LMFDB - Embedded newform 1840.2.m.g.1839.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.m (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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1839.5
Character	$\chi$	$=$	1840.1839
Dual form			1840.2.m.g.1839.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49894 q^{3} +(-2.19676 + 0.417445i) q^{5} -3.04922i q^{7} +3.24468 q^{9} +O(q^{10})$	$q-2.49894 q^{3} +(-2.19676 + 0.417445i) q^{5} -3.04922i q^{7} +3.24468 q^{9} -5.87962 q^{11} -6.00311i q^{13} +(5.48956 - 1.04317i) q^{15} -0.968611 q^{17} -1.72585 q^{19} +7.61980i q^{21} +(-3.79127 - 2.93705i) q^{23} +(4.65148 - 1.83405i) q^{25} -0.611451 q^{27} -3.89743 q^{29} -9.01605i q^{31} +14.6928 q^{33} +(1.27288 + 6.69839i) q^{35} +2.15559 q^{37} +15.0014i q^{39} +6.17642 q^{41} -9.08209i q^{43} +(-7.12778 + 1.35448i) q^{45} -9.73942 q^{47} -2.29773 q^{49} +2.42050 q^{51} -9.18153 q^{53} +(12.9161 - 2.45442i) q^{55} +4.31279 q^{57} +1.17122i q^{59} -0.951354i q^{61} -9.89375i q^{63} +(2.50597 + 13.1874i) q^{65} +9.83203i q^{67} +(9.47415 + 7.33950i) q^{69} -8.99828i q^{71} +8.53105i q^{73} +(-11.6238 + 4.58317i) q^{75} +17.9282i q^{77} -5.59242 q^{79} -8.20608 q^{81} +5.97096i q^{83} +(2.12780 - 0.404342i) q^{85} +9.73942 q^{87} +7.22303i q^{89} -18.3048 q^{91} +22.5305i q^{93} +(3.79127 - 0.720447i) q^{95} -7.51772 q^{97} -19.0775 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 80 q^{9}+O(q^{10}) $ 40 * q + 80 * q^9	$ 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) $ 40 * q + 80 * q^9 - 24 * q^25 + 24 * q^41 - 16 * q^49 + 80 * q^69 + 40 * q^81 - 8 * q^85

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
$3$	−2.49894			−1.44276			−0.721381	−	0.692539i	$-0.756492\pi$
$3$	−2.49894			−1.44276			−0.721381	+	0.692539i	$0.756492\pi$
$4$		0			0
$5$	−2.19676	+	0.417445i	−0.982419	+	0.186687i
$5$	−2.19676	+	0.417445i	−0.982419	+	0.186687i
$6$		0			0
$7$		−	3.04922i		−	1.15250i	−0.817275	−	0.576248i	$-0.804516\pi$
$7$		−	3.04922i		−	1.15250i	0.817275	−	0.576248i	$-0.195484\pi$
$8$		0			0
$9$	3.24468			1.08156
$10$		0			0
$11$	−5.87962			−1.77277			−0.886386	−	0.462948i	$-0.846792\pi$
$11$	−5.87962			−1.77277			−0.886386	+	0.462948i	$0.846792\pi$
$12$		0			0
$13$		−	6.00311i		−	1.66496i	−0.554052	−	0.832482i	$-0.686919\pi$
$13$		−	6.00311i		−	1.66496i	0.554052	−	0.832482i	$-0.313081\pi$
$14$		0			0
$15$	5.48956	−	1.04317i	1.41740	−	0.269345i
$16$		0			0
$17$	−0.968611			−0.234923			−0.117461	−	0.993077i	$-0.537476\pi$
$17$	−0.968611			−0.234923			−0.117461	+	0.993077i	$0.537476\pi$
$18$		0			0
$19$	−1.72585			−0.395937			−0.197969	−	0.980208i	$-0.563434\pi$
$19$	−1.72585			−0.395937			−0.197969	+	0.980208i	$0.563434\pi$
$20$		0			0
$21$			7.61980i			1.66278i
$22$		0			0
$23$	−3.79127	−	2.93705i	−0.790535	−	0.612417i
$23$	−3.79127	−	2.93705i	−0.790535	−	0.612417i
$24$		0			0
$25$	4.65148	−	1.83405i	0.930296	−	0.366810i
$26$		0			0
$27$	−0.611451			−0.117674
$28$		0			0
$29$	−3.89743			−0.723734			−0.361867	−	0.932230i	$-0.617861\pi$
$29$	−3.89743			−0.723734			−0.361867	+	0.932230i	$0.617861\pi$
$30$		0			0
$31$		−	9.01605i		−	1.61933i	−0.586892	−	0.809665i	$-0.699649\pi$
$31$		−	9.01605i		−	1.61933i	0.586892	−	0.809665i	$-0.300351\pi$
$32$		0			0
$33$	14.6928			2.55769
$34$		0			0
$35$	1.27288	+	6.69839i	0.215156	+	1.13223i
$36$		0			0
$37$	2.15559			0.354377			0.177189	−	0.984177i	$-0.443300\pi$
$37$	2.15559			0.354377			0.177189	+	0.984177i	$0.443300\pi$
$38$		0			0
$39$			15.0014i			2.40215i
$40$		0			0
$41$	6.17642			0.964595			0.482297	−	0.876008i	$-0.339802\pi$
$41$	6.17642			0.964595			0.482297	+	0.876008i	$0.339802\pi$
$42$		0			0
$43$		−	9.08209i		−	1.38501i	−0.721415	−	0.692503i	$-0.756508\pi$
$43$		−	9.08209i		−	1.38501i	0.721415	−	0.692503i	$-0.243492\pi$
$44$		0			0
$45$	−7.12778	+	1.35448i	−1.06255	+	0.201914i
$46$		0			0
$47$	−9.73942			−1.42064			−0.710320	−	0.703879i	$-0.751450\pi$
$47$	−9.73942			−1.42064			−0.710320	+	0.703879i	$0.751450\pi$
$48$		0			0
$49$	−2.29773			−0.328247
$50$		0			0
$51$	2.42050			0.338937
$52$		0			0
$53$	−9.18153			−1.26118			−0.630590	−	0.776116i	$-0.717187\pi$
$53$	−9.18153			−1.26118			−0.630590	+	0.776116i	$0.717187\pi$
$54$		0			0
$55$	12.9161	−	2.45442i	1.74161	−	0.330953i
$56$		0			0
$57$	4.31279			0.571243
$58$		0			0
$59$			1.17122i			0.152480i	0.997089	+	0.0762401i	$0.0242916\pi$
$59$			1.17122i			0.152480i	−0.997089	+	0.0762401i	$0.975708\pi$
$60$		0			0
$61$		−	0.951354i		−	0.121808i	−0.998144	−	0.0609042i	$-0.980602\pi$
$61$		−	0.951354i		−	0.121808i	0.998144	−	0.0609042i	$-0.0193984\pi$
$62$		0			0
$63$		−	9.89375i		−	1.24650i
$64$		0			0
$65$	2.50597	+	13.1874i	0.310827	+	1.63569i
$66$		0			0
$67$			9.83203i			1.20117i	0.799559	+	0.600587i	$0.205066\pi$
$67$			9.83203i			1.20117i	−0.799559	+	0.600587i	$0.794934\pi$
$68$		0			0
$69$	9.47415	+	7.33950i	1.14055	+	0.883572i
$70$		0			0
$71$		−	8.99828i		−	1.06790i	−0.845516	−	0.533949i	$-0.820707\pi$
$71$		−	8.99828i		−	1.06790i	0.845516	−	0.533949i	$-0.179293\pi$
$72$		0			0
$73$			8.53105i			0.998483i	0.866463	+	0.499242i	$0.166388\pi$
$73$			8.53105i			0.998483i	−0.866463	+	0.499242i	$0.833612\pi$
$74$		0			0
$75$	−11.6238	+	4.58317i	−1.34220	+	0.529219i
$76$		0			0
$77$			17.9282i			2.04311i
$78$		0			0
$79$	−5.59242			−0.629196			−0.314598	−	0.949225i	$-0.601870\pi$
$79$	−5.59242			−0.629196			−0.314598	+	0.949225i	$0.601870\pi$
$80$		0			0
$81$	−8.20608			−0.911786
$82$		0			0
$83$			5.97096i			0.655397i	0.944782	+	0.327699i	$0.106273\pi$
$83$			5.97096i			0.655397i	−0.944782	+	0.327699i	$0.893727\pi$
$84$		0			0
$85$	2.12780	−	0.404342i	0.230793	−	0.0438570i
$86$		0			0
$87$	9.73942			1.04418
$88$		0			0
$89$			7.22303i			0.765640i	0.923823	+	0.382820i	$0.125047\pi$
$89$			7.22303i			0.765640i	−0.923823	+	0.382820i	$0.874953\pi$
$90$		0			0
$91$	−18.3048			−1.91886
$92$		0			0
$93$			22.5305i			2.33631i
$94$		0			0
$95$	3.79127	−	0.720447i	0.388976	−	0.0739163i
$96$		0			0
$97$	−7.51772			−0.763309			−0.381654	−	0.924305i	$-0.624646\pi$
$97$	−7.51772			−0.763309			−0.381654	+	0.924305i	$0.624646\pi$
$98$		0			0
$99$	−19.0775			−1.91736
$100$		0			0
$101$	12.8797			1.28158			0.640788	−	0.767718i	$-0.278608\pi$
$101$	12.8797			1.28158			0.640788	+	0.767718i	$0.278608\pi$
$102$		0			0
$103$			16.2773i			1.60385i	0.597427	+	0.801923i	$0.296190\pi$
$103$			16.2773i			1.60385i	−0.597427	+	0.801923i	$0.703810\pi$
$104$		0			0
$105$	−3.18085	−	16.7389i	−0.310419	−	1.63354i
$106$		0			0
$107$		−	3.07686i		−	0.297452i	−0.988878	−	0.148726i	$-0.952483\pi$
$107$		−	3.07686i		−	0.297452i	0.988878	−	0.148726i	$-0.0475172\pi$
$108$		0			0
$109$		−	4.89197i		−	0.468566i	−0.972168	−	0.234283i	$-0.924726\pi$
$109$		−	4.89197i		−	0.468566i	0.972168	−	0.234283i	$-0.0752742\pi$
$110$		0			0
$111$	−5.38669			−0.511282
$112$		0			0
$113$	−1.92574			−0.181158			−0.0905791	−	0.995889i	$-0.528872\pi$
$113$	−1.92574			−0.181158			−0.0905791	+	0.995889i	$0.528872\pi$
$114$		0			0
$115$	9.55456	+	4.86933i	0.890967	+	0.454068i
$116$		0			0
$117$		−	19.4782i		−	1.80076i
$118$		0			0
$119$			2.95350i			0.270747i
$120$		0			0
$121$	23.5699			2.14272
$122$		0			0
$123$	−15.4345			−1.39168
$124$		0			0
$125$	−9.45255	+	5.97070i	−0.845462	+	0.534035i
$126$		0			0
$127$	13.5081			1.19865			0.599324	−	0.800506i	$-0.295436\pi$
$127$	13.5081			1.19865			0.599324	+	0.800506i	$0.295436\pi$
$128$		0			0
$129$			22.6956i			1.99823i
$130$		0			0
$131$		−	3.50575i		−	0.306299i	−0.988203	−	0.153149i	$-0.951058\pi$
$131$		−	3.50575i		−	0.306299i	0.988203	−	0.153149i	$-0.0489416\pi$
$132$		0			0
$133$			5.26249i			0.456316i
$134$		0			0
$135$	1.34321	−	0.255247i	0.115605	−	0.0219682i
$136$		0			0
$137$	−0.570160			−0.0487121			−0.0243560	−	0.999703i	$-0.507754\pi$
$137$	−0.570160			−0.0487121			−0.0243560	+	0.999703i	$0.507754\pi$
$138$		0			0
$139$			3.54130i			0.300369i	0.988658	+	0.150185i	$0.0479868\pi$
$139$			3.54130i			0.300369i	−0.988658	+	0.150185i	$0.952013\pi$
$140$		0			0
$141$	24.3382			2.04965
$142$		0			0
$143$			35.2960i			2.95160i
$144$		0			0
$145$	8.56170	−	1.62696i	0.711010	−	0.135112i
$146$		0			0
$147$	5.74188			0.473583
$148$		0			0
$149$		−	18.0365i		−	1.47761i	−0.673919	−	0.738805i	$-0.735390\pi$
$149$		−	18.0365i		−	1.47761i	0.673919	−	0.738805i	$-0.264610\pi$
$150$		0			0
$151$			10.5790i			0.860909i	0.902612	+	0.430454i	$0.141647\pi$
$151$			10.5790i			0.860909i	−0.902612	+	0.430454i	$0.858353\pi$
$152$		0			0
$153$	−3.14284			−0.254083
$154$		0			0
$155$	3.76371	+	19.8061i	0.302308	+	1.59086i
$156$		0			0
$157$	−16.3320			−1.30344			−0.651719	−	0.758460i	$-0.725952\pi$
$157$	−16.3320			−1.30344			−0.651719	+	0.758460i	$0.725952\pi$
$158$		0			0
$159$	22.9441			1.81958
$160$		0			0
$161$	−8.95570	+	11.5604i	−0.705808	+	0.911089i
$162$		0			0
$163$	14.1664			1.10959			0.554797	−	0.831985i	$-0.312796\pi$
$163$	14.1664			1.10959			0.554797	+	0.831985i	$0.312796\pi$
$164$		0			0
$165$	−32.2765	+	6.13343i	−2.51272	+	0.477487i
$166$		0			0
$167$	−2.54576			−0.196997			−0.0984985	−	0.995137i	$-0.531404\pi$
$167$	−2.54576			−0.196997			−0.0984985	+	0.995137i	$0.531404\pi$
$168$		0			0
$169$	−23.0374			−1.77210
$170$		0			0
$171$	−5.59984			−0.428230
$172$		0			0
$173$			4.52121i			0.343741i	0.985120	+	0.171870i	$0.0549810\pi$
$173$			4.52121i			0.343741i	−0.985120	+	0.171870i	$0.945019\pi$
$174$		0			0
$175$	−5.59242	−	14.1834i	−0.422747	−	1.07216i
$176$		0			0
$177$		−	2.92681i		−	0.219993i
$178$		0			0
$179$			20.1473i			1.50588i	0.658088	+	0.752941i	$0.271365\pi$
$179$			20.1473i			1.50588i	−0.658088	+	0.752941i	$0.728635\pi$
$180$		0			0
$181$		−	14.1244i		−	1.04986i	−0.851146	−	0.524930i	$-0.824092\pi$
$181$		−	14.1244i		−	1.04986i	0.851146	−	0.524930i	$-0.175908\pi$
$182$		0			0
$183$			2.37737i			0.175741i
$184$		0			0
$185$	−4.73532	+	0.899842i	−0.348147	+	0.0661577i
$186$		0			0
$187$	5.69506			0.416464
$188$		0			0
$189$			1.86445i			0.135619i
$190$		0			0
$191$	−7.52741			−0.544664			−0.272332	−	0.962203i	$-0.587795\pi$
$191$	−7.52741			−0.544664			−0.272332	+	0.962203i	$0.587795\pi$
$192$		0			0
$193$		−	16.0605i		−	1.15606i	−0.816016	−	0.578030i	$-0.803822\pi$
$193$		−	16.0605i		−	1.15606i	0.816016	−	0.578030i	$-0.196178\pi$
$194$		0			0
$195$	−6.26226	−	32.9544i	−0.448450	−	2.35992i
$196$		0			0
$197$		−	23.6707i		−	1.68647i	−0.537546	−	0.843234i	$-0.680649\pi$
$197$		−	23.6707i		−	1.68647i	0.537546	−	0.843234i	$-0.319351\pi$
$198$		0			0
$199$	20.5976			1.46013			0.730064	−	0.683379i	$-0.239490\pi$
$199$	20.5976			1.46013			0.730064	+	0.683379i	$0.239490\pi$
$200$		0			0
$201$		−	24.5696i		−	1.73301i
$202$		0			0
$203$			11.8841i			0.834100i
$204$		0			0
$205$	−13.5681	+	2.57832i	−0.947636	+	0.180077i
$206$		0			0
$207$	−12.3015	−	9.52979i	−0.855012	−	0.662367i
$208$		0			0
$209$	10.1473			0.701906
$210$		0			0
$211$		−	0.254864i		−	0.0175456i	−0.999962	−	0.00877279i	$-0.997207\pi$
$211$		−	0.254864i		−	0.0175456i	0.999962	−	0.00877279i	$-0.00279250\pi$
$212$		0			0
$213$			22.4861i			1.54072i
$214$		0			0
$215$	3.79127	+	19.9511i	0.258563	+	1.36066i
$216$		0			0
$217$	−27.4919			−1.86627
$218$		0			0
$219$		−	21.3185i		−	1.44057i
$220$		0			0
$221$			5.81468i			0.391138i
$222$		0			0
$223$	−3.90122			−0.261245			−0.130623	−	0.991432i	$-0.541698\pi$
$223$	−3.90122			−0.261245			−0.130623	+	0.991432i	$0.541698\pi$
$224$		0			0
$225$	15.0926	−	5.95091i	1.00617	−	0.396728i
$226$		0			0
$227$		−	17.6776i		−	1.17331i	−0.809839	−	0.586653i	$-0.800445\pi$
$227$		−	17.6776i		−	1.17331i	0.809839	−	0.586653i	$-0.199555\pi$
$228$		0			0
$229$			26.1981i			1.73122i	0.500722	+	0.865608i	$0.333068\pi$
$229$			26.1981i			1.73122i	−0.500722	+	0.865608i	$0.666932\pi$
$230$		0			0
$231$		−	44.8015i		−	2.94772i
$232$		0			0
$233$			13.2589i			0.868618i	0.900764	+	0.434309i	$0.143007\pi$
$233$			13.2589i			0.868618i	−0.900764	+	0.434309i	$0.856993\pi$
$234$		0			0
$235$	21.3951	−	4.06567i	1.39566	−	0.265215i
$236$		0			0
$237$	13.9751			0.907780
$238$		0			0
$239$		−	8.29573i		−	0.536606i	−0.963335	−	0.268303i	$-0.913537\pi$
$239$		−	8.29573i		−	0.536606i	0.963335	−	0.268303i	$-0.0864629\pi$
$240$		0			0
$241$		−	24.9949i		−	1.61006i	−0.593232	−	0.805031i	$-0.702148\pi$
$241$		−	24.9949i		−	1.61006i	0.593232	−	0.805031i	$-0.297852\pi$
$242$		0			0
$243$	22.3408			1.43316
$244$		0			0
$245$	5.04756	−	0.959176i	0.322477	−	0.0612795i
$246$		0			0
$247$			10.3605i			0.659221i
$248$		0			0
$249$		−	14.9210i		−	0.945582i
$250$		0			0
$251$	−10.5642			−0.666809			−0.333405	−	0.942784i	$-0.608197\pi$
$251$	−10.5642			−0.666809			−0.333405	+	0.942784i	$0.608197\pi$
$252$		0			0
$253$	22.2912	+	17.2687i	1.40144	+	1.08568i
$254$		0			0
$255$	−5.31724	+	1.01042i	−0.332979	+	0.0632752i
$256$		0			0
$257$		−	12.9404i		−	0.807202i	−0.914935	−	0.403601i	$-0.867758\pi$
$257$		−	12.9404i		−	0.807202i	0.914935	−	0.403601i	$-0.132242\pi$
$258$		0			0
$259$		−	6.57288i		−	0.408419i
$260$		0			0
$261$	−12.6459			−0.782763
$262$		0			0
$263$		−	22.5116i		−	1.38813i	−0.719914	−	0.694063i	$-0.755819\pi$
$263$		−	22.5116i		−	1.38813i	0.719914	−	0.694063i	$-0.244181\pi$
$264$		0			0
$265$	20.1696	−	3.83279i	1.23901	−	0.235446i
$266$		0			0
$267$		−	18.0499i		−	1.10464i
$268$		0			0
$269$	27.1013			1.65240			0.826199	−	0.563379i	$-0.190499\pi$
$269$	27.1013			1.65240			0.826199	+	0.563379i	$0.190499\pi$
$270$		0			0
$271$			18.3787i			1.11643i	0.829697	+	0.558214i	$0.188513\pi$
$271$			18.3787i			1.11643i	−0.829697	+	0.558214i	$0.811487\pi$
$272$		0			0
$273$	45.7425			2.76846
$274$		0			0
$275$	−27.3489	+	10.7835i	−1.64920	+	0.650270i
$276$		0			0
$277$			4.28386i			0.257392i	0.991684	+	0.128696i	$0.0410792\pi$
$277$			4.28386i			0.257392i	−0.991684	+	0.128696i	$0.958921\pi$
$278$		0			0
$279$		−	29.2542i		−	1.75141i
$280$		0			0
$281$		−	20.2609i		−	1.20866i	−0.796733	−	0.604332i	$-0.793440\pi$
$281$		−	20.2609i		−	1.20866i	0.796733	−	0.604332i	$-0.206560\pi$
$282$		0			0
$283$			7.40056i			0.439918i	0.975509	+	0.219959i	$0.0705923\pi$
$283$			7.40056i			0.439918i	−0.975509	+	0.219959i	$0.929408\pi$
$284$		0			0
$285$	−9.47415	+	1.80035i	−0.561200	+	0.106644i
$286$		0			0
$287$		−	18.8332i		−	1.11169i
$288$		0			0
$289$	−16.0618			−0.944811
$290$		0			0
$291$	18.7863			1.10127
$292$		0			0
$293$	25.0574			1.46387			0.731933	−	0.681376i	$-0.238618\pi$
$293$	25.0574			1.46387			0.731933	+	0.681376i	$0.238618\pi$
$294$		0			0
$295$	−0.488921	−	2.57289i	−0.0284661	−	0.149800i
$296$		0			0
$297$	3.59510			0.208609
$298$		0			0
$299$	−17.6314	+	22.7594i	−1.01965	+	1.31621i
$300$		0			0
$301$	−27.6933			−1.59621
$302$		0			0
$303$	−32.1855			−1.84901
$304$		0			0
$305$	0.397138	+	2.08989i	0.0227401	+	0.119667i
$306$		0			0
$307$	−8.46339			−0.483031			−0.241515	−	0.970397i	$-0.577644\pi$
$307$	−8.46339			−0.483031			−0.241515	+	0.970397i	$0.577644\pi$
$308$		0			0
$309$		−	40.6759i		−	2.31397i
$310$		0			0
$311$		−	21.5882i		−	1.22416i	−0.790798	−	0.612078i	$-0.790334\pi$
$311$		−	21.5882i		−	1.22416i	0.790798	−	0.612078i	$-0.209666\pi$
$312$		0			0
$313$	−12.7441			−0.720338			−0.360169	−	0.932887i	$-0.617281\pi$
$313$	−12.7441			−0.720338			−0.360169	+	0.932887i	$0.617281\pi$
$314$		0			0
$315$	4.13010	+	21.7342i	0.232705	+	1.22458i
$316$		0			0
$317$		−	20.3313i		−	1.14192i	−0.820978	−	0.570960i	$-0.806571\pi$
$317$		−	20.3313i		−	1.14192i	0.820978	−	0.570960i	$-0.193429\pi$
$318$		0			0
$319$	22.9154			1.28301
$320$		0			0
$321$			7.68889i			0.429152i
$322$		0			0
$323$	1.67168			0.0930146
$324$		0			0
$325$	−11.0100	−	27.9234i	−0.610725	−	1.54891i
$326$		0			0
$327$			12.2247i			0.676029i
$328$		0			0
$329$			29.6976i			1.63728i
$330$		0			0
$331$		−	0.989723i		−	0.0544001i	−0.999630	−	0.0272000i	$-0.991341\pi$
$331$		−	0.989723i		−	0.0544001i	0.999630	−	0.0272000i	$-0.00865911\pi$
$332$		0			0
$333$	6.99422			0.383281
$334$		0			0
$335$	−4.10433	−	21.5986i	−0.224244	−	1.18006i
$336$		0			0
$337$	25.4823			1.38811			0.694056	−	0.719921i	$-0.255822\pi$
$337$	25.4823			1.38811			0.694056	+	0.719921i	$0.255822\pi$
$338$		0			0
$339$	4.81230			0.261368
$340$		0			0
$341$			53.0109i			2.87070i
$342$		0			0
$343$		−	14.3382i		−	0.774192i
$344$		0			0
$345$	−23.8762	−	12.1682i	−1.28545	−	0.655111i
$346$		0			0
$347$	19.9535			1.07116			0.535579	−	0.844485i	$-0.320093\pi$
$347$	19.9535			1.07116			0.535579	+	0.844485i	$0.320093\pi$
$348$		0			0
$349$	−22.9641			−1.22924			−0.614620	−	0.788823i	$-0.710691\pi$
$349$	−22.9641			−1.22924			−0.614620	+	0.788823i	$0.710691\pi$
$350$		0			0
$351$			3.67061i			0.195923i
$352$		0			0
$353$			13.0078i			0.692337i	0.938172	+	0.346168i	$0.112517\pi$
$353$			13.0078i			0.692337i	−0.938172	+	0.346168i	$0.887483\pi$
$354$		0			0
$355$	3.75628	+	19.7670i	0.199363	+	1.04912i
$356$		0			0
$357$		−	7.38062i		−	0.390624i
$358$		0			0
$359$	24.2551			1.28013			0.640066	−	0.768320i	$-0.278907\pi$
$359$	24.2551			1.28013			0.640066	+	0.768320i	$0.278907\pi$
$360$		0			0
$361$	−16.0214			−0.843234
$362$		0			0
$363$	−58.8997			−3.09143
$364$		0			0
$365$	−3.56124	−	18.7406i	−0.186404	−	0.980929i
$366$		0			0
$367$			25.0126i			1.30565i	0.757511	+	0.652823i	$0.226415\pi$
$367$			25.0126i			1.30565i	−0.757511	+	0.652823i	$0.773585\pi$
$368$		0			0
$369$	20.0405			1.04327
$370$		0			0
$371$			27.9965i			1.45351i
$372$		0			0
$373$	−5.79181			−0.299889			−0.149944	−	0.988694i	$-0.547909\pi$
$373$	−5.79181			−0.299889			−0.149944	+	0.988694i	$0.547909\pi$
$374$		0			0
$375$	23.6213	−	14.9204i	1.21980	−	0.770486i
$376$		0			0
$377$			23.3967i			1.20499i
$378$		0			0
$379$	37.1025			1.90583			0.952914	−	0.303240i	$-0.0980682\pi$
$379$	37.1025			1.90583			0.952914	+	0.303240i	$0.0980682\pi$
$380$		0			0
$381$	−33.7558			−1.72936
$382$		0			0
$383$			9.75029i			0.498216i	0.968476	+	0.249108i	$0.0801375\pi$
$383$			9.75029i			0.498216i	−0.968476	+	0.249108i	$0.919863\pi$
$384$		0			0
$385$	−7.48405	−	39.3840i	−0.381423	−	2.00719i
$386$		0			0
$387$		−	29.4685i		−	1.49797i
$388$		0			0
$389$			12.3075i			0.624014i	0.950080	+	0.312007i	$0.101001\pi$
$389$			12.3075i			0.624014i	−0.950080	+	0.312007i	$0.898999\pi$
$390$		0			0
$391$	3.67227	+	2.84486i	0.185715	+	0.143871i
$392$		0			0
$393$			8.76065i			0.441916i
$394$		0			0
$395$	12.2852	−	2.33453i	0.618135	−	0.117463i
$396$		0			0
$397$			8.73767i			0.438531i	0.975665	+	0.219266i	$0.0703661\pi$
$397$			8.73767i			0.438531i	−0.975665	+	0.219266i	$0.929634\pi$
$398$		0			0
$399$		−	13.1506i		−	0.658355i
$400$		0			0
$401$			1.58414i			0.0791083i	0.999217	+	0.0395541i	$0.0125938\pi$
$401$			1.58414i			0.0791083i	−0.999217	+	0.0395541i	$0.987406\pi$
$402$		0			0
$403$	−54.1244			−2.69613
$404$		0			0
$405$	18.0268	−	3.42559i	0.895757	−	0.170219i
$406$		0			0
$407$	−12.6741			−0.628230
$408$		0			0
$409$	−20.0682			−0.992306			−0.496153	−	0.868235i	$-0.665255\pi$
$409$	−20.0682			−0.992306			−0.496153	+	0.868235i	$0.665255\pi$
$410$		0			0
$411$	1.42479			0.0702799
$412$		0			0
$413$	3.57131			0.175733
$414$		0			0
$415$	−2.49255	−	13.1167i	−0.122354	−	0.643875i
$416$		0			0
$417$		−	8.84949i		−	0.433361i
$418$		0			0
$419$	−22.4700			−1.09773			−0.548865	−	0.835911i	$-0.684940\pi$
$419$	−22.4700			−1.09773			−0.548865	+	0.835911i	$0.684940\pi$
$420$		0			0
$421$		−	10.3190i		−	0.502917i	−0.967868	−	0.251458i	$-0.919090\pi$
$421$		−	10.3190i		−	0.502917i	0.967868	−	0.251458i	$-0.0809102\pi$
$422$		0			0
$423$	−31.6013			−1.53651
$424$		0			0
$425$	−4.50547	+	1.77648i	−0.218548	+	0.0861719i
$426$		0			0
$427$	−2.90089			−0.140384
$428$		0			0
$429$		−	88.2025i		−	4.25846i
$430$		0			0
$431$	27.5506			1.32707			0.663534	−	0.748146i	$-0.269056\pi$
$431$	27.5506			1.32707			0.663534	+	0.748146i	$0.269056\pi$
$432$		0			0
$433$	9.84191			0.472972			0.236486	−	0.971635i	$-0.424004\pi$
$433$	9.84191			0.472972			0.236486	+	0.971635i	$0.424004\pi$
$434$		0			0
$435$	−21.3951	+	4.06567i	−1.02582	+	0.194934i
$436$		0			0
$437$	6.54317	+	5.06890i	0.313002	+	0.242479i
$438$		0			0
$439$		−	22.5128i		−	1.07448i	−0.843431	−	0.537238i	$-0.819468\pi$
$439$		−	22.5128i		−	1.07448i	0.843431	−	0.537238i	$-0.180532\pi$
$440$		0			0
$441$	−7.45541			−0.355020
$442$		0			0
$443$	31.0216			1.47388			0.736941	−	0.675957i	$-0.236270\pi$
$443$	31.0216			1.47388			0.736941	+	0.675957i	$0.236270\pi$
$444$		0			0
$445$	−3.01522	−	15.8672i	−0.142935	−	0.752180i
$446$		0			0
$447$			45.0721i			2.13184i
$448$		0			0
$449$	−22.2768			−1.05131			−0.525654	−	0.850698i	$-0.676179\pi$
$449$	−22.2768			−1.05131			−0.525654	+	0.850698i	$0.676179\pi$
$450$		0			0
$451$	−36.3150			−1.71001
$452$		0			0
$453$		−	26.4363i		−	1.24209i
$454$		0			0
$455$	40.2112	−	7.64125i	1.88513	−	0.358227i
$456$		0			0
$457$	−16.0377			−0.750210			−0.375105	−	0.926982i	$-0.622393\pi$
$457$	−16.0377			−0.750210			−0.375105	+	0.926982i	$0.622393\pi$
$458$		0			0
$459$	0.592258			0.0276442
$460$		0			0
$461$	12.8075			0.596507			0.298253	−	0.954487i	$-0.403596\pi$
$461$	12.8075			0.596507			0.298253	+	0.954487i	$0.403596\pi$
$462$		0			0
$463$	12.0225			0.558735			0.279368	−	0.960184i	$-0.409875\pi$
$463$	12.0225			0.558735			0.279368	+	0.960184i	$0.409875\pi$
$464$		0			0
$465$	−9.40526	−	49.4941i	−0.436159	−	2.29523i
$466$		0			0
$467$		−	0.645389i		−	0.0298650i	−0.999889	−	0.0149325i	$-0.995247\pi$
$467$		−	0.645389i		−	0.0298650i	0.999889	−	0.0149325i	$-0.00475334\pi$
$468$		0			0
$469$	29.9800			1.38435
$470$		0			0
$471$	40.8127			1.88055
$472$		0			0
$473$			53.3992i			2.45530i
$474$		0			0
$475$	−8.02776	+	3.16530i	−0.368339	+	0.145234i
$476$		0			0
$477$	−29.7912			−1.36404
$478$		0			0
$479$	−32.9225			−1.50427			−0.752134	−	0.659011i	$-0.770975\pi$
$479$	−32.9225			−1.50427			−0.752134	+	0.659011i	$0.770975\pi$
$480$		0			0
$481$		−	12.9403i		−	0.590026i
$482$		0			0
$483$	22.3797	−	28.8888i	1.01831	−	1.31448i
$484$		0			0
$485$	16.5146	−	3.13823i	0.749889	−	0.142500i
$486$		0			0
$487$	4.57008			0.207090			0.103545	−	0.994625i	$-0.466981\pi$
$487$	4.57008			0.207090			0.103545	+	0.994625i	$0.466981\pi$
$488$		0			0
$489$	−35.4008			−1.60088
$490$		0			0
$491$		−	26.5326i		−	1.19740i	−0.800974	−	0.598700i	$-0.795684\pi$
$491$		−	26.5326i		−	1.19740i	0.800974	−	0.598700i	$-0.204316\pi$
$492$		0			0
$493$	3.77509			0.170021
$494$		0			0
$495$	41.9086	−	7.96381i	1.88365	−	0.357947i
$496$		0			0
$497$	−27.4377			−1.23075
$498$		0			0
$499$		−	4.30379i		−	0.192664i	−0.995349	−	0.0963322i	$-0.969289\pi$
$499$		−	4.30379i		−	0.192664i	0.995349	−	0.0963322i	$-0.0307111\pi$
$500$		0			0
$501$	6.36170			0.284220
$502$		0			0
$503$		−	3.17002i		−	0.141344i	−0.997500	−	0.0706720i	$-0.977486\pi$
$503$		−	3.17002i		−	0.141344i	0.997500	−	0.0706720i	$-0.0225144\pi$
$504$		0			0
$505$	−28.2935	+	5.37656i	−1.25905	+	0.239254i
$506$		0			0
$507$	57.5689			2.55673
$508$		0			0
$509$	26.0671			1.15540			0.577702	−	0.816248i	$-0.303950\pi$
$509$	26.0671			1.15540			0.577702	+	0.816248i	$0.303950\pi$
$510$		0			0
$511$	26.0130			1.15075
$512$		0			0
$513$	1.05527			0.0465914
$514$		0			0
$515$	−6.79486	−	35.7572i	−0.299417	−	1.57565i
$516$		0			0
$517$	57.2641			2.51847
$518$		0			0
$519$		−	11.2982i		−	0.495936i
$520$		0			0
$521$		−	20.7148i		−	0.907532i	−0.891121	−	0.453766i	$-0.850080\pi$
$521$		−	20.7148i		−	0.907532i	0.891121	−	0.453766i	$-0.149920\pi$
$522$		0			0
$523$		−	2.82308i		−	0.123445i	−0.998093	−	0.0617223i	$-0.980341\pi$
$523$		−	2.82308i		−	0.123445i	0.998093	−	0.0617223i	$-0.0196593\pi$
$524$		0			0
$525$	13.9751	+	35.4434i	0.609923	+	1.54687i
$526$		0			0
$527$			8.73304i			0.380417i
$528$		0			0
$529$	5.74750	+	22.2703i	0.249891	+	0.968274i
$530$		0			0
$531$			3.80025i			0.164917i
$532$		0			0
$533$		−	37.0777i		−	1.60602i
$534$		0			0
$535$	1.28442	+	6.75912i	0.0555304	+	0.292222i
$536$		0			0
$537$		−	50.3469i		−	2.17263i
$538$		0			0
$539$	13.5098			0.581907
$540$		0			0
$541$	−26.8743			−1.15542			−0.577709	−	0.816243i	$-0.696053\pi$
$541$	−26.8743			−1.15542			−0.577709	+	0.816243i	$0.696053\pi$
$542$		0			0
$543$			35.2960i			1.51470i
$544$		0			0
$545$	2.04213	+	10.7465i	0.0874751	+	0.460328i
$546$		0			0
$547$	−1.62485			−0.0694737			−0.0347368	−	0.999396i	$-0.511059\pi$
$547$	−1.62485			−0.0694737			−0.0347368	+	0.999396i	$0.511059\pi$
$548$		0			0
$549$		−	3.08684i		−	0.131743i
$550$		0			0
$551$	6.72637			0.286553
$552$		0			0
$553$			17.0525i			0.725146i
$554$		0			0
$555$	11.8333	−	2.24865i	0.502294	−	0.0954498i
$556$		0			0
$557$	−11.8048			−0.500186			−0.250093	−	0.968222i	$-0.580461\pi$
$557$	−11.8048			−0.500186			−0.250093	+	0.968222i	$0.580461\pi$
$558$		0			0
$559$	−54.5208			−2.30598
$560$		0			0
$561$	−14.2316			−0.600858
$562$		0			0
$563$		−	7.58816i		−	0.319803i	−0.987133	−	0.159901i	$-0.948882\pi$
$563$		−	7.58816i		−	0.319803i	0.987133	−	0.159901i	$-0.0511176\pi$
$564$		0			0
$565$	4.23038	−	0.803889i	0.177973	−	0.0338199i
$566$		0			0
$567$			25.0221i			1.05083i
$568$		0			0
$569$		−	4.86024i		−	0.203752i	−0.994797	−	0.101876i	$-0.967516\pi$
$569$		−	4.86024i		−	0.203752i	0.994797	−	0.101876i	$-0.0324844\pi$
$570$		0			0
$571$	−5.68775			−0.238025			−0.119012	−	0.992893i	$-0.537973\pi$
$571$	−5.68775			−0.238025			−0.119012	+	0.992893i	$0.537973\pi$
$572$		0			0
$573$	18.8105			0.785821
$574$		0			0
$575$	−23.0217	−	6.70824i	−0.960072	−	0.279753i
$576$		0			0
$577$			44.1269i			1.83703i	0.395391	+	0.918513i	$0.370609\pi$
$577$			44.1269i			1.83703i	−0.395391	+	0.918513i	$0.629391\pi$
$578$		0			0
$579$			40.1341i			1.66792i
$580$		0			0
$581$	18.2067			0.755343
$582$		0			0
$583$	53.9839			2.23578
$584$		0			0
$585$	8.13108	+	42.7889i	0.336179	+	1.76910i
$586$		0			0
$587$	−11.8106			−0.487474			−0.243737	−	0.969841i	$-0.578373\pi$
$587$	−11.8106			−0.487474			−0.243737	+	0.969841i	$0.578373\pi$
$588$		0			0
$589$			15.5604i			0.641153i
$590$		0			0
$591$			59.1516i			2.43317i
$592$		0			0
$593$			19.7606i			0.811472i	0.913990	+	0.405736i	$0.132985\pi$
$593$			19.7606i			0.811472i	−0.913990	+	0.405736i	$0.867015\pi$
$594$		0			0
$595$	−1.23293	−	6.48813i	−0.0505450	−	0.265987i
$596$		0			0
$597$	−51.4722			−2.10662
$598$		0			0
$599$			22.0799i			0.902160i	0.892483	+	0.451080i	$0.148961\pi$
$599$			22.0799i			0.902160i	−0.892483	+	0.451080i	$0.851039\pi$
$600$		0			0
$601$	−21.2764			−0.867883			−0.433941	−	0.900941i	$-0.642877\pi$
$601$	−21.2764			−0.867883			−0.433941	+	0.900941i	$0.642877\pi$
$602$		0			0
$603$			31.9019i			1.29914i
$604$		0			0
$605$	−51.7773	+	9.83914i	−2.10505	+	0.400018i
$606$		0			0
$607$	5.66921			0.230106			0.115053	−	0.993359i	$-0.463296\pi$
$607$	5.66921			0.230106			0.115053	+	0.993359i	$0.463296\pi$
$608$		0			0
$609$		−	29.6976i		−	1.20341i
$610$		0			0
$611$			58.4668i			2.36532i
$612$		0			0
$613$	12.4953			0.504678			0.252339	−	0.967639i	$-0.418800\pi$
$613$	12.4953			0.504678			0.252339	+	0.967639i	$0.418800\pi$
$614$		0			0
$615$	33.9058	−	6.44305i	1.36721	−	0.259809i
$616$		0			0
$617$	14.4637			0.582286			0.291143	−	0.956680i	$-0.405965\pi$
$617$	14.4637			0.582286			0.291143	+	0.956680i	$0.405965\pi$
$618$		0			0
$619$	−16.0292			−0.644270			−0.322135	−	0.946694i	$-0.604400\pi$
$619$	−16.0292			−0.644270			−0.322135	+	0.946694i	$0.604400\pi$
$620$		0			0
$621$	2.31818	+	1.79586i	0.0930253	+	0.0720654i
$622$		0			0
$623$	22.0246			0.882397
$624$		0			0
$625$	18.2725	−	17.0621i	0.730901	−	0.682484i
$626$		0			0
$627$	−25.3576			−1.01268
$628$		0			0
$629$	−2.08793			−0.0832513
$630$		0			0
$631$	17.9112			0.713034			0.356517	−	0.934289i	$-0.383964\pi$
$631$	17.9112			0.713034			0.356517	+	0.934289i	$0.383964\pi$
$632$		0			0
$633$			0.636889i			0.0253141i
$634$		0			0
$635$	−29.6740	+	5.63888i	−1.17758	+	0.223772i
$636$		0			0
$637$			13.7935i			0.546520i
$638$		0			0
$639$		−	29.1966i		−	1.15500i
$640$		0			0
$641$			12.2324i			0.483150i	0.970382	+	0.241575i	$0.0776639\pi$
$641$			12.2324i			0.483150i	−0.970382	+	0.241575i	$0.922336\pi$
$642$		0			0
$643$			40.2616i			1.58776i	0.608072	+	0.793882i	$0.291943\pi$
$643$			40.2616i			1.58776i	−0.608072	+	0.793882i	$0.708057\pi$
$644$		0			0
$645$	−9.47415	−	49.8566i	−0.373044	−	1.96310i
$646$		0			0
$647$	0.217415			0.00854748			0.00427374	−	0.999991i	$-0.498640\pi$
$647$	0.217415			0.00854748			0.00427374	+	0.999991i	$0.498640\pi$
$648$		0			0
$649$		−	6.88634i		−	0.270313i
$650$		0			0
$651$	68.7005			2.69259
$652$		0			0
$653$			1.08204i			0.0423434i	0.999776	+	0.0211717i	$0.00673966\pi$
$653$			1.08204i			0.0423434i	−0.999776	+	0.0211717i	$0.993260\pi$
$654$		0			0
$655$	1.46346	+	7.70128i	0.0571820	+	0.300914i
$656$		0			0
$657$			27.6806i			1.07992i
$658$		0			0
$659$	10.7166			0.417460			0.208730	−	0.977973i	$-0.433067\pi$
$659$	10.7166			0.417460			0.208730	+	0.977973i	$0.433067\pi$
$660$		0			0
$661$		−	46.7678i		−	1.81905i	−0.415644	−	0.909527i	$-0.636444\pi$
$661$		−	46.7678i		−	1.81905i	0.415644	−	0.909527i	$-0.363556\pi$
$662$		0			0
$663$		−	14.5305i		−	0.564318i
$664$		0			0
$665$	−2.19680	−	11.5604i	−0.0851883	−	0.448294i
$666$		0			0
$667$	14.7762	+	11.4469i	0.572137	+	0.443227i
$668$		0			0
$669$	9.74891			0.376915
$670$		0			0
$671$			5.59360i			0.215938i
$672$		0			0
$673$		−	22.4945i		−	0.867100i	−0.901129	−	0.433550i	$-0.857261\pi$
$673$		−	22.4945i		−	0.867100i	0.901129	−	0.433550i	$-0.142739\pi$
$674$		0			0
$675$	−2.84415	+	1.12143i	−0.109471	+	0.0431639i
$676$		0			0
$677$	21.4298			0.823614			0.411807	−	0.911271i	$-0.364898\pi$
$677$	21.4298			0.823614			0.411807	+	0.911271i	$0.364898\pi$
$678$		0			0
$679$			22.9232i			0.879710i
$680$		0			0
$681$			44.1753i			1.69280i
$682$		0			0
$683$	−23.7273			−0.907899			−0.453949	−	0.891027i	$-0.649985\pi$
$683$	−23.7273			−0.907899			−0.453949	+	0.891027i	$0.649985\pi$
$684$		0			0
$685$	1.25250	−	0.238011i	0.0478557	−	0.00909392i
$686$		0			0
$687$		−	65.4673i		−	2.49773i
$688$		0			0
$689$			55.1178i			2.09982i
$690$		0			0
$691$			13.4839i			0.512952i	0.966551	+	0.256476i	$0.0825615\pi$
$691$			13.4839i			0.512952i	−0.966551	+	0.256476i	$0.917439\pi$
$692$		0			0
$693$			58.1715i			2.20975i
$694$		0			0
$695$	−1.47830	−	7.77938i	−0.0560751	−	0.295089i
$696$		0			0
$697$	−5.98255			−0.226605
$698$		0			0
$699$		−	33.1331i		−	1.25321i
$700$		0			0
$701$			5.04107i			0.190399i	0.995458	+	0.0951993i	$0.0303489\pi$
$701$			5.04107i			0.190399i	−0.995458	+	0.0951993i	$0.969651\pi$
$702$		0			0
$703$	−3.72023			−0.140311
$704$		0			0
$705$	−53.4651	+	10.1599i	−2.01361	+	0.382642i
$706$		0			0
$707$		−	39.2730i		−	1.47701i
$708$		0			0
$709$		−	20.7675i		−	0.779938i	−0.920828	−	0.389969i	$-0.872486\pi$
$709$		−	20.7675i		−	0.779938i	0.920828	−	0.389969i	$-0.127514\pi$
$710$		0			0
$711$	−18.1456			−0.680514
$712$		0			0
$713$	−26.4806	+	34.1823i	−0.991705	+	1.28014i
$714$		0			0
$715$	−14.7341	−	77.5367i	−0.551026	−	2.89971i
$716$		0			0
$717$			20.7305i			0.774195i
$718$		0			0
$719$			39.1171i			1.45882i	0.684077	+	0.729410i	$0.260205\pi$
$719$			39.1171i			1.45882i	−0.684077	+	0.729410i	$0.739795\pi$
$720$		0			0
$721$	49.6329			1.84843
$722$		0			0
$723$			62.4607i			2.32294i
$724$		0			0
$725$	−18.1288	+	7.14807i	−0.673287	+	0.265473i
$726$		0			0
$727$			17.8782i			0.663065i	0.943444	+	0.331533i	$0.107566\pi$
$727$			17.8782i			0.663065i	−0.943444	+	0.331533i	$0.892434\pi$
$728$		0			0
$729$	−31.2101			−1.15593
$730$		0			0
$731$			8.79701i			0.325369i
$732$		0			0
$733$	−12.7920			−0.472482			−0.236241	−	0.971694i	$-0.575916\pi$
$733$	−12.7920			−0.472482			−0.236241	+	0.971694i	$0.575916\pi$
$734$		0			0
$735$	−12.6135	+	2.39692i	−0.465257	+	0.0884118i
$736$		0			0
$737$		−	57.8086i		−	2.12941i
$738$		0			0
$739$		−	5.81963i		−	0.214079i	−0.994255	−	0.107039i	$-0.965863\pi$
$739$		−	5.81963i		−	0.214079i	0.994255	−	0.107039i	$-0.0341371\pi$
$740$		0			0
$741$		−	25.8902i		−	0.951099i
$742$		0			0
$743$		−	0.0511796i		−	0.00187760i	−1.00000		0.000938799i	$-0.999701\pi$
$743$		−	0.0511796i		−	0.00187760i	1.00000		0.000938799i	$-0.000298829\pi$
$744$		0			0
$745$	7.52926	+	39.6219i	0.275851	+	1.45163i
$746$		0			0
$747$			19.3739i			0.708853i
$748$		0			0
$749$	−9.38203			−0.342812
$750$		0			0
$751$	−35.4547			−1.29376			−0.646881	−	0.762591i	$-0.723927\pi$
$751$	−35.4547			−1.29376			−0.646881	+	0.762591i	$0.723927\pi$
$752$		0			0
$753$	26.3994			0.962047
$754$		0			0
$755$	−4.41616	−	23.2395i	−0.160721	−	0.845773i
$756$		0			0
$757$	47.5666			1.72884			0.864418	−	0.502773i	$-0.167687\pi$
$757$	47.5666			1.72884			0.864418	+	0.502773i	$0.167687\pi$
$758$		0			0
$759$	−55.7044	−	43.1534i	−2.02194	−	1.56637i
$760$		0			0
$761$	18.2647			0.662095			0.331047	−	0.943614i	$-0.392598\pi$
$761$	18.2647			0.662095			0.331047	+	0.943614i	$0.392598\pi$
$762$		0			0
$763$	−14.9167			−0.540020
$764$		0			0
$765$	6.90405	−	1.31196i	0.249616	−	0.0474340i
$766$		0			0
$767$	7.03098			0.253874
$768$		0			0
$769$		−	31.5027i		−	1.13602i	−0.823022	−	0.568009i	$-0.807714\pi$
$769$		−	31.5027i		−	1.13602i	0.823022	−	0.568009i	$-0.192286\pi$
$770$		0			0
$771$			32.3373i			1.16460i
$772$		0			0
$773$	16.3913			0.589554			0.294777	−	0.955566i	$-0.404755\pi$
$773$	16.3913			0.589554			0.294777	+	0.955566i	$0.404755\pi$
$774$		0			0
$775$	−16.5359	−	41.9380i	−0.593987	−	1.50646i
$776$		0			0
$777$			16.4252i			0.589251i
$778$		0			0
$779$	−10.6596			−0.381919
$780$		0			0
$781$			52.9064i			1.89314i
$782$		0			0
$783$	2.38309			0.0851645
$784$		0			0
$785$	35.8775	−	6.81773i	1.28052	−	0.243335i
$786$		0			0
$787$			38.6427i			1.37746i	0.725017	+	0.688731i	$0.241832\pi$
$787$			38.6427i			1.37746i	−0.725017	+	0.688731i	$0.758168\pi$
$788$		0			0
$789$			56.2551i			2.00273i
$790$		0			0
$791$			5.87199i			0.208784i
$792$		0			0
$793$	−5.71109			−0.202807
$794$		0			0
$795$	−50.4025	+	9.57789i	−1.78759	+	0.339693i
$796$		0			0
$797$	−47.5892			−1.68569			−0.842847	−	0.538153i	$-0.819122\pi$
$797$	−47.5892			−1.68569			−0.842847	+	0.538153i	$0.819122\pi$
$798$		0			0
$799$	9.43371			0.333741
$800$		0			0
$801$			23.4365i			0.828087i
$802$		0			0
$803$		−	50.1593i		−	1.77008i
$804$		0			0
$805$	14.8477	−	29.1339i	0.523311	−	1.02684i
$806$		0			0
$807$	−67.7245			−2.38402
$808$		0			0
$809$	5.89661			0.207314			0.103657	−	0.994613i	$-0.466946\pi$
$809$	5.89661			0.207314			0.103657	+	0.994613i	$0.466946\pi$
$810$		0			0
$811$		−	43.5295i		−	1.52853i	−0.644904	−	0.764264i	$-0.723103\pi$
$811$		−	43.5295i		−	1.52853i	0.644904	−	0.764264i	$-0.276897\pi$
$812$		0			0
$813$		−	45.9272i		−	1.61074i
$814$		0			0
$815$	−31.1200	+	5.91368i	−1.09009	+	0.207147i
$816$		0			0
$817$			15.6743i			0.548375i
$818$		0			0
$819$	−59.3933			−2.07537
$820$		0			0
$821$	−4.03114			−0.140688			−0.0703439	−	0.997523i	$-0.522410\pi$
$821$	−4.03114			−0.140688			−0.0703439	+	0.997523i	$0.522410\pi$
$822$		0			0
$823$	−17.1442			−0.597609			−0.298804	−	0.954314i	$-0.596588\pi$
$823$	−17.1442			−0.597609			−0.298804	+	0.954314i	$0.596588\pi$
$824$		0			0
$825$	68.3432	−	26.9473i	2.37941	−	0.938185i
$826$		0			0
$827$		−	19.2312i		−	0.668735i	−0.942443	−	0.334368i	$-0.891477\pi$
$827$		−	19.2312i		−	0.668735i	0.942443	−	0.334368i	$-0.108523\pi$
$828$		0			0
$829$	−2.83105			−0.0983263			−0.0491632	−	0.998791i	$-0.515655\pi$
$829$	−2.83105			−0.0983263			−0.0491632	+	0.998791i	$0.515655\pi$
$830$		0			0
$831$		−	10.7051i		−	0.371356i
$832$		0			0
$833$	2.22561			0.0771127
$834$		0			0
$835$	5.59242	−	1.06272i	0.193534	−	0.0367768i
$836$		0			0
$837$			5.51287i			0.190553i
$838$		0			0
$839$	52.6216			1.81670			0.908349	−	0.418212i	$-0.137343\pi$
$839$	52.6216			1.81670			0.908349	+	0.418212i	$0.137343\pi$
$840$		0			0
$841$	−13.8101			−0.476209
$842$		0			0
$843$			50.6307i			1.74381i
$844$		0			0
$845$	50.6075	−	9.61683i	1.74095	−	0.330829i
$846$		0			0
$847$		−	71.8698i		−	2.46947i
$848$		0			0
$849$		−	18.4935i		−	0.634696i
$850$		0			0
$851$	−8.17244	−	6.33108i	−0.280148	−	0.217027i
$852$		0			0
$853$			32.4984i			1.11272i	0.830940	+	0.556362i	$0.187803\pi$
$853$			32.4984i			1.11272i	−0.830940	+	0.556362i	$0.812197\pi$
$854$		0			0
$855$	12.3015	−	2.33762i	0.420702	−	0.0799451i
$856$		0			0
$857$		−	42.6503i		−	1.45691i	−0.685096	−	0.728453i	$-0.740240\pi$
$857$		−	42.6503i		−	1.45691i	0.685096	−	0.728453i	$-0.259760\pi$
$858$		0			0
$859$			19.4645i			0.664122i	0.943258	+	0.332061i	$0.107744\pi$
$859$			19.4645i			0.664122i	−0.943258	+	0.332061i	$0.892256\pi$
$860$		0			0
$861$			47.0631i			1.60391i
$862$		0			0
$863$	−19.6963			−0.670468			−0.335234	−	0.942135i	$-0.608815\pi$
$863$	−19.6963			−0.670468			−0.335234	+	0.942135i	$0.608815\pi$
$864$		0			0
$865$	−1.88735	−	9.93199i	−0.0641720	−	0.337698i
$866$		0			0
$867$	40.1374			1.36314
$868$		0			0
$869$	32.8813			1.11542
$870$		0			0
$871$	59.0228			1.99991
$872$		0			0
$873$	−24.3926			−0.825565
$874$		0			0
$875$	18.2060	+	28.8229i	0.615474	+	0.974392i
$876$		0			0
$877$		−	29.4682i		−	0.995070i	−0.867444	−	0.497535i	$-0.834239\pi$
$877$		−	29.4682i		−	0.995070i	0.867444	−	0.497535i	$-0.165761\pi$
$878$		0			0
$879$	−62.6168			−2.11201
$880$		0			0
$881$			47.8258i			1.61129i	0.592397	+	0.805646i	$0.298182\pi$
$881$			47.8258i			1.61129i	−0.592397	+	0.805646i	$0.701818\pi$
$882$		0			0
$883$	−6.36009			−0.214034			−0.107017	−	0.994257i	$-0.534130\pi$
$883$	−6.36009			−0.214034			−0.107017	+	0.994257i	$0.534130\pi$
$884$		0			0
$885$	1.22178	+	6.42949i	0.0410698	+	0.216125i
$886$		0			0
$887$	−58.2351			−1.95534			−0.977672	−	0.210138i	$-0.932609\pi$
$887$	−58.2351			−1.95534			−0.977672	+	0.210138i	$0.932609\pi$
$888$		0			0
$889$		−	41.1891i		−	1.38144i
$890$		0			0
$891$	48.2486			1.61639
$892$		0			0
$893$	16.8088			0.562484
$894$		0			0
$895$	−8.41040	−	44.2588i	−0.281129	−	1.47941i
$896$		0			0
$897$	44.0598	−	56.8744i	1.47111	−	1.89898i
$898$		0			0
$899$			35.1394i			1.17196i
$900$		0			0
$901$	8.89333			0.296280
$902$		0			0
$903$	69.2037			2.30296
$904$		0			0
$905$	5.89616	+	31.0279i	0.195995	+	1.03140i
$906$		0			0
$907$		−	29.7495i		−	0.987816i	−0.869514	−	0.493908i	$-0.835568\pi$
$907$		−	29.7495i		−	0.987816i	0.869514	−	0.493908i	$-0.164432\pi$
$908$		0			0
$909$	41.7905			1.38610
$910$		0			0
$911$	−9.21507			−0.305309			−0.152654	−	0.988280i	$-0.548782\pi$
$911$	−9.21507			−0.305309			−0.152654	+	0.988280i	$0.548782\pi$
$912$		0			0
$913$		−	35.1069i		−	1.16187i
$914$		0			0
$915$	−0.992423	−	5.22251i	−0.0328085	−	0.172651i
$916$		0			0
$917$	−10.6898			−0.353008
$918$		0			0
$919$	19.6543			0.648337			0.324168	−	0.945999i	$-0.394916\pi$
$919$	19.6543			0.648337			0.324168	+	0.945999i	$0.394916\pi$
$920$		0			0
$921$	21.1495			0.696899
$922$		0			0
$923$	−54.0177			−1.77801
$924$		0			0
$925$	10.0267	−	3.95347i	0.329676	−	0.129989i
$926$		0			0
$927$			52.8146i			1.73466i
$928$		0			0
$929$	4.30520			0.141249			0.0706245	−	0.997503i	$-0.477501\pi$
$929$	4.30520			0.141249			0.0706245	+	0.997503i	$0.477501\pi$
$930$		0			0
$931$	3.96554			0.129965
$932$		0			0
$933$			53.9476i			1.76616i
$934$		0			0
$935$	−12.5107	+	2.37737i	−0.409142	+	0.0777484i
$936$		0			0
$937$	29.4172			0.961018			0.480509	−	0.876990i	$-0.340452\pi$
$937$	29.4172			0.961018			0.480509	+	0.876990i	$0.340452\pi$
$938$		0			0
$939$	31.8467			1.03928
$940$		0			0
$941$			56.6435i			1.84653i	0.384169	+	0.923263i	$0.374488\pi$
$941$			56.6435i			1.84653i	−0.384169	+	0.923263i	$0.625512\pi$
$942$		0			0
$943$	−23.4165	−	18.1404i	−0.762546	−	0.590734i
$944$		0			0
$945$	−0.778304	−	4.09574i	−0.0253182	−	0.133234i
$946$		0			0
$947$	43.9839			1.42928			0.714642	−	0.699490i	$-0.246590\pi$
$947$	43.9839			1.42928			0.714642	+	0.699490i	$0.246590\pi$
$948$		0			0
$949$	51.2128			1.66244
$950$		0			0
$951$			50.8067i			1.64752i
$952$		0			0
$953$	−55.1773			−1.78737			−0.893685	−	0.448696i	$-0.851889\pi$
$953$	−55.1773			−1.78737			−0.893685	+	0.448696i	$0.851889\pi$
$954$		0			0
$955$	16.5359	−	3.14228i	0.535089	−	0.101682i
$956$		0			0
$957$	−57.2641			−1.85108
$958$		0			0
$959$			1.73854i			0.0561405i
$960$		0			0
$961$	−50.2892			−1.62223
$962$		0			0
$963$		−	9.98346i		−	0.321712i
$964$		0			0
$965$	6.70437	+	35.2810i	0.215821	+	1.13574i
$966$		0			0
$967$	−55.7346			−1.79230			−0.896151	−	0.443749i	$-0.853648\pi$
$967$	−55.7346			−1.79230			−0.896151	+	0.443749i	$0.853648\pi$
$968$		0			0
$969$	−4.17741			−0.134198
$970$		0			0
$971$	−19.0223			−0.610454			−0.305227	−	0.952280i	$-0.598732\pi$
$971$	−19.0223			−0.610454			−0.305227	+	0.952280i	$0.598732\pi$
$972$		0			0
$973$	10.7982			0.346175
$974$		0			0
$975$	27.5133	+	69.7787i	0.881131	+	2.23471i
$976$		0			0
$977$	4.17270			0.133496			0.0667482	−	0.997770i	$-0.478738\pi$
$977$	4.17270			0.133496			0.0667482	+	0.997770i	$0.478738\pi$
$978$		0			0
$979$		−	42.4687i		−	1.35730i
$980$		0			0
$981$		−	15.8729i		−	0.506783i
$982$		0			0
$983$			35.3376i			1.12710i	0.826084	+	0.563548i	$0.190564\pi$
$983$			35.3376i			1.12710i	−0.826084	+	0.563548i	$0.809436\pi$
$984$		0			0
$985$	9.88122	+	51.9988i	0.314842	+	1.65682i
$986$		0			0
$987$		−	74.2125i		−	2.36221i
$988$		0			0
$989$	−26.6745	+	34.4327i	−0.848201	+	1.09490i
$990$		0			0
$991$			56.0708i			1.78115i	0.454839	+	0.890574i	$0.349697\pi$
$991$			56.0708i			1.78115i	−0.454839	+	0.890574i	$0.650303\pi$
$992$		0			0
$993$			2.47325i			0.0784863i
$994$		0			0
$995$	−45.2480	+	8.59838i	−1.43446	+	0.272587i
$996$		0			0
$997$			1.80102i			0.0570389i	0.999593	+	0.0285195i	$0.00907926\pi$
$997$			1.80102i			0.0570389i	−0.999593	+	0.0285195i	$0.990921\pi$
$998$		0			0
$999$	−1.31804			−0.0417009

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.m.g.1839.5	✓	40
4.3	odd	2	inner	1840.2.m.g.1839.36	yes	40
5.4	even	2	inner	1840.2.m.g.1839.35	yes	40
20.19	odd	2	inner	1840.2.m.g.1839.6	yes	40
23.22	odd	2	inner	1840.2.m.g.1839.7	yes	40
92.91	even	2	inner	1840.2.m.g.1839.34	yes	40
115.114	odd	2	inner	1840.2.m.g.1839.33	yes	40
460.459	even	2	inner	1840.2.m.g.1839.8	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.m.g.1839.5	✓	40	1.1	even	1	trivial
1840.2.m.g.1839.6	yes	40	20.19	odd	2	inner
1840.2.m.g.1839.7	yes	40	23.22	odd	2	inner
1840.2.m.g.1839.8	yes	40	460.459	even	2	inner
1840.2.m.g.1839.33	yes	40	115.114	odd	2	inner
1840.2.m.g.1839.34	yes	40	92.91	even	2	inner
1840.2.m.g.1839.35	yes	40	5.4	even	2	inner
1840.2.m.g.1839.36	yes	40	4.3	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.49894 q^{3} +(-2.19676 + 0.417445i) q^{5} -3.04922i q^{7} +3.24468 q^{9} +O(q^{10})\)	\(q-2.49894 q^{3} +(-2.19676 + 0.417445i) q^{5} -3.04922i q^{7} +3.24468 q^{9} -5.87962 q^{11} -6.00311i q^{13} +(5.48956 - 1.04317i) q^{15} -0.968611 q^{17} -1.72585 q^{19} +7.61980i q^{21} +(-3.79127 - 2.93705i) q^{23} +(4.65148 - 1.83405i) q^{25} -0.611451 q^{27} -3.89743 q^{29} -9.01605i q^{31} +14.6928 q^{33} +(1.27288 + 6.69839i) q^{35} +2.15559 q^{37} +15.0014i q^{39} +6.17642 q^{41} -9.08209i q^{43} +(-7.12778 + 1.35448i) q^{45} -9.73942 q^{47} -2.29773 q^{49} +2.42050 q^{51} -9.18153 q^{53} +(12.9161 - 2.45442i) q^{55} +4.31279 q^{57} +1.17122i q^{59} -0.951354i q^{61} -9.89375i q^{63} +(2.50597 + 13.1874i) q^{65} +9.83203i q^{67} +(9.47415 + 7.33950i) q^{69} -8.99828i q^{71} +8.53105i q^{73} +(-11.6238 + 4.58317i) q^{75} +17.9282i q^{77} -5.59242 q^{79} -8.20608 q^{81} +5.97096i q^{83} +(2.12780 - 0.404342i) q^{85} +9.73942 q^{87} +7.22303i q^{89} -18.3048 q^{91} +22.5305i q^{93} +(3.79127 - 0.720447i) q^{95} -7.51772 q^{97} -19.0775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 80 q^{9}+O(q^{10}) \) 40 * q + 80 * q^9	\( 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) \) 40 * q + 80 * q^9 - 24 * q^25 + 24 * q^41 - 16 * q^49 + 80 * q^69 + 40 * q^81 - 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more