LMFDB - Embedded newform 7800.2.a.bx.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

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:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.461844.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} - 12x + 6 $ x^4 - x^3 - 14x^2 - 12x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.32424$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.32424 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.32424 q^{7} +1.00000 q^{9} +4.84395 q^{11} +1.00000 q^{13} -1.11760 q^{17} +4.05060 q^{19} +3.32424 q^{21} -4.16820 q^{23} +1.00000 q^{27} +5.64849 q^{29} -7.49244 q^{31} +4.84395 q^{33} -8.69908 q^{37} +1.00000 q^{39} +5.63731 q^{41} +12.8167 q^{43} -6.89455 q^{47} +4.05060 q^{49} -1.11760 q^{51} +12.2300 q^{53} +4.05060 q^{57} +0.910956 q^{59} +2.46911 q^{61} +3.32424 q^{63} +10.0233 q^{67} -4.16820 q^{69} +8.05060 q^{71} +10.4034 q^{73} +16.1025 q^{77} -10.6991 q^{79} +1.00000 q^{81} +2.67576 q^{83} +5.64849 q^{87} -3.68791 q^{89} +3.32424 q^{91} -7.49244 q^{93} -6.64849 q^{97} +4.84395 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9} + 5 q^{11} + 4 q^{13} + 7 q^{17} + 3 q^{19} + 3 q^{21} + 8 q^{23} + 4 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - q^{37} + 4 q^{39} + 7 q^{41} + 6 q^{43} + 3 q^{49} + 7 q^{51} + 6 q^{53} + 3 q^{57} - 9 q^{59} + 19 q^{61} + 3 q^{63} - 4 q^{67} + 8 q^{69} + 19 q^{71} - 6 q^{73} - 17 q^{77} - 9 q^{79} + 4 q^{81} + 21 q^{83} + 2 q^{87} + 14 q^{89} + 3 q^{91} + 5 q^{93} - 6 q^{97} + 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9 + 5 * q^11 + 4 * q^13 + 7 * q^17 + 3 * q^19 + 3 * q^21 + 8 * q^23 + 4 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - q^37 + 4 * q^39 + 7 * q^41 + 6 * q^43 + 3 * q^49 + 7 * q^51 + 6 * q^53 + 3 * q^57 - 9 * q^59 + 19 * q^61 + 3 * q^63 - 4 * q^67 + 8 * q^69 + 19 * q^71 - 6 * q^73 - 17 * q^77 - 9 * q^79 + 4 * q^81 + 21 * q^83 + 2 * q^87 + 14 * q^89 + 3 * q^91 + 5 * q^93 - 6 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.32424	1.25645	0.628223	−	0.778033i	$-0.283783\pi$
$7$	3.32424	1.25645	0.628223	+	0.778033i	$0.283783\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.84395	1.46051	0.730253	−	0.683176i	$-0.239402\pi$
$11$	4.84395	1.46051	0.730253	+	0.683176i	$0.239402\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.11760	−0.271058	−0.135529	−	0.990773i	$-0.543273\pi$
$17$	−1.11760	−0.271058	−0.135529	+	0.990773i	$0.543273\pi$
$18$	0	0
$19$	4.05060	0.929271	0.464635	−	0.885502i	$-0.346185\pi$
$19$	4.05060	0.929271	0.464635	+	0.885502i	$0.346185\pi$
$20$	0	0
$21$	3.32424	0.725409
$22$	0	0
$23$	−4.16820	−0.869129	−0.434565	−	0.900641i	$-0.643098\pi$
$23$	−4.16820	−0.869129	−0.434565	+	0.900641i	$0.643098\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.64849	1.04890	0.524449	−	0.851442i	$-0.324271\pi$
$29$	5.64849	1.04890	0.524449	+	0.851442i	$0.324271\pi$
$30$	0	0
$31$	−7.49244	−1.34568	−0.672841	−	0.739787i	$-0.734926\pi$
$31$	−7.49244	−1.34568	−0.672841	+	0.739787i	$0.734926\pi$
$32$	0	0
$33$	4.84395	0.843224
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.69908	−1.43012	−0.715060	−	0.699063i	$-0.753601\pi$
$37$	−8.69908	−1.43012	−0.715060	+	0.699063i	$0.753601\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	5.63731	0.880400	0.440200	−	0.897900i	$-0.354908\pi$
$41$	5.63731	0.880400	0.440200	+	0.897900i	$0.354908\pi$
$42$	0	0
$43$	12.8167	1.95453	0.977263	−	0.212030i	$-0.0680076\pi$
$43$	12.8167	1.95453	0.977263	+	0.212030i	$0.0680076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.89455	−1.00567	−0.502837	−	0.864381i	$-0.667710\pi$
$47$	−6.89455	−1.00567	−0.502837	+	0.864381i	$0.667710\pi$
$48$	0	0
$49$	4.05060	0.578657
$50$	0	0
$51$	−1.11760	−0.156495
$52$	0	0
$53$	12.2300	1.67992	0.839958	−	0.542651i	$-0.182580\pi$
$53$	12.2300	1.67992	0.839958	+	0.542651i	$0.182580\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.05060	0.536515
$58$	0	0
$59$	0.910956	0.118596	0.0592982	−	0.998240i	$-0.481114\pi$
$59$	0.910956	0.118596	0.0592982	+	0.998240i	$0.481114\pi$
$60$	0	0
$61$	2.46911	0.316137	0.158069	−	0.987428i	$-0.449473\pi$
$61$	2.46911	0.316137	0.158069	+	0.987428i	$0.449473\pi$
$62$	0	0
$63$	3.32424	0.418815
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.0233	1.22454	0.612272	−	0.790647i	$-0.290256\pi$
$67$	10.0233	1.22454	0.612272	+	0.790647i	$0.290256\pi$
$68$	0	0
$69$	−4.16820	−0.501792
$70$	0	0
$71$	8.05060	0.955430	0.477715	−	0.878515i	$-0.341465\pi$
$71$	8.05060	0.955430	0.477715	+	0.878515i	$0.341465\pi$
$72$	0	0
$73$	10.4034	1.21763	0.608813	−	0.793314i	$-0.291646\pi$
$73$	10.4034	1.21763	0.608813	+	0.793314i	$0.291646\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.1025	1.83505
$78$	0	0
$79$	−10.6991	−1.20374	−0.601871	−	0.798594i	$-0.705578\pi$
$79$	−10.6991	−1.20374	−0.601871	+	0.798594i	$0.705578\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.67576	0.293702	0.146851	−	0.989159i	$-0.453086\pi$
$83$	2.67576	0.293702	0.146851	+	0.989159i	$0.453086\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.64849	0.605581
$88$	0	0
$89$	−3.68791	−0.390917	−0.195459	−	0.980712i	$-0.562620\pi$
$89$	−3.68791	−0.390917	−0.195459	+	0.980712i	$0.562620\pi$
$90$	0	0
$91$	3.32424	0.348475
$92$	0	0
$93$	−7.49244	−0.776930
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.64849	−0.675052	−0.337526	−	0.941316i	$-0.609590\pi$
$97$	−6.64849	−0.675052	−0.337526	+	0.941316i	$0.609590\pi$
$98$	0	0
$99$	4.84395	0.486836
$100$	0	0
$101$	11.6991	1.16410	0.582051	−	0.813152i	$-0.302250\pi$
$101$	11.6991	1.16410	0.582051	+	0.813152i	$0.302250\pi$
$102$	0	0
$103$	−19.1531	−1.88721	−0.943604	−	0.331075i	$-0.892589\pi$
$103$	−19.1531	−1.88721	−0.943604	+	0.331075i	$0.892589\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.8673	−1.24393	−0.621964	−	0.783046i	$-0.713665\pi$
$107$	−12.8673	−1.24393	−0.621964	+	0.783046i	$0.713665\pi$
$108$	0	0
$109$	−11.1124	−1.06437	−0.532186	−	0.846627i	$-0.678629\pi$
$109$	−11.1124	−1.06437	−0.532186	+	0.846627i	$0.678629\pi$
$110$	0	0
$111$	−8.69908	−0.825681
$112$	0	0
$113$	−10.7497	−1.01125	−0.505623	−	0.862755i	$-0.668737\pi$
$113$	−10.7497	−1.01125	−0.505623	+	0.862755i	$0.668737\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−3.71518	−0.340570
$120$	0	0
$121$	12.4639	1.13308
$122$	0	0
$123$	5.63731	0.508299
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.3476	−1.00693	−0.503467	−	0.864014i	$-0.667942\pi$
$127$	−11.3476	−1.00693	−0.503467	+	0.864014i	$0.667942\pi$
$128$	0	0
$129$	12.8167	1.12845
$130$	0	0
$131$	−8.38699	−0.732775	−0.366387	−	0.930462i	$-0.619406\pi$
$131$	−8.38699	−0.732775	−0.366387	+	0.930462i	$0.619406\pi$
$132$	0	0
$133$	13.4652	1.16758
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.80551	0.837741	0.418870	−	0.908046i	$-0.362426\pi$
$137$	9.80551	0.837741	0.418870	+	0.908046i	$0.362426\pi$
$138$	0	0
$139$	−10.2352	−0.868138	−0.434069	−	0.900880i	$-0.642923\pi$
$139$	−10.2352	−0.868138	−0.434069	+	0.900880i	$0.642923\pi$
$140$	0	0
$141$	−6.89455	−0.580626
$142$	0	0
$143$	4.84395	0.405072
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.05060	0.334088
$148$	0	0
$149$	10.6485	0.872358	0.436179	−	0.899860i	$-0.356331\pi$
$149$	10.6485	0.872358	0.436179	+	0.899860i	$0.356331\pi$
$150$	0	0
$151$	−5.42544	−0.441516	−0.220758	−	0.975329i	$-0.570853\pi$
$151$	−5.42544	−0.441516	−0.220758	+	0.975329i	$0.570853\pi$
$152$	0	0
$153$	−1.11760	−0.0903526
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.94940	−0.554623	−0.277311	−	0.960780i	$-0.589443\pi$
$157$	−6.94940	−0.554623	−0.277311	+	0.960780i	$0.589443\pi$
$158$	0	0
$159$	12.2300	0.969900
$160$	0	0
$161$	−13.8561	−1.09201
$162$	0	0
$163$	18.2694	1.43097	0.715485	−	0.698629i	$-0.246206\pi$
$163$	18.2694	1.43097	0.715485	+	0.698629i	$0.246206\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.9961	1.39258	0.696288	−	0.717762i	$-0.254833\pi$
$167$	17.9961	1.39258	0.696288	+	0.717762i	$0.254833\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.05060	0.309757
$172$	0	0
$173$	−18.9573	−1.44130	−0.720648	−	0.693301i	$-0.756156\pi$
$173$	−18.9573	−1.44130	−0.720648	+	0.693301i	$0.756156\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.910956	0.0684717
$178$	0	0
$179$	−11.7773	−0.880274	−0.440137	−	0.897931i	$-0.645070\pi$
$179$	−11.7773	−0.880274	−0.440137	+	0.897931i	$0.645070\pi$
$180$	0	0
$181$	11.5979	0.862064	0.431032	−	0.902337i	$-0.358150\pi$
$181$	11.5979	0.862064	0.431032	+	0.902337i	$0.358150\pi$
$182$	0	0
$183$	2.46911	0.182522
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.41360	−0.395882
$188$	0	0
$189$	3.32424	0.241803
$190$	0	0
$191$	23.2970	1.68571	0.842855	−	0.538141i	$-0.180873\pi$
$191$	23.2970	1.68571	0.842855	+	0.538141i	$0.180873\pi$
$192$	0	0
$193$	−12.6386	−0.909746	−0.454873	−	0.890556i	$-0.650315\pi$
$193$	−12.6386	−0.909746	−0.454873	+	0.890556i	$0.650315\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.0913	−1.71643	−0.858217	−	0.513287i	$-0.828428\pi$
$197$	−24.0913	−1.71643	−0.858217	+	0.513287i	$0.828428\pi$
$198$	0	0
$199$	−12.1176	−0.858994	−0.429497	−	0.903068i	$-0.641309\pi$
$199$	−12.1176	−0.858994	−0.429497	+	0.903068i	$0.641309\pi$
$200$	0	0
$201$	10.0233	0.706991
$202$	0	0
$203$	18.7770	1.31788
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.16820	−0.289710
$208$	0	0
$209$	19.6209	1.35721
$210$	0	0
$211$	−0.168197	−0.0115792	−0.00578959	−	0.999983i	$-0.501843\pi$
$211$	−0.168197	−0.0115792	−0.00578959	+	0.999983i	$0.501843\pi$
$212$	0	0
$213$	8.05060	0.551618
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.9067	−1.69078
$218$	0	0
$219$	10.4034	0.702996
$220$	0	0
$221$	−1.11760	−0.0751779
$222$	0	0
$223$	−25.5440	−1.71055	−0.855277	−	0.518171i	$-0.826613\pi$
$223$	−25.5440	−1.71055	−0.855277	+	0.518171i	$0.826613\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.70995	0.445355	0.222677	−	0.974892i	$-0.428520\pi$
$227$	6.70995	0.445355	0.222677	+	0.974892i	$0.428520\pi$
$228$	0	0
$229$	2.92182	0.193079	0.0965396	−	0.995329i	$-0.469223\pi$
$229$	2.92182	0.193079	0.0965396	+	0.995329i	$0.469223\pi$
$230$	0	0
$231$	16.1025	1.05947
$232$	0	0
$233$	−1.68791	−0.110578	−0.0552892	−	0.998470i	$-0.517608\pi$
$233$	−1.68791	−0.110578	−0.0552892	+	0.998470i	$0.517608\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.6991	−0.694980
$238$	0	0
$239$	−16.2079	−1.04840	−0.524202	−	0.851594i	$-0.675636\pi$
$239$	−16.2079	−1.04840	−0.524202	+	0.851594i	$0.675636\pi$
$240$	0	0
$241$	12.1682	0.783822	0.391911	−	0.920003i	$-0.371814\pi$
$241$	12.1682	0.783822	0.391911	+	0.920003i	$0.371814\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.05060	0.257733
$248$	0	0
$249$	2.67576	0.169569
$250$	0	0
$251$	15.6373	0.987018	0.493509	−	0.869741i	$-0.335714\pi$
$251$	15.6373	0.987018	0.493509	+	0.869741i	$0.335714\pi$
$252$	0	0
$253$	−20.1906	−1.26937
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.5210	−0.718660	−0.359330	−	0.933211i	$-0.616995\pi$
$257$	−11.5210	−0.718660	−0.359330	+	0.933211i	$0.616995\pi$
$258$	0	0
$259$	−28.9179	−1.79687
$260$	0	0
$261$	5.64849	0.349633
$262$	0	0
$263$	1.41852	0.0874694	0.0437347	−	0.999043i	$-0.486074\pi$
$263$	1.41852	0.0874694	0.0437347	+	0.999043i	$0.486074\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.68791	−0.225696
$268$	0	0
$269$	26.3982	1.60952	0.804762	−	0.593597i	$-0.202293\pi$
$269$	26.3982	1.60952	0.804762	+	0.593597i	$0.202293\pi$
$270$	0	0
$271$	27.1133	1.64702	0.823509	−	0.567303i	$-0.192013\pi$
$271$	27.1133	1.64702	0.823509	+	0.567303i	$0.192013\pi$
$272$	0	0
$273$	3.32424	0.201192
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.0861	0.666098	0.333049	−	0.942910i	$-0.391923\pi$
$277$	11.0861	0.666098	0.333049	+	0.942910i	$0.391923\pi$
$278$	0	0
$279$	−7.49244	−0.448561
$280$	0	0
$281$	9.17938	0.547596	0.273798	−	0.961787i	$-0.411720\pi$
$281$	9.17938	0.547596	0.273798	+	0.961787i	$0.411720\pi$
$282$	0	0
$283$	31.8686	1.89439	0.947195	−	0.320658i	$-0.103904\pi$
$283$	31.8686	1.89439	0.947195	+	0.320658i	$0.103904\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.7398	1.10617
$288$	0	0
$289$	−15.7510	−0.926528
$290$	0	0
$291$	−6.64849	−0.389741
$292$	0	0
$293$	9.14061	0.534000	0.267000	−	0.963696i	$-0.413968\pi$
$293$	9.14061	0.534000	0.267000	+	0.963696i	$0.413968\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.84395	0.281075
$298$	0	0
$299$	−4.16820	−0.241053
$300$	0	0
$301$	42.6058	2.45576
$302$	0	0
$303$	11.6991	0.672095
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.40211	−0.422461	−0.211230	−	0.977436i	$-0.567747\pi$
$307$	−7.40211	−0.422461	−0.211230	+	0.977436i	$0.567747\pi$
$308$	0	0
$309$	−19.1531	−1.08958
$310$	0	0
$311$	21.5098	1.21971	0.609855	−	0.792513i	$-0.291228\pi$
$311$	21.5098	1.21971	0.609855	+	0.792513i	$0.291228\pi$
$312$	0	0
$313$	11.1458	0.630000	0.315000	−	0.949092i	$-0.397995\pi$
$313$	11.1458	0.630000	0.315000	+	0.949092i	$0.397995\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.0960	−1.52186	−0.760931	−	0.648833i	$-0.775257\pi$
$317$	−27.0960	−1.52186	−0.760931	+	0.648833i	$0.775257\pi$
$318$	0	0
$319$	27.3610	1.53192
$320$	0	0
$321$	−12.8673	−0.718182
$322$	0	0
$323$	−4.52695	−0.251886
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.1124	−0.614516
$328$	0	0
$329$	−22.9192	−1.26357
$330$	0	0
$331$	10.0670	0.553333	0.276666	−	0.960966i	$-0.410770\pi$
$331$	10.0670	0.553333	0.276666	+	0.960966i	$0.410770\pi$
$332$	0	0
$333$	−8.69908	−0.476707
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−35.0749	−1.91065	−0.955326	−	0.295555i	$-0.904495\pi$
$337$	−35.0749	−1.91065	−0.955326	+	0.295555i	$0.904495\pi$
$338$	0	0
$339$	−10.7497	−0.583843
$340$	0	0
$341$	−36.2930	−1.96538
$342$	0	0
$343$	−9.80453	−0.529395
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.15702	0.169478	0.0847388	−	0.996403i	$-0.472994\pi$
$347$	3.15702	0.169478	0.0847388	+	0.996403i	$0.472994\pi$
$348$	0	0
$349$	31.3883	1.68018	0.840088	−	0.542450i	$-0.182503\pi$
$349$	31.3883	1.68018	0.840088	+	0.542450i	$0.182503\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	24.3325	1.29509	0.647543	−	0.762029i	$-0.275797\pi$
$353$	24.3325	1.29509	0.647543	+	0.762029i	$0.275797\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.71518	−0.196628
$358$	0	0
$359$	26.1915	1.38234	0.691168	−	0.722694i	$-0.257096\pi$
$359$	26.1915	1.38234	0.691168	+	0.722694i	$0.257096\pi$
$360$	0	0
$361$	−2.59266	−0.136456
$362$	0	0
$363$	12.4639	0.654184
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.2806	1.00644	0.503219	−	0.864159i	$-0.332149\pi$
$367$	19.2806	1.00644	0.503219	+	0.864159i	$0.332149\pi$
$368$	0	0
$369$	5.63731	0.293467
$370$	0	0
$371$	40.6554	2.11072
$372$	0	0
$373$	−19.8331	−1.02692	−0.513459	−	0.858114i	$-0.671636\pi$
$373$	−19.8331	−1.02692	−0.513459	+	0.858114i	$0.671636\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.64849	0.290912
$378$	0	0
$379$	−21.5936	−1.10919	−0.554595	−	0.832120i	$-0.687127\pi$
$379$	−21.5936	−1.10919	−0.554595	+	0.832120i	$0.687127\pi$
$380$	0	0
$381$	−11.3476	−0.581354
$382$	0	0
$383$	7.81540	0.399348	0.199674	−	0.979862i	$-0.436012\pi$
$383$	7.81540	0.399348	0.199674	+	0.979862i	$0.436012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.8167	0.651509
$388$	0	0
$389$	19.7043	0.999048	0.499524	−	0.866300i	$-0.333508\pi$
$389$	19.7043	0.999048	0.499524	+	0.866300i	$0.333508\pi$
$390$	0	0
$391$	4.65838	0.235584
$392$	0	0
$393$	−8.38699	−0.423068
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.96848	0.249361	0.124680	−	0.992197i	$-0.460209\pi$
$397$	4.96848	0.249361	0.124680	+	0.992197i	$0.460209\pi$
$398$	0	0
$399$	13.4652	0.674102
$400$	0	0
$401$	−25.6209	−1.27945	−0.639723	−	0.768605i	$-0.720951\pi$
$401$	−25.6209	−1.27945	−0.639723	+	0.768605i	$0.720951\pi$
$402$	0	0
$403$	−7.49244	−0.373225
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−42.1380	−2.08870
$408$	0	0
$409$	13.6761	0.676238	0.338119	−	0.941103i	$-0.390209\pi$
$409$	13.6761	0.676238	0.338119	+	0.941103i	$0.390209\pi$
$410$	0	0
$411$	9.80551	0.483670
$412$	0	0
$413$	3.02824	0.149010
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.2352	−0.501220
$418$	0	0
$419$	7.50330	0.366560	0.183280	−	0.983061i	$-0.441328\pi$
$419$	7.50330	0.366560	0.183280	+	0.983061i	$0.441328\pi$
$420$	0	0
$421$	−9.39817	−0.458039	−0.229019	−	0.973422i	$-0.573552\pi$
$421$	−9.39817	−0.458039	−0.229019	+	0.973422i	$0.573552\pi$
$422$	0	0
$423$	−6.89455	−0.335225
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.20793	0.397210
$428$	0	0
$429$	4.84395	0.233868
$430$	0	0
$431$	4.77598	0.230051	0.115025	−	0.993363i	$-0.463305\pi$
$431$	4.77598	0.230051	0.115025	+	0.993363i	$0.463305\pi$
$432$	0	0
$433$	−2.50853	−0.120552	−0.0602762	−	0.998182i	$-0.519198\pi$
$433$	−2.50853	−0.120552	−0.0602762	+	0.998182i	$0.519198\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.8837	−0.807656
$438$	0	0
$439$	−13.0900	−0.624752	−0.312376	−	0.949958i	$-0.601125\pi$
$439$	−13.0900	−0.624752	−0.312376	+	0.949958i	$0.601125\pi$
$440$	0	0
$441$	4.05060	0.192886
$442$	0	0
$443$	38.2089	1.81536	0.907680	−	0.419663i	$-0.137852\pi$
$443$	38.2089	1.81536	0.907680	+	0.419663i	$0.137852\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6485	0.503656
$448$	0	0
$449$	9.29103	0.438471	0.219235	−	0.975672i	$-0.429644\pi$
$449$	9.29103	0.438471	0.219235	+	0.975672i	$0.429644\pi$
$450$	0	0
$451$	27.3069	1.28583
$452$	0	0
$453$	−5.42544	−0.254909
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.05189	0.236317	0.118159	−	0.992995i	$-0.462301\pi$
$457$	5.05189	0.236317	0.118159	+	0.992995i	$0.462301\pi$
$458$	0	0
$459$	−1.11760	−0.0521651
$460$	0	0
$461$	−9.89881	−0.461033	−0.230517	−	0.973068i	$-0.574042\pi$
$461$	−9.89881	−0.461033	−0.230517	+	0.973068i	$0.574042\pi$
$462$	0	0
$463$	24.8906	1.15676	0.578382	−	0.815766i	$-0.303684\pi$
$463$	24.8906	1.15676	0.578382	+	0.815766i	$0.303684\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.7333	−1.23707	−0.618534	−	0.785758i	$-0.712273\pi$
$467$	−26.7333	−1.23707	−0.618534	+	0.785758i	$0.712273\pi$
$468$	0	0
$469$	33.3200	1.53857
$470$	0	0
$471$	−6.94940	−0.320212
$472$	0	0
$473$	62.0834	2.85460
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.2300	0.559972
$478$	0	0
$479$	10.8827	0.497244	0.248622	−	0.968601i	$-0.420022\pi$
$479$	10.8827	0.497244	0.248622	+	0.968601i	$0.420022\pi$
$480$	0	0
$481$	−8.69908	−0.396644
$482$	0	0
$483$	−13.8561	−0.630475
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.88337	−0.0853438	−0.0426719	−	0.999089i	$-0.513587\pi$
$487$	−1.88337	−0.0853438	−0.0426719	+	0.999089i	$0.513587\pi$
$488$	0	0
$489$	18.2694	0.826170
$490$	0	0
$491$	−32.2050	−1.45339	−0.726695	−	0.686960i	$-0.758945\pi$
$491$	−32.2050	−1.45339	−0.726695	+	0.686960i	$0.758945\pi$
$492$	0	0
$493$	−6.31275	−0.284312
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.7621	1.20045
$498$	0	0
$499$	8.03516	0.359703	0.179852	−	0.983694i	$-0.442438\pi$
$499$	8.03516	0.359703	0.179852	+	0.983694i	$0.442438\pi$
$500$	0	0
$501$	17.9961	0.804005
$502$	0	0
$503$	2.40340	0.107162	0.0535811	−	0.998564i	$-0.482936\pi$
$503$	2.40340	0.107162	0.0535811	+	0.998564i	$0.482936\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−1.91064	−0.0846877	−0.0423438	−	0.999103i	$-0.513482\pi$
$509$	−1.91064	−0.0846877	−0.0423438	+	0.999103i	$0.513482\pi$
$510$	0	0
$511$	34.5834	1.52988
$512$	0	0
$513$	4.05060	0.178838
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−33.3969	−1.46879
$518$	0	0
$519$	−18.9573	−0.832133
$520$	0	0
$521$	−15.0618	−0.659868	−0.329934	−	0.944004i	$-0.607027\pi$
$521$	−15.0618	−0.659868	−0.329934	+	0.944004i	$0.607027\pi$
$522$	0	0
$523$	−33.1649	−1.45020	−0.725100	−	0.688643i	$-0.758207\pi$
$523$	−33.1649	−1.45020	−0.725100	+	0.688643i	$0.758207\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.37355	0.364758
$528$	0	0
$529$	−5.62613	−0.244614
$530$	0	0
$531$	0.910956	0.0395321
$532$	0	0
$533$	5.63731	0.244179
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.7773	−0.508227
$538$	0	0
$539$	19.6209	0.845132
$540$	0	0
$541$	−6.50459	−0.279654	−0.139827	−	0.990176i	$-0.544655\pi$
$541$	−6.50459	−0.279654	−0.139827	+	0.990176i	$0.544655\pi$
$542$	0	0
$543$	11.5979	0.497713
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−41.5118	−1.77492	−0.887459	−	0.460887i	$-0.847531\pi$
$547$	−41.5118	−1.77492	−0.887459	+	0.460887i	$0.847531\pi$
$548$	0	0
$549$	2.46911	0.105379
$550$	0	0
$551$	22.8797	0.974710
$552$	0	0
$553$	−35.5664	−1.51244
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13.7004	−0.580503	−0.290252	−	0.956950i	$-0.593739\pi$
$557$	−13.7004	−0.580503	−0.290252	+	0.956950i	$0.593739\pi$
$558$	0	0
$559$	12.8167	0.542088
$560$	0	0
$561$	−5.41360	−0.228562
$562$	0	0
$563$	35.3476	1.48972	0.744861	−	0.667219i	$-0.232516\pi$
$563$	35.3476	1.48972	0.744861	+	0.667219i	$0.232516\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.32424	0.139605
$568$	0	0
$569$	36.2483	1.51961	0.759804	−	0.650152i	$-0.225295\pi$
$569$	36.2483	1.51961	0.759804	+	0.650152i	$0.225295\pi$
$570$	0	0
$571$	7.96581	0.333359	0.166679	−	0.986011i	$-0.446696\pi$
$571$	7.96581	0.333359	0.166679	+	0.986011i	$0.446696\pi$
$572$	0	0
$573$	23.2970	0.973245
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−21.8219	−0.908458	−0.454229	−	0.890885i	$-0.650085\pi$
$577$	−21.8219	−0.908458	−0.454229	+	0.890885i	$0.650085\pi$
$578$	0	0
$579$	−12.6386	−0.525242
$580$	0	0
$581$	8.89487	0.369021
$582$	0	0
$583$	59.2414	2.45353
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.3518	−1.08766	−0.543828	−	0.839197i	$-0.683026\pi$
$587$	−26.3518	−1.08766	−0.543828	+	0.839197i	$0.683026\pi$
$588$	0	0
$589$	−30.3489	−1.25050
$590$	0	0
$591$	−24.0913	−0.990984
$592$	0	0
$593$	−21.8522	−0.897361	−0.448680	−	0.893692i	$-0.648106\pi$
$593$	−21.8522	−0.897361	−0.448680	+	0.893692i	$0.648106\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.1176	−0.495940
$598$	0	0
$599$	20.4822	0.836881	0.418441	−	0.908244i	$-0.362577\pi$
$599$	20.4822	0.836881	0.418441	+	0.908244i	$0.362577\pi$
$600$	0	0
$601$	31.1380	1.27014	0.635072	−	0.772453i	$-0.280970\pi$
$601$	31.1380	1.27014	0.635072	+	0.772453i	$0.280970\pi$
$602$	0	0
$603$	10.0233	0.408181
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−11.6715	−0.473732	−0.236866	−	0.971542i	$-0.576120\pi$
$607$	−11.6715	−0.473732	−0.236866	+	0.971542i	$0.576120\pi$
$608$	0	0
$609$	18.7770	0.760880
$610$	0	0
$611$	−6.89455	−0.278924
$612$	0	0
$613$	−41.3758	−1.67115	−0.835577	−	0.549374i	$-0.814866\pi$
$613$	−41.3758	−1.67115	−0.835577	+	0.549374i	$0.814866\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.7359	1.72048	0.860240	−	0.509889i	$-0.170313\pi$
$617$	42.7359	1.72048	0.860240	+	0.509889i	$0.170313\pi$
$618$	0	0
$619$	7.44087	0.299074	0.149537	−	0.988756i	$-0.452222\pi$
$619$	7.44087	0.299074	0.149537	+	0.988756i	$0.452222\pi$
$620$	0	0
$621$	−4.16820	−0.167264
$622$	0	0
$623$	−12.2595	−0.491167
$624$	0	0
$625$	0	0
$626$	0	0
$627$	19.6209	0.783583
$628$	0	0
$629$	9.72210	0.387645
$630$	0	0
$631$	−17.7221	−0.705506	−0.352753	−	0.935717i	$-0.614754\pi$
$631$	−17.7221	−0.705506	−0.352753	+	0.935717i	$0.614754\pi$
$632$	0	0
$633$	−0.168197	−0.00668524
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.05060	0.160491
$638$	0	0
$639$	8.05060	0.318477
$640$	0	0
$641$	−31.4764	−1.24324	−0.621621	−	0.783319i	$-0.713525\pi$
$641$	−31.4764	−1.24324	−0.621621	+	0.783319i	$0.713525\pi$
$642$	0	0
$643$	−40.7234	−1.60597	−0.802987	−	0.595997i	$-0.796757\pi$
$643$	−40.7234	−1.60597	−0.802987	+	0.595997i	$0.796757\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.1255	1.73475	0.867376	−	0.497653i	$-0.165805\pi$
$647$	44.1255	1.73475	0.867376	+	0.497653i	$0.165805\pi$
$648$	0	0
$649$	4.41263	0.173211
$650$	0	0
$651$	−24.9067	−0.976171
$652$	0	0
$653$	6.02301	0.235699	0.117849	−	0.993031i	$-0.462400\pi$
$653$	6.02301	0.235699	0.117849	+	0.993031i	$0.462400\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.4034	0.405875
$658$	0	0
$659$	−5.26810	−0.205216	−0.102608	−	0.994722i	$-0.532719\pi$
$659$	−5.26810	−0.205216	−0.102608	+	0.994722i	$0.532719\pi$
$660$	0	0
$661$	8.72144	0.339225	0.169612	−	0.985511i	$-0.445748\pi$
$661$	8.72144	0.339225	0.169612	+	0.985511i	$0.445748\pi$
$662$	0	0
$663$	−1.11760	−0.0434040
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.5440	−0.911628
$668$	0	0
$669$	−25.5440	−0.987589
$670$	0	0
$671$	11.9603	0.461721
$672$	0	0
$673$	6.04931	0.233184	0.116592	−	0.993180i	$-0.462803\pi$
$673$	6.04931	0.233184	0.116592	+	0.993180i	$0.462803\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.6636	−0.525135	−0.262568	−	0.964914i	$-0.584569\pi$
$677$	−13.6636	−0.525135	−0.262568	+	0.964914i	$0.584569\pi$
$678$	0	0
$679$	−22.1012	−0.848166
$680$	0	0
$681$	6.70995	0.257126
$682$	0	0
$683$	−3.41555	−0.130692	−0.0653462	−	0.997863i	$-0.520815\pi$
$683$	−3.41555	−0.130692	−0.0653462	+	0.997863i	$0.520815\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.92182	0.111474
$688$	0	0
$689$	12.2300	0.465925
$690$	0	0
$691$	20.6153	0.784242	0.392121	−	0.919914i	$-0.371741\pi$
$691$	20.6153	0.784242	0.392121	+	0.919914i	$0.371741\pi$
$692$	0	0
$693$	16.1025	0.611683
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.30026	−0.238639
$698$	0	0
$699$	−1.68791	−0.0638425
$700$	0	0
$701$	−13.5861	−0.513138	−0.256569	−	0.966526i	$-0.582592\pi$
$701$	−13.5861	−0.513138	−0.256569	+	0.966526i	$0.582592\pi$
$702$	0	0
$703$	−35.2365	−1.32897
$704$	0	0
$705$	0	0
$706$	0	0
$707$	38.8906	1.46263
$708$	0	0
$709$	−1.39616	−0.0524339	−0.0262169	−	0.999656i	$-0.508346\pi$
$709$	−1.39616	−0.0524339	−0.0262169	+	0.999656i	$0.508346\pi$
$710$	0	0
$711$	−10.6991	−0.401247
$712$	0	0
$713$	31.2300	1.16957
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.2079	−0.605296
$718$	0	0
$719$	−37.6130	−1.40273	−0.701365	−	0.712803i	$-0.747425\pi$
$719$	−37.6130	−1.40273	−0.701365	+	0.712803i	$0.747425\pi$
$720$	0	0
$721$	−63.6695	−2.37118
$722$	0	0
$723$	12.1682	0.452540
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.0191	1.29879	0.649393	−	0.760453i	$-0.275023\pi$
$727$	35.0191	1.29879	0.649393	+	0.760453i	$0.275023\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.3239	−0.529790
$732$	0	0
$733$	−17.8824	−0.660502	−0.330251	−	0.943893i	$-0.607133\pi$
$733$	−17.8824	−0.660502	−0.330251	+	0.943893i	$0.607133\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	48.5525	1.78846
$738$	0	0
$739$	34.2034	1.25819	0.629095	−	0.777328i	$-0.283425\pi$
$739$	34.2034	1.25819	0.629095	+	0.777328i	$0.283425\pi$
$740$	0	0
$741$	4.05060	0.148802
$742$	0	0
$743$	−4.96155	−0.182022	−0.0910109	−	0.995850i	$-0.529010\pi$
$743$	−4.96155	−0.182022	−0.0910109	+	0.995850i	$0.529010\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.67576	0.0979008
$748$	0	0
$749$	−42.7740	−1.56293
$750$	0	0
$751$	−30.8574	−1.12600	−0.563001	−	0.826456i	$-0.690353\pi$
$751$	−30.8574	−1.12600	−0.563001	+	0.826456i	$0.690353\pi$
$752$	0	0
$753$	15.6373	0.569855
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.9534	−1.27040	−0.635201	−	0.772347i	$-0.719083\pi$
$757$	−34.9534	−1.27040	−0.635201	+	0.772347i	$0.719083\pi$
$758$	0	0
$759$	−20.1906	−0.732871
$760$	0	0
$761$	−9.79304	−0.354997	−0.177499	−	0.984121i	$-0.556801\pi$
$761$	−9.79304	−0.354997	−0.177499	+	0.984121i	$0.556801\pi$
$762$	0	0
$763$	−36.9402	−1.33733
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.910956	0.0328927
$768$	0	0
$769$	1.66555	0.0600613	0.0300306	−	0.999549i	$-0.490440\pi$
$769$	1.66555	0.0600613	0.0300306	+	0.999549i	$0.490440\pi$
$770$	0	0
$771$	−11.5210	−0.414919
$772$	0	0
$773$	−32.7273	−1.17712	−0.588560	−	0.808454i	$-0.700305\pi$
$773$	−32.7273	−1.17712	−0.588560	+	0.808454i	$0.700305\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−28.9179	−1.03742
$778$	0	0
$779$	22.8345	0.818130
$780$	0	0
$781$	38.9967	1.39541
$782$	0	0
$783$	5.64849	0.201860
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.4484	0.479382	0.239691	−	0.970849i	$-0.422954\pi$
$787$	13.4484	0.479382	0.239691	+	0.970849i	$0.422954\pi$
$788$	0	0
$789$	1.41852	0.0505005
$790$	0	0
$791$	−35.7346	−1.27057
$792$	0	0
$793$	2.46911	0.0876808
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.6544	−0.908727	−0.454363	−	0.890816i	$-0.650133\pi$
$797$	−25.6544	−0.908727	−0.454363	+	0.890816i	$0.650133\pi$
$798$	0	0
$799$	7.70535	0.272596
$800$	0	0
$801$	−3.68791	−0.130306
$802$	0	0
$803$	50.3936	1.77835
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.3982	0.929260
$808$	0	0
$809$	−29.9915	−1.05444	−0.527222	−	0.849727i	$-0.676766\pi$
$809$	−29.9915	−1.05444	−0.527222	+	0.849727i	$0.676766\pi$
$810$	0	0
$811$	11.1133	0.390242	0.195121	−	0.980779i	$-0.437490\pi$
$811$	11.1133	0.390242	0.195121	+	0.980779i	$0.437490\pi$
$812$	0	0
$813$	27.1133	0.950907
$814$	0	0
$815$	0	0
$816$	0	0
$817$	51.9152	1.81628
$818$	0	0
$819$	3.32424	0.116158
$820$	0	0
$821$	−1.62090	−0.0565699	−0.0282850	−	0.999600i	$-0.509005\pi$
$821$	−1.62090	−0.0565699	−0.0282850	+	0.999600i	$0.509005\pi$
$822$	0	0
$823$	−40.5007	−1.41176	−0.705882	−	0.708329i	$-0.749449\pi$
$823$	−40.5007	−1.41176	−0.705882	+	0.708329i	$0.749449\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.95560	−0.172323	−0.0861616	−	0.996281i	$-0.527460\pi$
$827$	−4.95560	−0.172323	−0.0861616	+	0.996281i	$0.527460\pi$
$828$	0	0
$829$	−26.2588	−0.912007	−0.456004	−	0.889978i	$-0.650720\pi$
$829$	−26.2588	−0.912007	−0.456004	+	0.889978i	$0.650720\pi$
$830$	0	0
$831$	11.0861	0.384572
$832$	0	0
$833$	−4.52695	−0.156849
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.49244	−0.258977
$838$	0	0
$839$	32.0565	1.10671	0.553357	−	0.832944i	$-0.313346\pi$
$839$	32.0565	1.10671	0.553357	+	0.832944i	$0.313346\pi$
$840$	0	0
$841$	2.90541	0.100187
$842$	0	0
$843$	9.17938	0.316154
$844$	0	0
$845$	0	0
$846$	0	0
$847$	41.4330	1.42365
$848$	0	0
$849$	31.8686	1.09373
$850$	0	0
$851$	36.2595	1.24296
$852$	0	0
$853$	−52.5775	−1.80022	−0.900110	−	0.435662i	$-0.856514\pi$
$853$	−52.5775	−1.80022	−0.900110	+	0.435662i	$0.856514\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.5940	−0.566839	−0.283419	−	0.958996i	$-0.591469\pi$
$857$	−16.5940	−0.566839	−0.283419	+	0.958996i	$0.591469\pi$
$858$	0	0
$859$	27.0742	0.923761	0.461881	−	0.886942i	$-0.347175\pi$
$859$	27.0742	0.923761	0.461881	+	0.886942i	$0.347175\pi$
$860$	0	0
$861$	18.7398	0.638650
$862$	0	0
$863$	−9.99903	−0.340371	−0.170185	−	0.985412i	$-0.554437\pi$
$863$	−9.99903	−0.340371	−0.170185	+	0.985412i	$0.554437\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15.7510	−0.534931
$868$	0	0
$869$	−51.8259	−1.75807
$870$	0	0
$871$	10.0233	0.339628
$872$	0	0
$873$	−6.64849	−0.225017
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.10248	0.104763	0.0523817	−	0.998627i	$-0.483319\pi$
$877$	3.10248	0.104763	0.0523817	+	0.998627i	$0.483319\pi$
$878$	0	0
$879$	9.14061	0.308305
$880$	0	0
$881$	8.19121	0.275969	0.137984	−	0.990434i	$-0.455938\pi$
$881$	8.19121	0.275969	0.137984	+	0.990434i	$0.455938\pi$
$882$	0	0
$883$	39.1649	1.31800	0.659002	−	0.752141i	$-0.270979\pi$
$883$	39.1649	1.31800	0.659002	+	0.752141i	$0.270979\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.2543	−0.713649	−0.356824	−	0.934172i	$-0.616141\pi$
$887$	−21.2543	−0.713649	−0.356824	+	0.934172i	$0.616141\pi$
$888$	0	0
$889$	−37.7221	−1.26516
$890$	0	0
$891$	4.84395	0.162279
$892$	0	0
$893$	−27.9270	−0.934543
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.16820	−0.139172
$898$	0	0
$899$	−42.3210	−1.41148
$900$	0	0
$901$	−13.6682	−0.455354
$902$	0	0
$903$	42.6058	1.41783
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25.0618	0.832163	0.416081	−	0.909327i	$-0.363403\pi$
$907$	25.0618	0.832163	0.416081	+	0.909327i	$0.363403\pi$
$908$	0	0
$909$	11.6991	0.388034
$910$	0	0
$911$	10.3246	0.342068	0.171034	−	0.985265i	$-0.445289\pi$
$911$	10.3246	0.342068	0.171034	+	0.985265i	$0.445289\pi$
$912$	0	0
$913$	12.9612	0.428954
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−27.8804	−0.920692
$918$	0	0
$919$	−20.0605	−0.661734	−0.330867	−	0.943677i	$-0.607341\pi$
$919$	−20.0605	−0.661734	−0.330867	+	0.943677i	$0.607341\pi$
$920$	0	0
$921$	−7.40211	−0.243908
$922$	0	0
$923$	8.05060	0.264989
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−19.1531	−0.629070
$928$	0	0
$929$	38.8692	1.27526	0.637629	−	0.770344i	$-0.279915\pi$
$929$	38.8692	1.27526	0.637629	+	0.770344i	$0.279915\pi$
$930$	0	0
$931$	16.4073	0.537729
$932$	0	0
$933$	21.5098	0.704200
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−53.5007	−1.74779	−0.873895	−	0.486115i	$-0.838414\pi$
$937$	−53.5007	−1.74779	−0.873895	+	0.486115i	$0.838414\pi$
$938$	0	0
$939$	11.1458	0.363731
$940$	0	0
$941$	−20.1906	−0.658193	−0.329097	−	0.944296i	$-0.606744\pi$
$941$	−20.1906	−0.658193	−0.329097	+	0.944296i	$0.606744\pi$
$942$	0	0
$943$	−23.4974	−0.765181
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.4839	−1.15307	−0.576536	−	0.817071i	$-0.695596\pi$
$947$	−35.4839	−1.15307	−0.576536	+	0.817071i	$0.695596\pi$
$948$	0	0
$949$	10.4034	0.337709
$950$	0	0
$951$	−27.0960	−0.878647
$952$	0	0
$953$	1.02630	0.0332450	0.0166225	−	0.999862i	$-0.494709\pi$
$953$	1.02630	0.0332450	0.0166225	+	0.999862i	$0.494709\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	27.3610	0.884456
$958$	0	0
$959$	32.5959	1.05258
$960$	0	0
$961$	25.1367	0.810860
$962$	0	0
$963$	−12.8673	−0.414642
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.8465	−0.541748	−0.270874	−	0.962615i	$-0.587313\pi$
$967$	−16.8465	−0.541748	−0.270874	+	0.962615i	$0.587313\pi$
$968$	0	0
$969$	−4.52695	−0.145427
$970$	0	0
$971$	51.3153	1.64679	0.823394	−	0.567471i	$-0.192078\pi$
$971$	51.3153	1.64679	0.823394	+	0.567471i	$0.192078\pi$
$972$	0	0
$973$	−34.0243	−1.09077
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.2733	−1.60839	−0.804193	−	0.594368i	$-0.797402\pi$
$977$	−50.2733	−1.60839	−0.804193	+	0.594368i	$0.797402\pi$
$978$	0	0
$979$	−17.8640	−0.570937
$980$	0	0
$981$	−11.1124	−0.354791
$982$	0	0
$983$	30.5319	0.973815	0.486908	−	0.873454i	$-0.338125\pi$
$983$	30.5319	0.973815	0.486908	+	0.873454i	$0.338125\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−22.9192	−0.729525
$988$	0	0
$989$	−53.4225	−1.69874
$990$	0	0
$991$	−20.8876	−0.663517	−0.331759	−	0.943364i	$-0.607642\pi$
$991$	−20.8876	−0.663517	−0.331759	+	0.943364i	$0.607642\pi$
$992$	0	0
$993$	10.0670	0.319467
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.7727	0.467856	0.233928	−	0.972254i	$-0.424842\pi$
$997$	14.7727	0.467856	0.233928	+	0.972254i	$0.424842\pi$
$998$	0	0
$999$	−8.69908	−0.275227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bx.1.4	yes	4
5.4	even	2		7800.2.a.bu.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bu.1.1	✓	4	5.4	even	2
7800.2.a.bx.1.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.32424 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.32424 q^{7} +1.00000 q^{9} +4.84395 q^{11} +1.00000 q^{13} -1.11760 q^{17} +4.05060 q^{19} +3.32424 q^{21} -4.16820 q^{23} +1.00000 q^{27} +5.64849 q^{29} -7.49244 q^{31} +4.84395 q^{33} -8.69908 q^{37} +1.00000 q^{39} +5.63731 q^{41} +12.8167 q^{43} -6.89455 q^{47} +4.05060 q^{49} -1.11760 q^{51} +12.2300 q^{53} +4.05060 q^{57} +0.910956 q^{59} +2.46911 q^{61} +3.32424 q^{63} +10.0233 q^{67} -4.16820 q^{69} +8.05060 q^{71} +10.4034 q^{73} +16.1025 q^{77} -10.6991 q^{79} +1.00000 q^{81} +2.67576 q^{83} +5.64849 q^{87} -3.68791 q^{89} +3.32424 q^{91} -7.49244 q^{93} -6.64849 q^{97} +4.84395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9} + 5 q^{11} + 4 q^{13} + 7 q^{17} + 3 q^{19} + 3 q^{21} + 8 q^{23} + 4 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - q^{37} + 4 q^{39} + 7 q^{41} + 6 q^{43} + 3 q^{49} + 7 q^{51} + 6 q^{53} + 3 q^{57} - 9 q^{59} + 19 q^{61} + 3 q^{63} - 4 q^{67} + 8 q^{69} + 19 q^{71} - 6 q^{73} - 17 q^{77} - 9 q^{79} + 4 q^{81} + 21 q^{83} + 2 q^{87} + 14 q^{89} + 3 q^{91} + 5 q^{93} - 6 q^{97} + 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 3 * q^7 + 4 * q^9 + 5 * q^11 + 4 * q^13 + 7 * q^17 + 3 * q^19 + 3 * q^21 + 8 * q^23 + 4 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - q^37 + 4 * q^39 + 7 * q^41 + 6 * q^43 + 3 * q^49 + 7 * q^51 + 6 * q^53 + 3 * q^57 - 9 * q^59 + 19 * q^61 + 3 * q^63 - 4 * q^67 + 8 * q^69 + 19 * q^71 - 6 * q^73 - 17 * q^77 - 9 * q^79 + 4 * q^81 + 21 * q^83 + 2 * q^87 + 14 * q^89 + 3 * q^91 + 5 * q^93 - 6 * q^97 + 5 * q^99

Introduction

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