LMFDB - Embedded newform 3800.2.d.i.3649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not 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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.i.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} -2.82843i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.41421i q^{3} -2.82843i q^{7} +1.00000 q^{9} -0.828427 q^{11} -3.41421i q^{13} +4.82843i q^{17} +1.00000 q^{19} +4.00000 q^{21} -4.00000i q^{23} +5.65685i q^{27} +0.828427 q^{29} -1.17157i q^{33} +10.2426i q^{37} +4.82843 q^{39} -0.828427 q^{41} -2.82843i q^{43} -8.48528i q^{47} -1.00000 q^{49} -6.82843 q^{51} -13.0711i q^{53} +1.41421i q^{57} -2.82843 q^{59} -1.65685 q^{61} -2.82843i q^{63} -9.41421i q^{67} +5.65685 q^{69} +15.3137 q^{71} -12.1421i q^{73} +2.34315i q^{77} +9.17157 q^{79} -5.00000 q^{81} -8.00000i q^{83} +1.17157i q^{87} +3.17157 q^{89} -9.65685 q^{91} -2.24264i q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} + 8 q^{11} + 4 q^{19} + 16 q^{21} - 8 q^{29} + 8 q^{39} + 8 q^{41} - 4 q^{49} - 16 q^{51} + 16 q^{61} + 16 q^{71} + 48 q^{79} - 20 q^{81} + 24 q^{89} - 16 q^{91} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^9 + 8 * q^11 + 4 * q^19 + 16 * q^21 - 8 * q^29 + 8 * q^39 + 8 * q^41 - 4 * q^49 - 16 * q^51 + 16 * q^61 + 16 * q^71 + 48 * q^79 - 20 * q^81 + 24 * q^89 - 16 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.41421i	0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$	1.41421i	0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	− 3.41421i	− 0.946932i	−0.880812	−	0.473466i	$-0.843003\pi$
$13$	− 3.41421i	− 0.946932i	0.880812	−	0.473466i	$-0.156997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843i	1.17107i	0.810649	+	0.585533i	$0.199115\pi$
$17$	4.82843i	1.17107i	−0.810649	+	0.585533i	$0.800885\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65685i	1.08866i
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	− 1.17157i	− 0.203945i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.2426i	1.68388i	0.539571	+	0.841940i	$0.318586\pi$
$37$	10.2426i	1.68388i	−0.539571	+	0.841940i	$0.681414\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	0	0
$43$	− 2.82843i	− 0.431331i	−0.976467	−	0.215666i	$-0.930808\pi$
$43$	− 2.82843i	− 0.431331i	0.976467	−	0.215666i	$-0.0691921\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.48528i	− 1.23771i	−0.785507	−	0.618853i	$-0.787598\pi$
$47$	− 8.48528i	− 1.23771i	0.785507	−	0.618853i	$-0.212402\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−6.82843	−0.956171
$52$	0	0
$53$	− 13.0711i	− 1.79545i	−0.440557	−	0.897725i	$-0.645219\pi$
$53$	− 13.0711i	− 1.79545i	0.440557	−	0.897725i	$-0.354781\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.41421i	0.187317i
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	−1.65685	−0.212138	−0.106069	−	0.994359i	$-0.533827\pi$
$61$	−1.65685	−0.212138	−0.106069	+	0.994359i	$0.533827\pi$
$62$	0	0
$63$	− 2.82843i	− 0.356348i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.41421i	− 1.15013i	−0.818108	−	0.575065i	$-0.804977\pi$
$67$	− 9.41421i	− 1.15013i	0.818108	−	0.575065i	$-0.195023\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	15.3137	1.81740	0.908701	−	0.417447i	$-0.137075\pi$
$71$	15.3137	1.81740	0.908701	+	0.417447i	$0.137075\pi$
$72$	0	0
$73$	− 12.1421i	− 1.42113i	−0.703632	−	0.710565i	$-0.748440\pi$
$73$	− 12.1421i	− 1.42113i	0.703632	−	0.710565i	$-0.251560\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.34315i	0.267026i
$78$	0	0
$79$	9.17157	1.03188	0.515941	−	0.856624i	$-0.327442\pi$
$79$	9.17157	1.03188	0.515941	+	0.856624i	$0.327442\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.17157i	0.125606i
$88$	0	0
$89$	3.17157	0.336186	0.168093	−	0.985771i	$-0.446239\pi$
$89$	3.17157	0.336186	0.168093	+	0.985771i	$0.446239\pi$
$90$	0	0
$91$	−9.65685	−1.01231
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.24264i	− 0.227706i	−0.993498	−	0.113853i	$-0.963681\pi$
$97$	− 2.24264i	− 0.227706i	0.993498	−	0.113853i	$-0.0363192\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	17.6569	1.75692	0.878461	−	0.477813i	$-0.158571\pi$
$101$	17.6569	1.75692	0.878461	+	0.477813i	$0.158571\pi$
$102$	0	0
$103$	− 8.24264i	− 0.812172i	−0.913835	−	0.406086i	$-0.866893\pi$
$103$	− 8.24264i	− 0.812172i	0.913835	−	0.406086i	$-0.133107\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.41421i	0.136717i	0.997661	+	0.0683586i	$0.0217762\pi$
$107$	1.41421i	0.136717i	−0.997661	+	0.0683586i	$0.978224\pi$
$108$	0	0
$109$	−3.65685	−0.350263	−0.175132	−	0.984545i	$-0.556035\pi$
$109$	−3.65685	−0.350263	−0.175132	+	0.984545i	$0.556035\pi$
$110$	0	0
$111$	−14.4853	−1.37488
$112$	0	0
$113$	14.7279i	1.38549i	0.721184	+	0.692743i	$0.243598\pi$
$113$	14.7279i	1.38549i	−0.721184	+	0.692743i	$0.756402\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.41421i	− 0.315644i
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	− 1.17157i	− 0.105637i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.4142i	− 1.19032i	−0.803608	−	0.595159i	$-0.797089\pi$
$127$	− 13.4142i	− 1.19032i	0.803608	−	0.595159i	$-0.202911\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	7.31371	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$132$	0	0
$133$	− 2.82843i	− 0.245256i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.17157i	− 0.612709i	−0.951918	−	0.306354i	$-0.900891\pi$
$137$	− 7.17157i	− 0.612709i	0.951918	−	0.306354i	$-0.0991093\pi$
$138$	0	0
$139$	3.17157	0.269009	0.134505	−	0.990913i	$-0.457056\pi$
$139$	3.17157	0.269009	0.134505	+	0.990913i	$0.457056\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	2.82843i	0.236525i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.41421i	− 0.116642i
$148$	0	0
$149$	−1.65685	−0.135735	−0.0678674	−	0.997694i	$-0.521619\pi$
$149$	−1.65685	−0.135735	−0.0678674	+	0.997694i	$0.521619\pi$
$150$	0	0
$151$	6.14214	0.499840	0.249920	−	0.968267i	$-0.419596\pi$
$151$	6.14214	0.499840	0.249920	+	0.968267i	$0.419596\pi$
$152$	0	0
$153$	4.82843i	0.390355i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.4853i	− 0.836817i	−0.908259	−	0.418408i	$-0.862588\pi$
$157$	− 10.4853i	− 0.836817i	0.908259	−	0.418408i	$-0.137412\pi$
$158$	0	0
$159$	18.4853	1.46598
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	− 14.8284i	− 1.16145i	−0.814099	−	0.580726i	$-0.802769\pi$
$163$	− 14.8284i	− 1.16145i	0.814099	−	0.580726i	$-0.197231\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.10051i	0.162542i	0.996692	+	0.0812710i	$0.0258979\pi$
$167$	2.10051i	0.162542i	−0.996692	+	0.0812710i	$0.974102\pi$
$168$	0	0
$169$	1.34315	0.103319
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	10.7279i	0.815629i	0.913065	+	0.407814i	$0.133709\pi$
$173$	10.7279i	0.815629i	−0.913065	+	0.407814i	$0.866291\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	0	0
$179$	−8.48528	−0.634220	−0.317110	−	0.948389i	$-0.602712\pi$
$179$	−8.48528	−0.634220	−0.317110	+	0.948389i	$0.602712\pi$
$180$	0	0
$181$	−5.31371	−0.394965	−0.197482	−	0.980306i	$-0.563277\pi$
$181$	−5.31371	−0.394965	−0.197482	+	0.980306i	$0.563277\pi$
$182$	0	0
$183$	− 2.34315i	− 0.173210i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	0	0
$193$	6.92893i	0.498755i	0.968406	+	0.249378i	$0.0802261\pi$
$193$	6.92893i	0.498755i	−0.968406	+	0.249378i	$0.919774\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.3137i	1.80353i	0.432230	+	0.901764i	$0.357727\pi$
$197$	25.3137i	1.80353i	−0.432230	+	0.901764i	$0.642273\pi$
$198$	0	0
$199$	5.65685	0.401004	0.200502	−	0.979693i	$-0.435743\pi$
$199$	5.65685	0.401004	0.200502	+	0.979693i	$0.435743\pi$
$200$	0	0
$201$	13.3137	0.939077
$202$	0	0
$203$	− 2.34315i	− 0.164457i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	−0.828427	−0.0573035
$210$	0	0
$211$	−16.4853	−1.13489	−0.567447	−	0.823410i	$-0.692069\pi$
$211$	−16.4853	−1.13489	−0.567447	+	0.823410i	$0.692069\pi$
$212$	0	0
$213$	21.6569i	1.48390i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	17.1716	1.16035
$220$	0	0
$221$	16.4853	1.10892
$222$	0	0
$223$	− 13.4142i	− 0.898282i	−0.893461	−	0.449141i	$-0.851730\pi$
$223$	− 13.4142i	− 0.898282i	0.893461	−	0.449141i	$-0.148270\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0416i	1.59570i	0.602857	+	0.797850i	$0.294029\pi$
$227$	24.0416i	1.59570i	−0.602857	+	0.797850i	$0.705971\pi$
$228$	0	0
$229$	−11.3137	−0.747631	−0.373815	−	0.927503i	$-0.621951\pi$
$229$	−11.3137	−0.747631	−0.373815	+	0.927503i	$0.621951\pi$
$230$	0	0
$231$	−3.31371	−0.218026
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.9706i	0.842529i
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	8.14214	0.524481	0.262241	−	0.965003i	$-0.415539\pi$
$241$	8.14214	0.524481	0.262241	+	0.965003i	$0.415539\pi$
$242$	0	0
$243$	9.89949i	0.635053i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.41421i	− 0.217241i
$248$	0	0
$249$	11.3137	0.716977
$250$	0	0
$251$	28.9706	1.82861	0.914303	−	0.405031i	$-0.132739\pi$
$251$	28.9706	1.82861	0.914303	+	0.405031i	$0.132739\pi$
$252$	0	0
$253$	3.31371i	0.208331i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.10051i	− 0.255782i	−0.991788	−	0.127891i	$-0.959179\pi$
$257$	− 4.10051i	− 0.255782i	0.991788	−	0.127891i	$-0.0408208\pi$
$258$	0	0
$259$	28.9706	1.80014
$260$	0	0
$261$	0.828427	0.0512784
$262$	0	0
$263$	− 27.3137i	− 1.68424i	−0.539294	−	0.842118i	$-0.681309\pi$
$263$	− 27.3137i	− 1.68424i	0.539294	−	0.842118i	$-0.318691\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.48528i	0.274495i
$268$	0	0
$269$	29.3137	1.78729	0.893644	−	0.448776i	$-0.148140\pi$
$269$	29.3137	1.78729	0.893644	+	0.448776i	$0.148140\pi$
$270$	0	0
$271$	−9.51472	−0.577978	−0.288989	−	0.957332i	$-0.593319\pi$
$271$	−9.51472	−0.577978	−0.288989	+	0.957332i	$0.593319\pi$
$272$	0	0
$273$	− 13.6569i	− 0.826550i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.1716i	1.15191i	0.817482	+	0.575954i	$0.195369\pi$
$277$	19.1716i	1.15191i	−0.817482	+	0.575954i	$0.804631\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.1716	0.666440	0.333220	−	0.942849i	$-0.391865\pi$
$281$	11.1716	0.666440	0.333220	+	0.942849i	$0.391865\pi$
$282$	0	0
$283$	− 6.34315i	− 0.377061i	−0.982067	−	0.188530i	$-0.939628\pi$
$283$	− 6.34315i	− 0.377061i	0.982067	−	0.188530i	$-0.0603724\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.34315i	0.138312i
$288$	0	0
$289$	−6.31371	−0.371395
$290$	0	0
$291$	3.17157	0.185921
$292$	0	0
$293$	− 32.3848i	− 1.89194i	−0.324256	−	0.945969i	$-0.605114\pi$
$293$	− 32.3848i	− 1.89194i	0.324256	−	0.945969i	$-0.394886\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.68629i	− 0.271926i
$298$	0	0
$299$	−13.6569	−0.789796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	24.9706i	1.43452i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.41421i	0.0807134i	0.999185	+	0.0403567i	$0.0128494\pi$
$307$	1.41421i	0.0807134i	−0.999185	+	0.0403567i	$0.987151\pi$
$308$	0	0
$309$	11.6569	0.663135
$310$	0	0
$311$	19.1716	1.08712	0.543560	−	0.839370i	$-0.317076\pi$
$311$	19.1716	1.08712	0.543560	+	0.839370i	$0.317076\pi$
$312$	0	0
$313$	5.31371i	0.300349i	0.988660	+	0.150174i	$0.0479835\pi$
$313$	5.31371i	0.300349i	−0.988660	+	0.150174i	$0.952017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.100505i	0.00564493i	0.999996	+	0.00282246i	$0.000898419\pi$
$317$	0.100505i	0.00564493i	−0.999996	+	0.00282246i	$0.999102\pi$
$318$	0	0
$319$	−0.686292	−0.0384249
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	4.82843i	0.268661i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.17157i	− 0.285989i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−17.4558	−0.959460	−0.479730	−	0.877416i	$-0.659265\pi$
$331$	−17.4558	−0.959460	−0.479730	+	0.877416i	$0.659265\pi$
$332$	0	0
$333$	10.2426i	0.561293i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 20.3848i	− 1.11043i	−0.831707	−	0.555215i	$-0.812636\pi$
$337$	− 20.3848i	− 1.11043i	0.831707	−	0.555215i	$-0.187364\pi$
$338$	0	0
$339$	−20.8284	−1.13124
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.2843i	− 1.08892i	−0.838788	−	0.544458i	$-0.816735\pi$
$347$	− 20.2843i	− 1.08892i	0.838788	−	0.544458i	$-0.183265\pi$
$348$	0	0
$349$	−12.3431	−0.660713	−0.330357	−	0.943856i	$-0.607169\pi$
$349$	−12.3431	−0.660713	−0.330357	+	0.943856i	$0.607169\pi$
$350$	0	0
$351$	19.3137	1.03089
$352$	0	0
$353$	12.1421i	0.646261i	0.946354	+	0.323130i	$0.104735\pi$
$353$	12.1421i	0.646261i	−0.946354	+	0.323130i	$0.895265\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	19.3137i	1.02219i
$358$	0	0
$359$	19.4558	1.02684	0.513420	−	0.858137i	$-0.328378\pi$
$359$	19.4558	1.02684	0.513420	+	0.858137i	$0.328378\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 14.5858i	− 0.765555i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.1127i	1.62407i	0.583609	+	0.812035i	$0.301640\pi$
$367$	31.1127i	1.62407i	−0.583609	+	0.812035i	$0.698360\pi$
$368$	0	0
$369$	−0.828427	−0.0431262
$370$	0	0
$371$	−36.9706	−1.91942
$372$	0	0
$373$	33.5563i	1.73748i	0.495267	+	0.868741i	$0.335070\pi$
$373$	33.5563i	1.73748i	−0.495267	+	0.868741i	$0.664930\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.82843i	− 0.145671i
$378$	0	0
$379$	−9.65685	−0.496039	−0.248020	−	0.968755i	$-0.579780\pi$
$379$	−9.65685	−0.496039	−0.248020	+	0.968755i	$0.579780\pi$
$380$	0	0
$381$	18.9706	0.971891
$382$	0	0
$383$	− 32.7279i	− 1.67232i	−0.548487	−	0.836159i	$-0.684796\pi$
$383$	− 32.7279i	− 1.67232i	0.548487	−	0.836159i	$-0.315204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.82843i	− 0.143777i
$388$	0	0
$389$	−9.31371	−0.472224	−0.236112	−	0.971726i	$-0.575873\pi$
$389$	−9.31371	−0.472224	−0.236112	+	0.971726i	$0.575873\pi$
$390$	0	0
$391$	19.3137	0.976736
$392$	0	0
$393$	10.3431i	0.521743i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	36.8284i	1.84837i	0.381950	+	0.924183i	$0.375253\pi$
$397$	36.8284i	1.84837i	−0.381950	+	0.924183i	$0.624747\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	14.9706	0.747594	0.373797	−	0.927510i	$-0.378056\pi$
$401$	14.9706	0.747594	0.373797	+	0.927510i	$0.378056\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.48528i	− 0.420600i
$408$	0	0
$409$	30.2843	1.49746	0.748730	−	0.662875i	$-0.230664\pi$
$409$	30.2843	1.49746	0.748730	+	0.662875i	$0.230664\pi$
$410$	0	0
$411$	10.1421	0.500275
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.48528i	0.219645i
$418$	0	0
$419$	−1.65685	−0.0809426	−0.0404713	−	0.999181i	$-0.512886\pi$
$419$	−1.65685	−0.0809426	−0.0404713	+	0.999181i	$0.512886\pi$
$420$	0	0
$421$	−20.8284	−1.01512	−0.507558	−	0.861618i	$-0.669452\pi$
$421$	−20.8284	−1.01512	−0.507558	+	0.861618i	$0.669452\pi$
$422$	0	0
$423$	− 8.48528i	− 0.412568i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.68629i	0.226786i
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−14.8284	−0.714260	−0.357130	−	0.934055i	$-0.616245\pi$
$431$	−14.8284	−0.714260	−0.357130	+	0.934055i	$0.616245\pi$
$432$	0	0
$433$	18.0416i	0.867025i	0.901147	+	0.433513i	$0.142726\pi$
$433$	18.0416i	0.867025i	−0.901147	+	0.433513i	$0.857274\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.00000i	− 0.191346i
$438$	0	0
$439$	−14.1421	−0.674967	−0.337484	−	0.941331i	$-0.609576\pi$
$439$	−14.1421	−0.674967	−0.337484	+	0.941331i	$0.609576\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 0.686292i	− 0.0326067i	−0.999867	−	0.0163033i	$-0.994810\pi$
$443$	− 0.686292i	− 0.0326067i	0.999867	−	0.0163033i	$-0.00518975\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2.34315i	− 0.110827i
$448$	0	0
$449$	−26.2843	−1.24043	−0.620216	−	0.784431i	$-0.712955\pi$
$449$	−26.2843	−1.24043	−0.620216	+	0.784431i	$0.712955\pi$
$450$	0	0
$451$	0.686292	0.0323162
$452$	0	0
$453$	8.68629i	0.408118i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 34.2843i	− 1.60375i	−0.597491	−	0.801875i	$-0.703836\pi$
$457$	− 34.2843i	− 1.60375i	0.597491	−	0.801875i	$-0.296164\pi$
$458$	0	0
$459$	−27.3137	−1.27489
$460$	0	0
$461$	−8.62742	−0.401819	−0.200909	−	0.979610i	$-0.564390\pi$
$461$	−8.62742	−0.401819	−0.200909	+	0.979610i	$0.564390\pi$
$462$	0	0
$463$	− 26.6274i	− 1.23748i	−0.785595	−	0.618741i	$-0.787643\pi$
$463$	− 26.6274i	− 1.23748i	0.785595	−	0.618741i	$-0.212357\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	0	0
$469$	−26.6274	−1.22954
$470$	0	0
$471$	14.8284	0.683258
$472$	0	0
$473$	2.34315i	0.107738i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 13.0711i	− 0.598483i
$478$	0	0
$479$	−23.4558	−1.07172	−0.535862	−	0.844305i	$-0.680013\pi$
$479$	−23.4558	−1.07172	−0.535862	+	0.844305i	$0.680013\pi$
$480$	0	0
$481$	34.9706	1.59452
$482$	0	0
$483$	− 16.0000i	− 0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.5563i	1.24870i	0.781146	+	0.624349i	$0.214636\pi$
$487$	27.5563i	1.24870i	−0.781146	+	0.624349i	$0.785364\pi$
$488$	0	0
$489$	20.9706	0.948322
$490$	0	0
$491$	−12.9706	−0.585353	−0.292677	−	0.956211i	$-0.594546\pi$
$491$	−12.9706	−0.585353	−0.292677	+	0.956211i	$0.594546\pi$
$492$	0	0
$493$	4.00000i	0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 43.3137i	− 1.94289i
$498$	0	0
$499$	−4.14214	−0.185427	−0.0927137	−	0.995693i	$-0.529554\pi$
$499$	−4.14214	−0.185427	−0.0927137	+	0.995693i	$0.529554\pi$
$500$	0	0
$501$	−2.97056	−0.132715
$502$	0	0
$503$	1.65685i	0.0738755i	0.999318	+	0.0369377i	$0.0117603\pi$
$503$	1.65685i	0.0738755i	−0.999318	+	0.0369377i	$0.988240\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.89949i	0.0843595i
$508$	0	0
$509$	15.4558	0.685068	0.342534	−	0.939505i	$-0.388715\pi$
$509$	15.4558	0.685068	0.342534	+	0.939505i	$0.388715\pi$
$510$	0	0
$511$	−34.3431	−1.51925
$512$	0	0
$513$	5.65685i	0.249756i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.02944i	0.309154i
$518$	0	0
$519$	−15.1716	−0.665958
$520$	0	0
$521$	14.6863	0.643418	0.321709	−	0.946839i	$-0.395743\pi$
$521$	14.6863	0.643418	0.321709	+	0.946839i	$0.395743\pi$
$522$	0	0
$523$	36.2426i	1.58478i	0.610015	+	0.792390i	$0.291163\pi$
$523$	36.2426i	1.58478i	−0.610015	+	0.792390i	$0.708837\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−2.82843	−0.122743
$532$	0	0
$533$	2.82843i	0.122513i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	0.828427	0.0356829
$540$	0	0
$541$	36.2843	1.55998	0.779991	−	0.625790i	$-0.215223\pi$
$541$	36.2843	1.55998	0.779991	+	0.625790i	$0.215223\pi$
$542$	0	0
$543$	− 7.51472i	− 0.322487i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 39.0711i	− 1.67056i	−0.549826	−	0.835279i	$-0.685306\pi$
$547$	− 39.0711i	− 1.67056i	0.549826	−	0.835279i	$-0.314694\pi$
$548$	0	0
$549$	−1.65685	−0.0707128
$550$	0	0
$551$	0.828427	0.0352922
$552$	0	0
$553$	− 25.9411i	− 1.10313i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.79899i	− 0.245711i	−0.992425	−	0.122856i	$-0.960795\pi$
$557$	− 5.79899i	− 0.245711i	0.992425	−	0.122856i	$-0.0392052\pi$
$558$	0	0
$559$	−9.65685	−0.408441
$560$	0	0
$561$	5.65685	0.238833
$562$	0	0
$563$	4.24264i	0.178806i	0.995996	+	0.0894030i	$0.0284959\pi$
$563$	4.24264i	0.178806i	−0.995996	+	0.0894030i	$0.971504\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	14.1421i	0.593914i
$568$	0	0
$569$	31.9411	1.33904	0.669521	−	0.742793i	$-0.266499\pi$
$569$	31.9411	1.33904	0.669521	+	0.742793i	$0.266499\pi$
$570$	0	0
$571$	−30.4853	−1.27577	−0.637885	−	0.770132i	$-0.720190\pi$
$571$	−30.4853	−1.27577	−0.637885	+	0.770132i	$0.720190\pi$
$572$	0	0
$573$	4.68629i	0.195773i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.6274i	1.85786i	0.370251	+	0.928932i	$0.379272\pi$
$577$	44.6274i	1.85786i	−0.370251	+	0.928932i	$0.620728\pi$
$578$	0	0
$579$	−9.79899	−0.407232
$580$	0	0
$581$	−22.6274	−0.938743
$582$	0	0
$583$	10.8284i	0.448468i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.65685i	− 0.233483i	−0.993162	−	0.116742i	$-0.962755\pi$
$587$	− 5.65685i	− 0.233483i	0.993162	−	0.116742i	$-0.0372450\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−35.7990	−1.47257
$592$	0	0
$593$	− 16.6274i	− 0.682806i	−0.939917	−	0.341403i	$-0.889098\pi$
$593$	− 16.6274i	− 0.682806i	0.939917	−	0.341403i	$-0.110902\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.00000i	0.327418i
$598$	0	0
$599$	−6.14214	−0.250961	−0.125480	−	0.992096i	$-0.540047\pi$
$599$	−6.14214	−0.250961	−0.125480	+	0.992096i	$0.540047\pi$
$600$	0	0
$601$	−14.4853	−0.590867	−0.295433	−	0.955363i	$-0.595464\pi$
$601$	−14.4853	−0.590867	−0.295433	+	0.955363i	$0.595464\pi$
$602$	0	0
$603$	− 9.41421i	− 0.383376i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.75736i	− 0.314862i	−0.987530	−	0.157431i	$-0.949679\pi$
$607$	− 7.75736i	− 0.314862i	0.987530	−	0.157431i	$-0.0503211\pi$
$608$	0	0
$609$	3.31371	0.134278
$610$	0	0
$611$	−28.9706	−1.17202
$612$	0	0
$613$	− 17.7990i	− 0.718894i	−0.933165	−	0.359447i	$-0.882965\pi$
$613$	− 17.7990i	− 0.718894i	0.933165	−	0.359447i	$-0.117035\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 37.7990i	− 1.52173i	−0.648910	−	0.760865i	$-0.724775\pi$
$617$	− 37.7990i	− 1.52173i	0.648910	−	0.760865i	$-0.275225\pi$
$618$	0	0
$619$	−40.1421	−1.61345	−0.806724	−	0.590928i	$-0.798762\pi$
$619$	−40.1421	−1.61345	−0.806724	+	0.590928i	$0.798762\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	− 8.97056i	− 0.359398i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.17157i	− 0.0467881i
$628$	0	0
$629$	−49.4558	−1.97193
$630$	0	0
$631$	−34.4853	−1.37284	−0.686419	−	0.727207i	$-0.740818\pi$
$631$	−34.4853	−1.37284	−0.686419	+	0.727207i	$0.740818\pi$
$632$	0	0
$633$	− 23.3137i	− 0.926637i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.41421i	0.135276i
$638$	0	0
$639$	15.3137	0.605801
$640$	0	0
$641$	27.4558	1.08444	0.542220	−	0.840236i	$-0.317584\pi$
$641$	27.4558	1.08444	0.542220	+	0.840236i	$0.317584\pi$
$642$	0	0
$643$	35.3137i	1.39264i	0.717733	+	0.696318i	$0.245180\pi$
$643$	35.3137i	1.39264i	−0.717733	+	0.696318i	$0.754820\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16.6863i	− 0.656006i	−0.944677	−	0.328003i	$-0.893624\pi$
$647$	− 16.6863i	− 0.656006i	0.944677	−	0.328003i	$-0.106376\pi$
$648$	0	0
$649$	2.34315	0.0919765
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 12.1421i	− 0.475158i	−0.971368	−	0.237579i	$-0.923646\pi$
$653$	− 12.1421i	− 0.475158i	0.971368	−	0.237579i	$-0.0763539\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 12.1421i	− 0.473710i
$658$	0	0
$659$	28.4853	1.10963	0.554815	−	0.831974i	$-0.312789\pi$
$659$	28.4853	1.10963	0.554815	+	0.831974i	$0.312789\pi$
$660$	0	0
$661$	8.82843	0.343386	0.171693	−	0.985151i	$-0.445076\pi$
$661$	8.82843	0.343386	0.171693	+	0.985151i	$0.445076\pi$
$662$	0	0
$663$	23.3137i	0.905429i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.31371i	− 0.128307i
$668$	0	0
$669$	18.9706	0.733444
$670$	0	0
$671$	1.37258	0.0529880
$672$	0	0
$673$	− 8.38478i	− 0.323209i	−0.986856	−	0.161605i	$-0.948333\pi$
$673$	− 8.38478i	− 0.323209i	0.986856	−	0.161605i	$-0.0516669\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.61522i	− 0.292677i	−0.989235	−	0.146338i	$-0.953251\pi$
$677$	− 7.61522i	− 0.292677i	0.989235	−	0.146338i	$-0.0467488\pi$
$678$	0	0
$679$	−6.34315	−0.243428
$680$	0	0
$681$	−34.0000	−1.30288
$682$	0	0
$683$	9.89949i	0.378794i	0.981901	+	0.189397i	$0.0606533\pi$
$683$	9.89949i	0.378794i	−0.981901	+	0.189397i	$0.939347\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 16.0000i	− 0.610438i
$688$	0	0
$689$	−44.6274	−1.70017
$690$	0	0
$691$	−35.1716	−1.33799	−0.668995	−	0.743267i	$-0.733275\pi$
$691$	−35.1716	−1.33799	−0.668995	+	0.743267i	$0.733275\pi$
$692$	0	0
$693$	2.34315i	0.0890087i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	−14.1421	−0.534905
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	10.2426i	0.386309i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 49.9411i	− 1.87823i
$708$	0	0
$709$	18.2843	0.686680	0.343340	−	0.939211i	$-0.388442\pi$
$709$	18.2843	0.686680	0.343340	+	0.939211i	$0.388442\pi$
$710$	0	0
$711$	9.17157	0.343961
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 11.3137i	− 0.422518i
$718$	0	0
$719$	31.4558	1.17311	0.586553	−	0.809911i	$-0.300485\pi$
$719$	31.4558	1.17311	0.586553	+	0.809911i	$0.300485\pi$
$720$	0	0
$721$	−23.3137	−0.868248
$722$	0	0
$723$	11.5147i	0.428237i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 41.4558i	− 1.53751i	−0.639542	−	0.768756i	$-0.720876\pi$
$727$	− 41.4558i	− 1.53751i	0.639542	−	0.768756i	$-0.279124\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	13.6569	0.505117
$732$	0	0
$733$	− 4.34315i	− 0.160418i	−0.996778	−	0.0802089i	$-0.974441\pi$
$733$	− 4.34315i	− 0.160418i	0.996778	−	0.0802089i	$-0.0255587\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.79899i	0.287279i
$738$	0	0
$739$	−0.686292	−0.0252456	−0.0126228	−	0.999920i	$-0.504018\pi$
$739$	−0.686292	−0.0252456	−0.0126228	+	0.999920i	$0.504018\pi$
$740$	0	0
$741$	4.82843	0.177377
$742$	0	0
$743$	15.7574i	0.578081i	0.957317	+	0.289041i	$0.0933363\pi$
$743$	15.7574i	0.578081i	−0.957317	+	0.289041i	$0.906664\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−44.4853	−1.62329	−0.811645	−	0.584150i	$-0.801428\pi$
$751$	−44.4853	−1.62329	−0.811645	+	0.584150i	$0.801428\pi$
$752$	0	0
$753$	40.9706i	1.49305i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22.2843i	− 0.809936i	−0.914331	−	0.404968i	$-0.867283\pi$
$757$	− 22.2843i	− 0.809936i	0.914331	−	0.404968i	$-0.132717\pi$
$758$	0	0
$759$	−4.68629	−0.170102
$760$	0	0
$761$	1.37258	0.0497561	0.0248780	−	0.999690i	$-0.492080\pi$
$761$	1.37258	0.0497561	0.0248780	+	0.999690i	$0.492080\pi$
$762$	0	0
$763$	10.3431i	0.374447i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.65685i	0.348689i
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	5.79899	0.208846
$772$	0	0
$773$	48.8701i	1.75773i	0.477067	+	0.878867i	$0.341700\pi$
$773$	48.8701i	1.75773i	−0.477067	+	0.878867i	$0.658300\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	40.9706i	1.46981i
$778$	0	0
$779$	−0.828427	−0.0296815
$780$	0	0
$781$	−12.6863	−0.453951
$782$	0	0
$783$	4.68629i	0.167474i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.41421i	− 0.0504113i	−0.999682	−	0.0252056i	$-0.991976\pi$
$787$	− 1.41421i	− 0.0504113i	0.999682	−	0.0252056i	$-0.00802405\pi$
$788$	0	0
$789$	38.6274	1.37517
$790$	0	0
$791$	41.6569	1.48115
$792$	0	0
$793$	5.65685i	0.200881i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.6985i	0.981131i	0.871404	+	0.490565i	$0.163210\pi$
$797$	27.6985i	0.981131i	−0.871404	+	0.490565i	$0.836790\pi$
$798$	0	0
$799$	40.9706	1.44943
$800$	0	0
$801$	3.17157	0.112062
$802$	0	0
$803$	10.0589i	0.354970i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	41.4558i	1.45931i
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	−0.686292	−0.0240990	−0.0120495	−	0.999927i	$-0.503836\pi$
$811$	−0.686292	−0.0240990	−0.0120495	+	0.999927i	$0.503836\pi$
$812$	0	0
$813$	− 13.4558i	− 0.471917i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.82843i	− 0.0989541i
$818$	0	0
$819$	−9.65685	−0.337438
$820$	0	0
$821$	−49.5980	−1.73098	−0.865491	−	0.500925i	$-0.832993\pi$
$821$	−49.5980	−1.73098	−0.865491	+	0.500925i	$0.832993\pi$
$822$	0	0
$823$	16.4853i	0.574641i	0.957835	+	0.287320i	$0.0927644\pi$
$823$	16.4853i	0.574641i	−0.957835	+	0.287320i	$0.907236\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.6985i	0.476343i	0.971223	+	0.238171i	$0.0765480\pi$
$827$	13.6985i	0.476343i	−0.971223	+	0.238171i	$0.923452\pi$
$828$	0	0
$829$	49.5980	1.72261	0.861305	−	0.508089i	$-0.169648\pi$
$829$	49.5980	1.72261	0.861305	+	0.508089i	$0.169648\pi$
$830$	0	0
$831$	−27.1127	−0.940529
$832$	0	0
$833$	− 4.82843i	− 0.167295i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−0.201010	−0.00693964	−0.00346982	−	0.999994i	$-0.501104\pi$
$839$	−0.201010	−0.00693964	−0.00346982	+	0.999994i	$0.501104\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	15.7990i	0.544146i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29.1716i	1.00235i
$848$	0	0
$849$	8.97056	0.307869
$850$	0	0
$851$	40.9706	1.40445
$852$	0	0
$853$	27.9411i	0.956686i	0.878173	+	0.478343i	$0.158762\pi$
$853$	27.9411i	0.956686i	−0.878173	+	0.478343i	$0.841238\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.2132i	1.06622i	0.846045	+	0.533111i	$0.178977\pi$
$857$	31.2132i	1.06622i	−0.846045	+	0.533111i	$0.821023\pi$
$858$	0	0
$859$	−6.34315	−0.216425	−0.108213	−	0.994128i	$-0.534513\pi$
$859$	−6.34315	−0.216425	−0.108213	+	0.994128i	$0.534513\pi$
$860$	0	0
$861$	−3.31371	−0.112931
$862$	0	0
$863$	44.0416i	1.49919i	0.661894	+	0.749597i	$0.269753\pi$
$863$	44.0416i	1.49919i	−0.661894	+	0.749597i	$0.730247\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.92893i	− 0.303242i
$868$	0	0
$869$	−7.59798	−0.257744
$870$	0	0
$871$	−32.1421	−1.08909
$872$	0	0
$873$	− 2.24264i	− 0.0759019i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.7279i	0.632397i	0.948693	+	0.316198i	$0.102407\pi$
$877$	18.7279i	0.632397i	−0.948693	+	0.316198i	$0.897593\pi$
$878$	0	0
$879$	45.7990	1.54476
$880$	0	0
$881$	−1.65685	−0.0558208	−0.0279104	−	0.999610i	$-0.508885\pi$
$881$	−1.65685	−0.0558208	−0.0279104	+	0.999610i	$0.508885\pi$
$882$	0	0
$883$	10.8284i	0.364406i	0.983261	+	0.182203i	$0.0583227\pi$
$883$	10.8284i	0.364406i	−0.983261	+	0.182203i	$0.941677\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 15.2721i	− 0.512786i	−0.966573	−	0.256393i	$-0.917466\pi$
$887$	− 15.2721i	− 0.512786i	0.966573	−	0.256393i	$-0.0825342\pi$
$888$	0	0
$889$	−37.9411	−1.27250
$890$	0	0
$891$	4.14214	0.138767
$892$	0	0
$893$	− 8.48528i	− 0.283949i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 19.3137i	− 0.644866i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	63.1127	2.10259
$902$	0	0
$903$	− 11.3137i	− 0.376497i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	48.0416i	1.59520i	0.603189	+	0.797598i	$0.293896\pi$
$907$	48.0416i	1.59520i	−0.603189	+	0.797598i	$0.706104\pi$
$908$	0	0
$909$	17.6569	0.585641
$910$	0	0
$911$	−9.17157	−0.303868	−0.151934	−	0.988391i	$-0.548550\pi$
$911$	−9.17157	−0.303868	−0.151934	+	0.988391i	$0.548550\pi$
$912$	0	0
$913$	6.62742i	0.219335i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 20.6863i	− 0.683122i
$918$	0	0
$919$	17.9411	0.591823	0.295912	−	0.955215i	$-0.404377\pi$
$919$	17.9411	0.591823	0.295912	+	0.955215i	$0.404377\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	− 52.2843i	− 1.72096i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.24264i	− 0.270724i
$928$	0	0
$929$	−41.3137	−1.35546	−0.677729	−	0.735311i	$-0.737036\pi$
$929$	−41.3137	−1.35546	−0.677729	+	0.735311i	$0.737036\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	27.1127i	0.887630i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	55.4558i	1.81166i	0.423638	+	0.905832i	$0.360753\pi$
$937$	55.4558i	1.81166i	−0.423638	+	0.905832i	$0.639247\pi$
$938$	0	0
$939$	−7.51472	−0.245234
$940$	0	0
$941$	21.5980	0.704074	0.352037	−	0.935986i	$-0.385489\pi$
$941$	21.5980	0.704074	0.352037	+	0.935986i	$0.385489\pi$
$942$	0	0
$943$	3.31371i	0.107909i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.3137i	1.53749i	0.639556	+	0.768744i	$0.279118\pi$
$947$	47.3137i	1.53749i	−0.639556	+	0.768744i	$0.720882\pi$
$948$	0	0
$949$	−41.4558	−1.34571
$950$	0	0
$951$	−0.142136	−0.00460906
$952$	0	0
$953$	34.7279i	1.12495i	0.826815	+	0.562474i	$0.190150\pi$
$953$	34.7279i	1.12495i	−0.826815	+	0.562474i	$0.809850\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 0.970563i	− 0.0313738i
$958$	0	0
$959$	−20.2843	−0.655013
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	1.41421i	0.0455724i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37.6569i	1.21096i	0.795859	+	0.605481i	$0.207019\pi$
$967$	37.6569i	1.21096i	−0.795859	+	0.605481i	$0.792981\pi$
$968$	0	0
$969$	−6.82843	−0.219361
$970$	0	0
$971$	25.9411	0.832490	0.416245	−	0.909252i	$-0.363346\pi$
$971$	25.9411	0.832490	0.416245	+	0.909252i	$0.363346\pi$
$972$	0	0
$973$	− 8.97056i	− 0.287583i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 53.8406i	− 1.72251i	−0.508170	−	0.861257i	$-0.669678\pi$
$977$	− 53.8406i	− 1.72251i	0.508170	−	0.861257i	$-0.330322\pi$
$978$	0	0
$979$	−2.62742	−0.0839726
$980$	0	0
$981$	−3.65685	−0.116754
$982$	0	0
$983$	− 4.04163i	− 0.128908i	−0.997921	−	0.0644540i	$-0.979469\pi$
$983$	− 4.04163i	− 0.128908i	0.997921	−	0.0644540i	$-0.0205306\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 33.9411i	− 1.08036i
$988$	0	0
$989$	−11.3137	−0.359755
$990$	0	0
$991$	44.7696	1.42215	0.711076	−	0.703115i	$-0.248208\pi$
$991$	44.7696	1.42215	0.711076	+	0.703115i	$0.248208\pi$
$992$	0	0
$993$	− 24.6863i	− 0.783396i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.8284i	0.406280i	0.979150	+	0.203140i	$0.0651147\pi$
$997$	12.8284i	0.406280i	−0.979150	+	0.203140i	$0.934885\pi$
$998$	0	0
$999$	−57.9411	−1.83318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.i.3649.3		4
5.2	odd	4		3800.2.a.n.1.2		2
5.3	odd	4		760.2.a.f.1.1	✓	2
5.4	even	2	inner	3800.2.d.i.3649.1		4
15.8	even	4		6840.2.a.z.1.1		2
20.3	even	4		1520.2.a.m.1.2		2
20.7	even	4		7600.2.a.ba.1.1		2
40.3	even	4		6080.2.a.bg.1.1		2
40.13	odd	4		6080.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.f.1.1	✓	2	5.3	odd	4
1520.2.a.m.1.2		2	20.3	even	4
3800.2.a.n.1.2		2	5.2	odd	4
3800.2.d.i.3649.1		4	5.4	even	2	inner
3800.2.d.i.3649.3		4	1.1	even	1	trivial
6080.2.a.bf.1.2		2	40.13	odd	4
6080.2.a.bg.1.1		2	40.3	even	4
6840.2.a.z.1.1		2	15.8	even	4
7600.2.a.ba.1.1		2	20.7	even	4

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} -2.82843i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{3} -2.82843i q^{7} +1.00000 q^{9} -0.828427 q^{11} -3.41421i q^{13} +4.82843i q^{17} +1.00000 q^{19} +4.00000 q^{21} -4.00000i q^{23} +5.65685i q^{27} +0.828427 q^{29} -1.17157i q^{33} +10.2426i q^{37} +4.82843 q^{39} -0.828427 q^{41} -2.82843i q^{43} -8.48528i q^{47} -1.00000 q^{49} -6.82843 q^{51} -13.0711i q^{53} +1.41421i q^{57} -2.82843 q^{59} -1.65685 q^{61} -2.82843i q^{63} -9.41421i q^{67} +5.65685 q^{69} +15.3137 q^{71} -12.1421i q^{73} +2.34315i q^{77} +9.17157 q^{79} -5.00000 q^{81} -8.00000i q^{83} +1.17157i q^{87} +3.17157 q^{89} -9.65685 q^{91} -2.24264i q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} + 8 q^{11} + 4 q^{19} + 16 q^{21} - 8 q^{29} + 8 q^{39} + 8 q^{41} - 4 q^{49} - 16 q^{51} + 16 q^{61} + 16 q^{71} + 48 q^{79} - 20 q^{81} + 24 q^{89} - 16 q^{91} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^9 + 8 * q^11 + 4 * q^19 + 16 * q^21 - 8 * q^29 + 8 * q^39 + 8 * q^41 - 4 * q^49 - 16 * q^51 + 16 * q^61 + 16 * q^71 + 48 * q^79 - 20 * q^81 + 24 * q^89 - 16 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more