$682$		0			0
$683$		−	31.4172i		−	1.20215i	−0.799194	−	0.601073i	$-0.794740\pi$
$683$		−	31.4172i		−	1.20215i	0.799194	−	0.601073i	$-0.205260\pi$
$684$		0			0
$685$	3.27030			0.124952
$686$		0			0
$687$	36.6934			1.39994
$688$		0			0
$689$		−	10.6504i		−	0.405746i
$690$		0			0
$691$			26.0502i			0.990995i	0.868609	+	0.495497i	$0.165014\pi$
$691$			26.0502i			0.990995i	−0.868609	+	0.495497i	$0.834986\pi$
$692$		0			0
$693$	11.4045			0.433222
$694$		0			0
$695$	16.7604			0.635756
$696$		0			0
$697$		−	52.3791i		−	1.98400i
$698$		0			0
$699$			36.5184i			1.38125i
$700$		0			0
$701$		−	14.0100i		−	0.529150i	−0.964365	−	0.264575i	$-0.914768\pi$
$701$		−	14.0100i		−	0.529150i	0.964365	−	0.264575i	$-0.0852317\pi$
$702$		0			0
$703$			16.0151i			0.604019i
$704$		0			0
$705$		−	0.539610i		−	0.0203229i
$706$		0			0
$707$	−2.32905			−0.0875929
$708$		0			0
$709$		−	9.79445i		−	0.367838i	−0.982941	−	0.183919i	$-0.941122\pi$
$709$		−	9.79445i		−	0.367838i	0.982941	−	0.183919i	$-0.0588784\pi$
$710$		0			0
$711$	31.4071			1.17786
$712$		0			0
$713$	−7.90545	+	18.2429i	−0.296061	+	0.683200i
$714$		0			0
$715$			7.65811i			0.286397i
$716$		0			0
$717$	−68.1145			−2.54378
$718$		0			0
$719$			14.1779i			0.528748i	0.964420	+	0.264374i	$0.0851652\pi$
$719$			14.1779i			0.528748i	−0.964420	+	0.264374i	$0.914835\pi$
$720$		0			0
$721$	−15.2703			−0.568696
$722$		0			0
$723$	30.0542			1.11773
$724$		0			0
$725$	−0.837348			−0.0310983
$726$		0			0
$727$	−33.3838			−1.23814			−0.619068	−	0.785337i	$-0.712489\pi$
$727$	−33.3838			−1.23814			−0.619068	+	0.785337i	$0.712489\pi$
$728$		0			0
$729$	−3.27017			−0.121117
$730$		0			0
$731$			58.7492i			2.17292i
$732$		0			0
$733$			36.2671i			1.33955i	0.742562	+	0.669777i	$0.233610\pi$
$733$			36.2671i			1.33955i	−0.742562	+	0.669777i	$0.766390\pi$
$734$		0			0
$735$	18.7065			0.690000
$736$		0			0
$737$	−16.9143			−0.623047
$738$		0			0
$739$			29.9020i			1.09996i	0.835177	+	0.549981i	$0.185365\pi$
$739$			29.9020i			1.09996i	−0.835177	+	0.549981i	$0.814635\pi$
$740$		0			0
$741$		−	66.9550i		−	2.45965i
$742$		0			0
$743$	−32.0736			−1.17667			−0.588333	−	0.808619i	$-0.700216\pi$
$743$	−32.0736			−1.17667			−0.588333	+	0.808619i	$0.700216\pi$
$744$		0			0
$745$	1.32518			0.0485508
$746$		0			0
$747$	80.9681			2.96247
$748$		0			0
$749$	−8.67147			−0.316849
$750$		0			0
$751$	31.6348			1.15437			0.577186	−	0.816613i	$-0.304151\pi$
$751$	31.6348			1.15437			0.577186	+	0.816613i	$0.304151\pi$
$752$		0			0
$753$			6.92078i			0.252207i
$754$		0			0
$755$	−6.57080			−0.239136
$756$		0			0
$757$		−	42.6024i		−	1.54841i	−0.632935	−	0.774205i	$-0.718150\pi$
$757$		−	42.6024i		−	1.54841i	0.632935	−	0.774205i	$-0.281850\pi$
$758$		0			0
$759$	9.86775	−	22.7711i	0.358177	−	0.826540i
$760$		0			0
$761$	0.622057			0.0225496			0.0112748	−	0.999936i	$-0.496411\pi$
$761$	0.622057			0.0225496			0.0112748	+	0.999936i	$0.496411\pi$
$762$		0			0
$763$			5.57501i			0.201829i
$764$		0			0
$765$	−47.4825			−1.71673
$766$		0			0
$767$		−	8.79816i		−	0.317683i
$768$		0			0
$769$		−	17.6263i		−	0.635619i	−0.948155	−	0.317810i	$-0.897053\pi$
$769$		−	17.6263i		−	0.635619i	0.948155	−	0.317810i	$-0.102947\pi$
$770$		0			0
$771$			48.7982i			1.75742i
$772$		0			0
$773$			39.9551i			1.43709i	0.695483	+	0.718543i	$0.255191\pi$
$773$			39.9551i			1.43709i	−0.695483	+	0.718543i	$0.744809\pi$
$774$		0			0
$775$		−	4.14570i		−	0.148918i
$776$		0			0
$777$	11.0243			0.395496
$778$		0			0
$779$	34.6459			1.24132
$780$		0			0
$781$			1.39005i			0.0497399i
$782$		0			0
$783$			9.99201i			0.357085i
$784$		0			0
$785$	18.4968			0.660180
$786$		0			0
$787$	11.4842			0.409367			0.204683	−	0.978828i	$-0.434384\pi$
$787$	11.4842			0.409367			0.204683	+	0.978828i	$0.434384\pi$
$788$		0			0
$789$			92.2740i			3.28504i
$790$		0			0
$791$		−	9.95586i		−	0.353990i
$792$		0			0
$793$			57.6847i			2.04844i
$794$		0			0
$795$		−	7.19669i		−	0.255240i
$796$		0			0
$797$			45.2769i			1.60379i	0.597464	+	0.801896i	$0.296175\pi$
$797$			45.2769i			1.60379i	−0.597464	+	0.801896i	$0.703825\pi$
$798$		0			0
$799$	−1.20127			−0.0424978
$800$		0			0
$801$			55.2948i			1.95374i
$802$		0			0
$803$	−21.9905			−0.776027
$804$		0			0
$805$	−1.93289	+	4.46039i	−0.0681253	+	0.157208i
$806$		0			0
$807$			96.2412i			3.38785i
$808$		0			0
$809$	47.4825			1.66940			0.834698	−	0.550708i	$-0.185642\pi$
$809$	47.4825			1.66940			0.834698	+	0.550708i	$0.185642\pi$
$810$		0			0
$811$		−	4.76872i		−	0.167452i	−0.996489	−	0.0837261i	$-0.973318\pi$
$811$		−	4.76872i		−	0.167452i	0.996489	−	0.0837261i	$-0.0266821\pi$
$812$		0			0
$813$	21.8658			0.766866
$814$		0			0
$815$	20.3892			0.714203
$816$		0			0
$817$	−38.8594			−1.35952
$818$		0			0
$819$	−31.9951			−1.11800
$820$		0			0
$821$	−20.2946			−0.708288			−0.354144	−	0.935191i	$-0.615228\pi$
$821$	−20.2946			−0.708288			−0.354144	+	0.935191i	$0.615228\pi$
$822$		0			0
$823$		−	25.9993i		−	0.906277i	−0.891440	−	0.453139i	$-0.850304\pi$
$823$		−	25.9993i		−	0.906277i	0.891440	−	0.453139i	$-0.149696\pi$
$824$		0			0
$825$			5.17476i			0.180162i
$826$		0			0
$827$	−37.8789			−1.31718			−0.658589	−	0.752503i	$-0.728847\pi$
$827$	−37.8789			−1.31718			−0.658589	+	0.752503i	$0.728847\pi$
$828$		0			0
$829$	24.6166			0.854969			0.427485	−	0.904023i	$-0.359400\pi$
$829$	24.6166			0.854969			0.427485	+	0.904023i	$0.359400\pi$
$830$		0			0
$831$			78.1579i			2.71127i
$832$		0			0
$833$		−	41.6440i		−	1.44288i
$834$		0			0
$835$	18.5518			0.642010
$836$		0			0
$837$	−49.4704			−1.70995
$838$		0			0
$839$	38.4975			1.32908			0.664540	−	0.747253i	$-0.268628\pi$
$839$	38.4975			1.32908			0.664540	+	0.747253i	$0.268628\pi$
$840$		0			0
$841$	−28.2988			−0.975822
$842$		0			0
$843$	37.7881			1.30149
$844$		0			0
$845$		−	8.48460i		−	0.291879i
$846$		0			0
$847$	8.38300			0.288043
$848$		0			0
$849$			3.84156i			0.131842i
$850$		0			0
$851$	6.62171	−	15.2805i	0.226989	−	0.523807i
$852$		0			0
$853$	−8.65036			−0.296183			−0.148091	−	0.988974i	$-0.547313\pi$
$853$	−8.65036			−0.296183			−0.148091	+	0.988974i	$0.547313\pi$
$854$		0			0
$855$		−	31.4071i		−	1.07410i
$856$		0			0
$857$	−51.7892			−1.76909			−0.884543	−	0.466458i	$-0.845530\pi$
$857$	−51.7892			−1.76909			−0.884543	+	0.466458i	$0.845530\pi$
$858$		0			0
$859$		−	7.60712i		−	0.259552i	−0.991543	−	0.129776i	$-0.958574\pi$
$859$		−	7.60712i		−	0.259552i	0.991543	−	0.129776i	$-0.0414258\pi$
$860$		0			0
$861$		−	23.8493i		−	0.812783i
$862$		0			0
$863$			25.9896i			0.884695i	0.896844	+	0.442347i	$0.145854\pi$
$863$			25.9896i			0.884695i	−0.896844	+	0.442347i	$0.854146\pi$
$864$		0			0
$865$			8.34952i			0.283892i
$866$		0			0
$867$			99.0256i			3.36309i
$868$		0			0
$869$	−7.61982			−0.258485
$870$		0			0
$871$	47.4527			1.60787
$872$		0			0
$873$		−	100.750i		−	3.40986i
$874$		0			0
$875$		−	1.01363i		−	0.0342669i
$876$		0			0
$877$	32.7801			1.10691			0.553453	−	0.832881i	$-0.313310\pi$
$877$	32.7801			1.10691			0.553453	+	0.832881i	$0.313310\pi$
$878$		0			0
$879$	49.0088			1.65303
$880$		0			0
$881$			2.00000i			0.0673817i	0.999432	+	0.0336909i	$0.0107262\pi$
$881$			2.00000i			0.0673817i	−0.999432	+	0.0336909i	$0.989274\pi$
$882$		0			0
$883$			39.2501i			1.32087i	0.750883	+	0.660435i	$0.229628\pi$
$883$			39.2501i			1.32087i	−0.750883	+	0.660435i	$0.770372\pi$
$884$		0			0
$885$		−	5.94512i		−	0.199843i
$886$		0			0
$887$		−	25.8169i		−	0.866846i	−0.901191	−	0.433423i	$-0.857306\pi$
$887$		−	25.8169i		−	0.866846i	0.901191	−	0.433423i	$-0.142694\pi$
$888$		0			0
$889$			16.8190i			0.564093i
$890$		0			0
$891$	27.9964			0.937915
$892$		0			0
$893$		−	0.794574i		−	0.0265894i
$894$		0			0
$895$	−23.5592			−0.787497
$896$		0			0
$897$	−27.6837	+	63.8838i	−0.924332	+	2.13302i
$898$		0			0
$899$			3.47140i			0.115778i
$900$		0			0
$901$	−16.0211			−0.533741
$902$		0			0
$903$			26.7498i			0.890178i
$904$		0			0
$905$	−1.84958			−0.0614822
$906$		0			0
$907$	27.6919			0.919495			0.459748	−	0.888050i	$-0.347940\pi$
$907$	27.6919			0.919495			0.459748	+	0.888050i	$0.347940\pi$
$908$		0			0
$909$	−15.6474			−0.518991
$910$		0			0
$911$	−23.2220			−0.769381			−0.384690	−	0.923046i	$-0.625692\pi$
$911$	−23.2220			−0.769381			−0.384690	+	0.923046i	$0.625692\pi$
$912$		0			0
$913$	−19.6440			−0.650122
$914$		0			0
$915$			38.9789i			1.28860i
$916$		0			0
$917$			10.7701i			0.355660i
$918$		0			0
$919$	27.4084			0.904121			0.452061	−	0.891987i	$-0.350689\pi$
$919$	27.4084			0.904121			0.452061	+	0.891987i	$0.350689\pi$
$920$		0			0
$921$	25.7234			0.847614
$922$		0			0
$923$		−	3.89975i		−	0.128362i
$924$		0			0
$925$			3.47250i			0.114175i
$926$		0			0
$927$	−102.591			−3.36954
$928$		0			0
$929$	−37.4573			−1.22893			−0.614467	−	0.788943i	$-0.710629\pi$
$929$	−37.4573			−1.22893			−0.614467	+	0.788943i	$0.710629\pi$
$930$		0			0
$931$	27.5453			0.902760
$932$		0			0
$933$	−56.0903			−1.83631
$934$		0			0
$935$	11.5199			0.376742
$936$		0			0
$937$			10.5551i			0.344819i	0.985025	+	0.172410i	$0.0551552\pi$
$937$			10.5551i			0.344819i	−0.985025	+	0.172410i	$0.944845\pi$
$938$		0			0
$939$	56.3677			1.83949
$940$		0			0
$941$		−	7.72338i		−	0.251775i	−0.992045	−	0.125887i	$-0.959822\pi$
$941$		−	7.72338i		−	0.251775i	0.992045	−	0.125887i	$-0.0401778\pi$
$942$		0			0
$943$	−33.0567	−	14.3250i	−1.07648	−	0.466485i
$944$		0			0
$945$	−12.0955			−0.393468
$946$		0			0
$947$		−	50.5192i		−	1.64165i	−0.571177	−	0.820827i	$-0.693513\pi$
$947$		−	50.5192i		−	1.64165i	0.571177	−	0.820827i	$-0.306487\pi$
$948$		0			0
$949$	61.6937			2.00266
$950$		0			0
$951$			30.0542i			0.974573i
$952$		0			0
$953$			46.2091i			1.49686i	0.663214	+	0.748430i	$0.269192\pi$
$953$			46.2091i			1.49686i	−0.663214	+	0.748430i	$0.730808\pi$
$954$		0			0
$955$			8.81837i			0.285356i
$956$		0			0
$957$		−	4.33307i		−	0.140068i
$958$		0			0
$959$		−	3.31486i		−	0.107043i
$960$		0			0
$961$	13.8131			0.445585
$962$		0			0
$963$	−58.2580			−1.87734
$964$		0			0
$965$		−	3.36485i		−	0.108318i
$966$		0			0
$967$			0.481811i			0.0154940i	0.999970	+	0.00774700i	$0.00246597\pi$
$967$			0.481811i			0.0154940i	−0.999970	+	0.00774700i	$0.997534\pi$
$968$		0			0
$969$	−100.719			−3.23556
$970$		0			0
$971$	−53.4281			−1.71459			−0.857295	−	0.514826i	$-0.827856\pi$
$971$	−53.4281			−1.71459			−0.857295	+	0.514826i	$0.827856\pi$
$972$		0			0
$973$		−	16.9888i		−	0.544634i
$974$		0			0
$975$		−	14.5176i		−	0.464937i
$976$		0			0
$977$		−	16.3626i		−	0.523486i	−0.965138	−	0.261743i	$-0.915703\pi$
$977$		−	16.3626i		−	0.523486i	0.965138	−	0.261743i	$-0.0842973\pi$
$978$		0			0
$979$		−	13.4153i		−	0.428755i
$980$		0			0
$981$			37.4549i			1.19584i
$982$		0			0
$983$	−34.9704			−1.11538			−0.557691	−	0.830048i	$-0.688313\pi$
$983$	−34.9704			−1.11538			−0.557691	+	0.830048i	$0.688313\pi$
$984$		0			0
$985$		−	23.2000i		−	0.739212i
$986$		0			0
$987$	−0.546964			−0.0174100
$988$		0			0
$989$	37.0770	+	16.0671i	1.17898	+	0.510904i
$990$		0			0
$991$		−	47.2475i		−	1.50087i	−0.660946	−	0.750433i	$-0.729845\pi$
$991$		−	47.2475i		−	1.50087i	0.660946	−	0.750433i	$-0.270155\pi$
$992$		0			0
$993$	−59.4155			−1.88549
$994$		0			0
$995$		−	21.3772i		−	0.677702i
$996$		0			0
$997$	−30.4857			−0.965492			−0.482746	−	0.875760i	$-0.660361\pi$
$997$	−30.4857			−0.965492			−0.482746	+	0.875760i	$0.660361\pi$
$998$		0			0
$999$	41.4370			1.31101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.b.1471.1	✓	16
4.3	odd	2	inner	1840.2.i.b.1471.15	yes	16
23.22	odd	2	inner	1840.2.i.b.1471.2	yes	16
92.91	even	2	inner	1840.2.i.b.1471.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.b.1471.1	✓	16	1.1	even	1	trivial
1840.2.i.b.1471.2	yes	16	23.22	odd	2	inner
1840.2.i.b.1471.15	yes	16	4.3	odd	2	inner
1840.2.i.b.1471.16	yes	16	92.91	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.13208i q^{3} -1.00000i q^{5} -1.01363 q^{7} -6.80991 q^{9} +O(q^{10})\)	\(q-3.13208i q^{3} -1.00000i q^{5} -1.01363 q^{7} -6.80991 q^{9} +1.65218 q^{11} -4.63515 q^{13} -3.13208 q^{15} +6.97256i q^{17} -4.61197 q^{19} +3.17476i q^{21} +(4.40042 + 1.90690i) q^{23} -1.00000 q^{25} +11.9329i q^{27} +0.837348 q^{29} +4.14570i q^{31} -5.17476i q^{33} +1.01363i q^{35} -3.47250i q^{37} +14.5176i q^{39} -7.51217 q^{41} +8.42577 q^{43} +6.80991i q^{45} +0.172285i q^{47} -5.97256 q^{49} +21.8386 q^{51} +2.29774i q^{53} -1.65218i q^{55} +14.4451i q^{57} +1.89814i q^{59} -12.4451i q^{61} +6.90271 q^{63} +4.63515i q^{65} -10.2376 q^{67} +(5.97256 - 13.7825i) q^{69} +0.841342i q^{71} -13.3100 q^{73} +3.13208i q^{75} -1.67470 q^{77} -4.61197 q^{79} +16.9451 q^{81} -11.8898 q^{83} +6.97256 q^{85} -2.62264i q^{87} -8.11975i q^{89} +4.69831 q^{91} +12.9847 q^{93} +4.61197i q^{95} +14.7946i q^{97} -11.2512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100}) \) 16 * q - 16 * q^9 - 8 * q^13 - 16 * q^25 + 36 * q^29 - 44 * q^41 + 20 * q^49 - 20 * q^69 - 48 * q^73 - 72 * q^77 + 40 * q^81 - 4 * q^85 + 88 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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