LMFDB - Embedded newform 7350.2.a.dg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.41421 q^{11} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{16} +5.82843 q^{17} -1.00000 q^{18} +1.41421 q^{19} -3.41421 q^{22} -0.414214 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} -2.41421 q^{31} -1.00000 q^{32} +3.41421 q^{33} -5.82843 q^{34} +1.00000 q^{36} +7.07107 q^{37} -1.41421 q^{38} +3.00000 q^{39} -5.82843 q^{41} +1.58579 q^{43} +3.41421 q^{44} +0.414214 q^{46} -1.65685 q^{47} +1.00000 q^{48} +5.82843 q^{51} +3.00000 q^{52} +12.6569 q^{53} -1.00000 q^{54} +1.41421 q^{57} -1.00000 q^{58} +7.24264 q^{59} -0.171573 q^{61} +2.41421 q^{62} +1.00000 q^{64} -3.41421 q^{66} -15.3137 q^{67} +5.82843 q^{68} -0.414214 q^{69} -9.31371 q^{71} -1.00000 q^{72} +15.0711 q^{73} -7.07107 q^{74} +1.41421 q^{76} -3.00000 q^{78} +1.07107 q^{79} +1.00000 q^{81} +5.82843 q^{82} -0.0710678 q^{83} -1.58579 q^{86} +1.00000 q^{87} -3.41421 q^{88} +4.00000 q^{89} -0.414214 q^{92} -2.41421 q^{93} +1.65685 q^{94} -1.00000 q^{96} -0.343146 q^{97} +3.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 4 q^{22} + 2 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{36} + 6 q^{39} - 6 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} + 8 q^{47} + 2 q^{48} + 6 q^{51} + 6 q^{52} + 14 q^{53} - 2 q^{54} - 2 q^{58} + 6 q^{59} - 6 q^{61} + 2 q^{62} + 2 q^{64} - 4 q^{66} - 8 q^{67} + 6 q^{68} + 2 q^{69} + 4 q^{71} - 2 q^{72} + 16 q^{73} - 6 q^{78} - 12 q^{79} + 2 q^{81} + 6 q^{82} + 14 q^{83} - 6 q^{86} + 2 q^{87} - 4 q^{88} + 8 q^{89} + 2 q^{92} - 2 q^{93} - 8 q^{94} - 2 q^{96} - 12 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^16 + 6 * q^17 - 2 * q^18 - 4 * q^22 + 2 * q^23 - 2 * q^24 - 6 * q^26 + 2 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^36 + 6 * q^39 - 6 * q^41 + 6 * q^43 + 4 * q^44 - 2 * q^46 + 8 * q^47 + 2 * q^48 + 6 * q^51 + 6 * q^52 + 14 * q^53 - 2 * q^54 - 2 * q^58 + 6 * q^59 - 6 * q^61 + 2 * q^62 + 2 * q^64 - 4 * q^66 - 8 * q^67 + 6 * q^68 + 2 * q^69 + 4 * q^71 - 2 * q^72 + 16 * q^73 - 6 * q^78 - 12 * q^79 + 2 * q^81 + 6 * q^82 + 14 * q^83 - 6 * q^86 + 2 * q^87 - 4 * q^88 + 8 * q^89 + 2 * q^92 - 2 * q^93 - 8 * q^94 - 2 * q^96 - 12 * q^97 + 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.41421	1.02942	0.514712	−	0.857363i	$-0.327899\pi$
$11$	3.41421	1.02942	0.514712	+	0.857363i	$0.327899\pi$
$12$	1.00000	0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	5.82843	1.41360	0.706801	−	0.707413i	$-0.250138\pi$
$17$	5.82843	1.41360	0.706801	+	0.707413i	$0.250138\pi$
$18$	−1.00000	−0.235702
$19$	1.41421	0.324443	0.162221	−	0.986754i	$-0.448134\pi$
$19$	1.41421	0.324443	0.162221	+	0.986754i	$0.448134\pi$
$20$	0	0
$21$	0	0
$22$	−3.41421	−0.727913
$23$	−0.414214	−0.0863695	−0.0431847	−	0.999067i	$-0.513750\pi$
$23$	−0.414214	−0.0863695	−0.0431847	+	0.999067i	$0.513750\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−3.00000	−0.588348
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−2.41421	−0.433606	−0.216803	−	0.976215i	$-0.569563\pi$
$31$	−2.41421	−0.433606	−0.216803	+	0.976215i	$0.569563\pi$
$32$	−1.00000	−0.176777
$33$	3.41421	0.594338
$34$	−5.82843	−0.999567
$35$	0	0
$36$	1.00000	0.166667
$37$	7.07107	1.16248	0.581238	−	0.813733i	$-0.302568\pi$
$37$	7.07107	1.16248	0.581238	+	0.813733i	$0.302568\pi$
$38$	−1.41421	−0.229416
$39$	3.00000	0.480384
$40$	0	0
$41$	−5.82843	−0.910247	−0.455124	−	0.890428i	$-0.650405\pi$
$41$	−5.82843	−0.910247	−0.455124	+	0.890428i	$0.650405\pi$
$42$	0	0
$43$	1.58579	0.241830	0.120915	−	0.992663i	$-0.461417\pi$
$43$	1.58579	0.241830	0.120915	+	0.992663i	$0.461417\pi$
$44$	3.41421	0.514712
$45$	0	0
$46$	0.414214	0.0610725
$47$	−1.65685	−0.241677	−0.120839	−	0.992672i	$-0.538558\pi$
$47$	−1.65685	−0.241677	−0.120839	+	0.992672i	$0.538558\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	5.82843	0.816143
$52$	3.00000	0.416025
$53$	12.6569	1.73855	0.869276	−	0.494326i	$-0.164585\pi$
$53$	12.6569	1.73855	0.869276	+	0.494326i	$0.164585\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	1.41421	0.187317
$58$	−1.00000	−0.131306
$59$	7.24264	0.942912	0.471456	−	0.881890i	$-0.343729\pi$
$59$	7.24264	0.942912	0.471456	+	0.881890i	$0.343729\pi$
$60$	0	0
$61$	−0.171573	−0.0219677	−0.0109838	−	0.999940i	$-0.503496\pi$
$61$	−0.171573	−0.0219677	−0.0109838	+	0.999940i	$0.503496\pi$
$62$	2.41421	0.306605
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.41421	−0.420261
$67$	−15.3137	−1.87087	−0.935434	−	0.353502i	$-0.884991\pi$
$67$	−15.3137	−1.87087	−0.935434	+	0.353502i	$0.884991\pi$
$68$	5.82843	0.706801
$69$	−0.414214	−0.0498655
$70$	0	0
$71$	−9.31371	−1.10533	−0.552667	−	0.833402i	$-0.686390\pi$
$71$	−9.31371	−1.10533	−0.552667	+	0.833402i	$0.686390\pi$
$72$	−1.00000	−0.117851
$73$	15.0711	1.76394	0.881968	−	0.471310i	$-0.156219\pi$
$73$	15.0711	1.76394	0.881968	+	0.471310i	$0.156219\pi$
$74$	−7.07107	−0.821995
$75$	0	0
$76$	1.41421	0.162221
$77$	0	0
$78$	−3.00000	−0.339683
$79$	1.07107	0.120505	0.0602523	−	0.998183i	$-0.480809\pi$
$79$	1.07107	0.120505	0.0602523	+	0.998183i	$0.480809\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.82843	0.643642
$83$	−0.0710678	−0.00780071	−0.00390035	−	0.999992i	$-0.501242\pi$
$83$	−0.0710678	−0.00780071	−0.00390035	+	0.999992i	$0.501242\pi$
$84$	0	0
$85$	0	0
$86$	−1.58579	−0.171000
$87$	1.00000	0.107211
$88$	−3.41421	−0.363956
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	−0.414214	−0.0431847
$93$	−2.41421	−0.250342
$94$	1.65685	0.170891
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	0	0
$99$	3.41421	0.343141
$100$	0	0
$101$	5.65685	0.562878	0.281439	−	0.959579i	$-0.409188\pi$
$101$	5.65685	0.562878	0.281439	+	0.959579i	$0.409188\pi$
$102$	−5.82843	−0.577100
$103$	−0.0710678	−0.00700252	−0.00350126	−	0.999994i	$-0.501114\pi$
$103$	−0.0710678	−0.00700252	−0.00350126	+	0.999994i	$0.501114\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−12.6569	−1.22934
$107$	6.48528	0.626956	0.313478	−	0.949595i	$-0.398506\pi$
$107$	6.48528	0.626956	0.313478	+	0.949595i	$0.398506\pi$
$108$	1.00000	0.0962250
$109$	−11.3137	−1.08366	−0.541828	−	0.840489i	$-0.682268\pi$
$109$	−11.3137	−1.08366	−0.541828	+	0.840489i	$0.682268\pi$
$110$	0	0
$111$	7.07107	0.671156
$112$	0	0
$113$	6.58579	0.619539	0.309769	−	0.950812i	$-0.399748\pi$
$113$	6.58579	0.619539	0.309769	+	0.950812i	$0.399748\pi$
$114$	−1.41421	−0.132453
$115$	0	0
$116$	1.00000	0.0928477
$117$	3.00000	0.277350
$118$	−7.24264	−0.666739
$119$	0	0
$120$	0	0
$121$	0.656854	0.0597140
$122$	0.171573	0.0155335
$123$	−5.82843	−0.525532
$124$	−2.41421	−0.216803
$125$	0	0
$126$	0	0
$127$	−3.17157	−0.281432	−0.140716	−	0.990050i	$-0.544940\pi$
$127$	−3.17157	−0.281432	−0.140716	+	0.990050i	$0.544940\pi$
$128$	−1.00000	−0.0883883
$129$	1.58579	0.139621
$130$	0	0
$131$	−18.8284	−1.64505	−0.822524	−	0.568731i	$-0.807435\pi$
$131$	−18.8284	−1.64505	−0.822524	+	0.568731i	$0.807435\pi$
$132$	3.41421	0.297169
$133$	0	0
$134$	15.3137	1.32290
$135$	0	0
$136$	−5.82843	−0.499784
$137$	−23.2132	−1.98324	−0.991619	−	0.129197i	$-0.958760\pi$
$137$	−23.2132	−1.98324	−0.991619	+	0.129197i	$0.958760\pi$
$138$	0.414214	0.0352602
$139$	−15.5563	−1.31947	−0.659736	−	0.751497i	$-0.729332\pi$
$139$	−15.5563	−1.31947	−0.659736	+	0.751497i	$0.729332\pi$
$140$	0	0
$141$	−1.65685	−0.139532
$142$	9.31371	0.781589
$143$	10.2426	0.856533
$144$	1.00000	0.0833333
$145$	0	0
$146$	−15.0711	−1.24729
$147$	0	0
$148$	7.07107	0.581238
$149$	4.65685	0.381504	0.190752	−	0.981638i	$-0.438907\pi$
$149$	4.65685	0.381504	0.190752	+	0.981638i	$0.438907\pi$
$150$	0	0
$151$	7.89949	0.642852	0.321426	−	0.946935i	$-0.395838\pi$
$151$	7.89949	0.642852	0.321426	+	0.946935i	$0.395838\pi$
$152$	−1.41421	−0.114708
$153$	5.82843	0.471200
$154$	0	0
$155$	0	0
$156$	3.00000	0.240192
$157$	−19.7990	−1.58013	−0.790066	−	0.613022i	$-0.789954\pi$
$157$	−19.7990	−1.58013	−0.790066	+	0.613022i	$0.789954\pi$
$158$	−1.07107	−0.0852096
$159$	12.6569	1.00375
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	5.24264	0.410635	0.205318	−	0.978695i	$-0.434177\pi$
$163$	5.24264	0.410635	0.205318	+	0.978695i	$0.434177\pi$
$164$	−5.82843	−0.455124
$165$	0	0
$166$	0.0710678	0.00551593
$167$	−8.82843	−0.683164	−0.341582	−	0.939852i	$-0.610963\pi$
$167$	−8.82843	−0.683164	−0.341582	+	0.939852i	$0.610963\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	1.41421	0.108148
$172$	1.58579	0.120915
$173$	14.2426	1.08285	0.541424	−	0.840750i	$-0.317885\pi$
$173$	14.2426	1.08285	0.541424	+	0.840750i	$0.317885\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	3.41421	0.257356
$177$	7.24264	0.544390
$178$	−4.00000	−0.299813
$179$	−5.07107	−0.379029	−0.189515	−	0.981878i	$-0.560691\pi$
$179$	−5.07107	−0.379029	−0.189515	+	0.981878i	$0.560691\pi$
$180$	0	0
$181$	−23.3137	−1.73289	−0.866447	−	0.499269i	$-0.833602\pi$
$181$	−23.3137	−1.73289	−0.866447	+	0.499269i	$0.833602\pi$
$182$	0	0
$183$	−0.171573	−0.0126830
$184$	0.414214	0.0305362
$185$	0	0
$186$	2.41421	0.177019
$187$	19.8995	1.45520
$188$	−1.65685	−0.120839
$189$	0	0
$190$	0	0
$191$	16.0711	1.16286	0.581431	−	0.813596i	$-0.302493\pi$
$191$	16.0711	1.16286	0.581431	+	0.813596i	$0.302493\pi$
$192$	1.00000	0.0721688
$193$	−6.82843	−0.491521	−0.245760	−	0.969331i	$-0.579038\pi$
$193$	−6.82843	−0.491521	−0.245760	+	0.969331i	$0.579038\pi$
$194$	0.343146	0.0246364
$195$	0	0
$196$	0	0
$197$	23.1421	1.64881	0.824404	−	0.566001i	$-0.191510\pi$
$197$	23.1421	1.64881	0.824404	+	0.566001i	$0.191510\pi$
$198$	−3.41421	−0.242638
$199$	19.6569	1.39344	0.696719	−	0.717344i	$-0.254643\pi$
$199$	19.6569	1.39344	0.696719	+	0.717344i	$0.254643\pi$
$200$	0	0
$201$	−15.3137	−1.08015
$202$	−5.65685	−0.398015
$203$	0	0
$204$	5.82843	0.408072
$205$	0	0
$206$	0.0710678	0.00495153
$207$	−0.414214	−0.0287898
$208$	3.00000	0.208013
$209$	4.82843	0.333989
$210$	0	0
$211$	−26.5563	−1.82821	−0.914107	−	0.405473i	$-0.867107\pi$
$211$	−26.5563	−1.82821	−0.914107	+	0.405473i	$0.867107\pi$
$212$	12.6569	0.869276
$213$	−9.31371	−0.638165
$214$	−6.48528	−0.443325
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	11.3137	0.766261
$219$	15.0711	1.01841
$220$	0	0
$221$	17.4853	1.17619
$222$	−7.07107	−0.474579
$223$	24.5563	1.64441	0.822207	−	0.569188i	$-0.192742\pi$
$223$	24.5563	1.64441	0.822207	+	0.569188i	$0.192742\pi$
$224$	0	0
$225$	0	0
$226$	−6.58579	−0.438080
$227$	25.3848	1.68485	0.842423	−	0.538816i	$-0.181128\pi$
$227$	25.3848	1.68485	0.842423	+	0.538816i	$0.181128\pi$
$228$	1.41421	0.0936586
$229$	29.3137	1.93710	0.968552	−	0.248810i	$-0.0800396\pi$
$229$	29.3137	1.93710	0.968552	+	0.248810i	$0.0800396\pi$
$230$	0	0
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	14.2426	0.933066	0.466533	−	0.884504i	$-0.345503\pi$
$233$	14.2426	0.933066	0.466533	+	0.884504i	$0.345503\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	7.24264	0.471456
$237$	1.07107	0.0695733
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−0.656854	−0.0422242
$243$	1.00000	0.0641500
$244$	−0.171573	−0.0109838
$245$	0	0
$246$	5.82843	0.371607
$247$	4.24264	0.269953
$248$	2.41421	0.153303
$249$	−0.0710678	−0.00450374
$250$	0	0
$251$	−1.92893	−0.121753	−0.0608766	−	0.998145i	$-0.519390\pi$
$251$	−1.92893	−0.121753	−0.0608766	+	0.998145i	$0.519390\pi$
$252$	0	0
$253$	−1.41421	−0.0889108
$254$	3.17157	0.199002
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.3431	0.832323	0.416161	−	0.909291i	$-0.363375\pi$
$257$	13.3431	0.832323	0.416161	+	0.909291i	$0.363375\pi$
$258$	−1.58579	−0.0987268
$259$	0	0
$260$	0	0
$261$	1.00000	0.0618984
$262$	18.8284	1.16322
$263$	−10.5563	−0.650932	−0.325466	−	0.945554i	$-0.605521\pi$
$263$	−10.5563	−0.650932	−0.325466	+	0.945554i	$0.605521\pi$
$264$	−3.41421	−0.210130
$265$	0	0
$266$	0	0
$267$	4.00000	0.244796
$268$	−15.3137	−0.935434
$269$	0.443651	0.0270499	0.0135249	−	0.999909i	$-0.495695\pi$
$269$	0.443651	0.0270499	0.0135249	+	0.999909i	$0.495695\pi$
$270$	0	0
$271$	−11.5147	−0.699469	−0.349735	−	0.936849i	$-0.613728\pi$
$271$	−11.5147	−0.699469	−0.349735	+	0.936849i	$0.613728\pi$
$272$	5.82843	0.353400
$273$	0	0
$274$	23.2132	1.40236
$275$	0	0
$276$	−0.414214	−0.0249327
$277$	−24.0416	−1.44452	−0.722261	−	0.691621i	$-0.756897\pi$
$277$	−24.0416	−1.44452	−0.722261	+	0.691621i	$0.756897\pi$
$278$	15.5563	0.933008
$279$	−2.41421	−0.144535
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	1.65685	0.0986642
$283$	−0.242641	−0.0144235	−0.00721175	−	0.999974i	$-0.502296\pi$
$283$	−0.242641	−0.0144235	−0.00721175	+	0.999974i	$0.502296\pi$
$284$	−9.31371	−0.552667
$285$	0	0
$286$	−10.2426	−0.605660
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	16.9706	0.998268
$290$	0	0
$291$	−0.343146	−0.0201156
$292$	15.0711	0.881968
$293$	−11.3137	−0.660954	−0.330477	−	0.943814i	$-0.607210\pi$
$293$	−11.3137	−0.660954	−0.330477	+	0.943814i	$0.607210\pi$
$294$	0	0
$295$	0	0
$296$	−7.07107	−0.410997
$297$	3.41421	0.198113
$298$	−4.65685	−0.269764
$299$	−1.24264	−0.0718638
$300$	0	0
$301$	0	0
$302$	−7.89949	−0.454565
$303$	5.65685	0.324978
$304$	1.41421	0.0811107
$305$	0	0
$306$	−5.82843	−0.333189
$307$	8.58579	0.490017	0.245008	−	0.969521i	$-0.421209\pi$
$307$	8.58579	0.490017	0.245008	+	0.969521i	$0.421209\pi$
$308$	0	0
$309$	−0.0710678	−0.00404291
$310$	0	0
$311$	−11.4142	−0.647241	−0.323620	−	0.946187i	$-0.604900\pi$
$311$	−11.4142	−0.647241	−0.323620	+	0.946187i	$0.604900\pi$
$312$	−3.00000	−0.169842
$313$	−16.0416	−0.906727	−0.453363	−	0.891326i	$-0.649776\pi$
$313$	−16.0416	−0.906727	−0.453363	+	0.891326i	$0.649776\pi$
$314$	19.7990	1.11732
$315$	0	0
$316$	1.07107	0.0602523
$317$	19.9706	1.12166	0.560829	−	0.827931i	$-0.310482\pi$
$317$	19.9706	1.12166	0.560829	+	0.827931i	$0.310482\pi$
$318$	−12.6569	−0.709761
$319$	3.41421	0.191159
$320$	0	0
$321$	6.48528	0.361973
$322$	0	0
$323$	8.24264	0.458633
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.24264	−0.290363
$327$	−11.3137	−0.625650
$328$	5.82843	0.321821
$329$	0	0
$330$	0	0
$331$	−24.0711	−1.32307	−0.661533	−	0.749916i	$-0.730094\pi$
$331$	−24.0711	−1.32307	−0.661533	+	0.749916i	$0.730094\pi$
$332$	−0.0710678	−0.00390035
$333$	7.07107	0.387492
$334$	8.82843	0.483070
$335$	0	0
$336$	0	0
$337$	4.79899	0.261418	0.130709	−	0.991421i	$-0.458275\pi$
$337$	4.79899	0.261418	0.130709	+	0.991421i	$0.458275\pi$
$338$	4.00000	0.217571
$339$	6.58579	0.357691
$340$	0	0
$341$	−8.24264	−0.446364
$342$	−1.41421	−0.0764719
$343$	0	0
$344$	−1.58579	−0.0854999
$345$	0	0
$346$	−14.2426	−0.765689
$347$	−9.79899	−0.526037	−0.263019	−	0.964791i	$-0.584718\pi$
$347$	−9.79899	−0.526037	−0.263019	+	0.964791i	$0.584718\pi$
$348$	1.00000	0.0536056
$349$	24.6569	1.31985	0.659926	−	0.751331i	$-0.270588\pi$
$349$	24.6569	1.31985	0.659926	+	0.751331i	$0.270588\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	−3.41421	−0.181978
$353$	20.4853	1.09032	0.545161	−	0.838332i	$-0.316469\pi$
$353$	20.4853	1.09032	0.545161	+	0.838332i	$0.316469\pi$
$354$	−7.24264	−0.384942
$355$	0	0
$356$	4.00000	0.212000
$357$	0	0
$358$	5.07107	0.268014
$359$	33.8701	1.78759	0.893797	−	0.448472i	$-0.148032\pi$
$359$	33.8701	1.78759	0.893797	+	0.448472i	$0.148032\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	23.3137	1.22534
$363$	0.656854	0.0344759
$364$	0	0
$365$	0	0
$366$	0.171573	0.00896826
$367$	0.757359	0.0395338	0.0197669	−	0.999805i	$-0.493708\pi$
$367$	0.757359	0.0395338	0.0197669	+	0.999805i	$0.493708\pi$
$368$	−0.414214	−0.0215924
$369$	−5.82843	−0.303416
$370$	0	0
$371$	0	0
$372$	−2.41421	−0.125171
$373$	2.34315	0.121323	0.0606617	−	0.998158i	$-0.480679\pi$
$373$	2.34315	0.121323	0.0606617	+	0.998158i	$0.480679\pi$
$374$	−19.8995	−1.02898
$375$	0	0
$376$	1.65685	0.0854457
$377$	3.00000	0.154508
$378$	0	0
$379$	−34.2132	−1.75741	−0.878707	−	0.477361i	$-0.841593\pi$
$379$	−34.2132	−1.75741	−0.878707	+	0.477361i	$0.841593\pi$
$380$	0	0
$381$	−3.17157	−0.162485
$382$	−16.0711	−0.822267
$383$	27.5563	1.40806	0.704032	−	0.710168i	$-0.251381\pi$
$383$	27.5563	1.40806	0.704032	+	0.710168i	$0.251381\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	6.82843	0.347558
$387$	1.58579	0.0806101
$388$	−0.343146	−0.0174206
$389$	−10.3431	−0.524418	−0.262209	−	0.965011i	$-0.584451\pi$
$389$	−10.3431	−0.524418	−0.262209	+	0.965011i	$0.584451\pi$
$390$	0	0
$391$	−2.41421	−0.122092
$392$	0	0
$393$	−18.8284	−0.949769
$394$	−23.1421	−1.16588
$395$	0	0
$396$	3.41421	0.171571
$397$	30.3137	1.52140	0.760701	−	0.649103i	$-0.224856\pi$
$397$	30.3137	1.52140	0.760701	+	0.649103i	$0.224856\pi$
$398$	−19.6569	−0.985309
$399$	0	0
$400$	0	0
$401$	12.1005	0.604270	0.302135	−	0.953265i	$-0.402301\pi$
$401$	12.1005	0.604270	0.302135	+	0.953265i	$0.402301\pi$
$402$	15.3137	0.763778
$403$	−7.24264	−0.360782
$404$	5.65685	0.281439
$405$	0	0
$406$	0	0
$407$	24.1421	1.19668
$408$	−5.82843	−0.288550
$409$	22.5858	1.11680	0.558398	−	0.829573i	$-0.311416\pi$
$409$	22.5858	1.11680	0.558398	+	0.829573i	$0.311416\pi$
$410$	0	0
$411$	−23.2132	−1.14502
$412$	−0.0710678	−0.00350126
$413$	0	0
$414$	0.414214	0.0203575
$415$	0	0
$416$	−3.00000	−0.147087
$417$	−15.5563	−0.761798
$418$	−4.82843	−0.236166
$419$	24.5563	1.19966	0.599828	−	0.800129i	$-0.295236\pi$
$419$	24.5563	1.19966	0.599828	+	0.800129i	$0.295236\pi$
$420$	0	0
$421$	−35.6569	−1.73781	−0.868904	−	0.494980i	$-0.835175\pi$
$421$	−35.6569	−1.73781	−0.868904	+	0.494980i	$0.835175\pi$
$422$	26.5563	1.29274
$423$	−1.65685	−0.0805590
$424$	−12.6569	−0.614671
$425$	0	0
$426$	9.31371	0.451251
$427$	0	0
$428$	6.48528	0.313478
$429$	10.2426	0.494519
$430$	0	0
$431$	33.0416	1.59156	0.795780	−	0.605586i	$-0.207061\pi$
$431$	33.0416	1.59156	0.795780	+	0.605586i	$0.207061\pi$
$432$	1.00000	0.0481125
$433$	28.6274	1.37575	0.687873	−	0.725831i	$-0.258545\pi$
$433$	28.6274	1.37575	0.687873	+	0.725831i	$0.258545\pi$
$434$	0	0
$435$	0	0
$436$	−11.3137	−0.541828
$437$	−0.585786	−0.0280220
$438$	−15.0711	−0.720123
$439$	1.58579	0.0756855	0.0378427	−	0.999284i	$-0.487951\pi$
$439$	1.58579	0.0756855	0.0378427	+	0.999284i	$0.487951\pi$
$440$	0	0
$441$	0	0
$442$	−17.4853	−0.831690
$443$	−14.3431	−0.681463	−0.340732	−	0.940161i	$-0.610675\pi$
$443$	−14.3431	−0.681463	−0.340732	+	0.940161i	$0.610675\pi$
$444$	7.07107	0.335578
$445$	0	0
$446$	−24.5563	−1.16278
$447$	4.65685	0.220262
$448$	0	0
$449$	−25.0711	−1.18318	−0.591588	−	0.806240i	$-0.701499\pi$
$449$	−25.0711	−1.18318	−0.591588	+	0.806240i	$0.701499\pi$
$450$	0	0
$451$	−19.8995	−0.937031
$452$	6.58579	0.309769
$453$	7.89949	0.371151
$454$	−25.3848	−1.19137
$455$	0	0
$456$	−1.41421	−0.0662266
$457$	−3.82843	−0.179086	−0.0895431	−	0.995983i	$-0.528541\pi$
$457$	−3.82843	−0.179086	−0.0895431	+	0.995983i	$0.528541\pi$
$458$	−29.3137	−1.36974
$459$	5.82843	0.272048
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	29.8995	1.38955	0.694774	−	0.719228i	$-0.255505\pi$
$463$	29.8995	1.38955	0.694774	+	0.719228i	$0.255505\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−14.2426	−0.659778
$467$	−1.24264	−0.0575026	−0.0287513	−	0.999587i	$-0.509153\pi$
$467$	−1.24264	−0.0575026	−0.0287513	+	0.999587i	$0.509153\pi$
$468$	3.00000	0.138675
$469$	0	0
$470$	0	0
$471$	−19.7990	−0.912289
$472$	−7.24264	−0.333370
$473$	5.41421	0.248946
$474$	−1.07107	−0.0491958
$475$	0	0
$476$	0	0
$477$	12.6569	0.579518
$478$	11.3137	0.517477
$479$	20.2426	0.924910	0.462455	−	0.886643i	$-0.346969\pi$
$479$	20.2426	0.924910	0.462455	+	0.886643i	$0.346969\pi$
$480$	0	0
$481$	21.2132	0.967239
$482$	6.00000	0.273293
$483$	0	0
$484$	0.656854	0.0298570
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	5.51472	0.249896	0.124948	−	0.992163i	$-0.460124\pi$
$487$	5.51472	0.249896	0.124948	+	0.992163i	$0.460124\pi$
$488$	0.171573	0.00776674
$489$	5.24264	0.237080
$490$	0	0
$491$	−0.142136	−0.00641449	−0.00320725	−	0.999995i	$-0.501021\pi$
$491$	−0.142136	−0.00641449	−0.00320725	+	0.999995i	$0.501021\pi$
$492$	−5.82843	−0.262766
$493$	5.82843	0.262499
$494$	−4.24264	−0.190885
$495$	0	0
$496$	−2.41421	−0.108401
$497$	0	0
$498$	0.0710678	0.00318462
$499$	−26.0711	−1.16710	−0.583551	−	0.812077i	$-0.698337\pi$
$499$	−26.0711	−1.16710	−0.583551	+	0.812077i	$0.698337\pi$
$500$	0	0
$501$	−8.82843	−0.394425
$502$	1.92893	0.0860925
$503$	−24.8701	−1.10890	−0.554451	−	0.832217i	$-0.687072\pi$
$503$	−24.8701	−1.10890	−0.554451	+	0.832217i	$0.687072\pi$
$504$	0	0
$505$	0	0
$506$	1.41421	0.0628695
$507$	−4.00000	−0.177646
$508$	−3.17157	−0.140716
$509$	−6.04163	−0.267791	−0.133895	−	0.990995i	$-0.542749\pi$
$509$	−6.04163	−0.267791	−0.133895	+	0.990995i	$0.542749\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	1.41421	0.0624391
$514$	−13.3431	−0.588541
$515$	0	0
$516$	1.58579	0.0698104
$517$	−5.65685	−0.248788
$518$	0	0
$519$	14.2426	0.625183
$520$	0	0
$521$	−10.6569	−0.466885	−0.233443	−	0.972371i	$-0.574999\pi$
$521$	−10.6569	−0.466885	−0.233443	+	0.972371i	$0.574999\pi$
$522$	−1.00000	−0.0437688
$523$	−24.2843	−1.06188	−0.530939	−	0.847410i	$-0.678160\pi$
$523$	−24.2843	−1.06188	−0.530939	+	0.847410i	$0.678160\pi$
$524$	−18.8284	−0.822524
$525$	0	0
$526$	10.5563	0.460279
$527$	−14.0711	−0.612945
$528$	3.41421	0.148585
$529$	−22.8284	−0.992540
$530$	0	0
$531$	7.24264	0.314304
$532$	0	0
$533$	−17.4853	−0.757372
$534$	−4.00000	−0.173097
$535$	0	0
$536$	15.3137	0.661451
$537$	−5.07107	−0.218833
$538$	−0.443651	−0.0191271
$539$	0	0
$540$	0	0
$541$	−34.2426	−1.47221	−0.736103	−	0.676869i	$-0.763336\pi$
$541$	−34.2426	−1.47221	−0.736103	+	0.676869i	$0.763336\pi$
$542$	11.5147	0.494600
$543$	−23.3137	−1.00049
$544$	−5.82843	−0.249892
$545$	0	0
$546$	0	0
$547$	33.8701	1.44818	0.724090	−	0.689706i	$-0.242260\pi$
$547$	33.8701	1.44818	0.724090	+	0.689706i	$0.242260\pi$
$548$	−23.2132	−0.991619
$549$	−0.171573	−0.00732255
$550$	0	0
$551$	1.41421	0.0602475
$552$	0.414214	0.0176301
$553$	0	0
$554$	24.0416	1.02143
$555$	0	0
$556$	−15.5563	−0.659736
$557$	−19.3137	−0.818348	−0.409174	−	0.912456i	$-0.634183\pi$
$557$	−19.3137	−0.818348	−0.409174	+	0.912456i	$0.634183\pi$
$558$	2.41421	0.102202
$559$	4.75736	0.201215
$560$	0	0
$561$	19.8995	0.840157
$562$	−20.0000	−0.843649
$563$	27.3848	1.15413	0.577065	−	0.816698i	$-0.304198\pi$
$563$	27.3848	1.15413	0.577065	+	0.816698i	$0.304198\pi$
$564$	−1.65685	−0.0697661
$565$	0	0
$566$	0.242641	0.0101989
$567$	0	0
$568$	9.31371	0.390795
$569$	−28.2426	−1.18399	−0.591997	−	0.805941i	$-0.701660\pi$
$569$	−28.2426	−1.18399	−0.591997	+	0.805941i	$0.701660\pi$
$570$	0	0
$571$	36.0122	1.50706	0.753532	−	0.657412i	$-0.228349\pi$
$571$	36.0122	1.50706	0.753532	+	0.657412i	$0.228349\pi$
$572$	10.2426	0.428266
$573$	16.0711	0.671378
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	19.6569	0.818326	0.409163	−	0.912461i	$-0.365821\pi$
$577$	19.6569	0.818326	0.409163	+	0.912461i	$0.365821\pi$
$578$	−16.9706	−0.705882
$579$	−6.82843	−0.283780
$580$	0	0
$581$	0	0
$582$	0.343146	0.0142238
$583$	43.2132	1.78971
$584$	−15.0711	−0.623645
$585$	0	0
$586$	11.3137	0.467365
$587$	−41.5269	−1.71400	−0.857000	−	0.515317i	$-0.827674\pi$
$587$	−41.5269	−1.71400	−0.857000	+	0.515317i	$0.827674\pi$
$588$	0	0
$589$	−3.41421	−0.140680
$590$	0	0
$591$	23.1421	0.951940
$592$	7.07107	0.290619
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	−3.41421	−0.140087
$595$	0	0
$596$	4.65685	0.190752
$597$	19.6569	0.804501
$598$	1.24264	0.0508154
$599$	−30.4142	−1.24269	−0.621346	−	0.783537i	$-0.713414\pi$
$599$	−30.4142	−1.24269	−0.621346	+	0.783537i	$0.713414\pi$
$600$	0	0
$601$	1.27208	0.0518891	0.0259446	−	0.999663i	$-0.491741\pi$
$601$	1.27208	0.0518891	0.0259446	+	0.999663i	$0.491741\pi$
$602$	0	0
$603$	−15.3137	−0.623622
$604$	7.89949	0.321426
$605$	0	0
$606$	−5.65685	−0.229794
$607$	39.9411	1.62116	0.810580	−	0.585628i	$-0.199152\pi$
$607$	39.9411	1.62116	0.810580	+	0.585628i	$0.199152\pi$
$608$	−1.41421	−0.0573539
$609$	0	0
$610$	0	0
$611$	−4.97056	−0.201087
$612$	5.82843	0.235600
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−8.58579	−0.346494
$615$	0	0
$616$	0	0
$617$	22.3848	0.901177	0.450589	−	0.892732i	$-0.351214\pi$
$617$	22.3848	0.901177	0.450589	+	0.892732i	$0.351214\pi$
$618$	0.0710678	0.00285877
$619$	−12.1421	−0.488034	−0.244017	−	0.969771i	$-0.578465\pi$
$619$	−12.1421	−0.488034	−0.244017	+	0.969771i	$0.578465\pi$
$620$	0	0
$621$	−0.414214	−0.0166218
$622$	11.4142	0.457668
$623$	0	0
$624$	3.00000	0.120096
$625$	0	0
$626$	16.0416	0.641153
$627$	4.82843	0.192829
$628$	−19.7990	−0.790066
$629$	41.2132	1.64328
$630$	0	0
$631$	−40.8284	−1.62535	−0.812677	−	0.582714i	$-0.801991\pi$
$631$	−40.8284	−1.62535	−0.812677	+	0.582714i	$0.801991\pi$
$632$	−1.07107	−0.0426048
$633$	−26.5563	−1.05552
$634$	−19.9706	−0.793132
$635$	0	0
$636$	12.6569	0.501877
$637$	0	0
$638$	−3.41421	−0.135170
$639$	−9.31371	−0.368445
$640$	0	0
$641$	31.7574	1.25434	0.627170	−	0.778882i	$-0.284213\pi$
$641$	31.7574	1.25434	0.627170	+	0.778882i	$0.284213\pi$
$642$	−6.48528	−0.255954
$643$	−13.7574	−0.542537	−0.271269	−	0.962504i	$-0.587443\pi$
$643$	−13.7574	−0.542537	−0.271269	+	0.962504i	$0.587443\pi$
$644$	0	0
$645$	0	0
$646$	−8.24264	−0.324302
$647$	4.87006	0.191462	0.0957309	−	0.995407i	$-0.469481\pi$
$647$	4.87006	0.191462	0.0957309	+	0.995407i	$0.469481\pi$
$648$	−1.00000	−0.0392837
$649$	24.7279	0.970656
$650$	0	0
$651$	0	0
$652$	5.24264	0.205318
$653$	−12.6863	−0.496453	−0.248226	−	0.968702i	$-0.579848\pi$
$653$	−12.6863	−0.496453	−0.248226	+	0.968702i	$0.579848\pi$
$654$	11.3137	0.442401
$655$	0	0
$656$	−5.82843	−0.227562
$657$	15.0711	0.587978
$658$	0	0
$659$	3.17157	0.123547	0.0617735	−	0.998090i	$-0.480324\pi$
$659$	3.17157	0.123547	0.0617735	+	0.998090i	$0.480324\pi$
$660$	0	0
$661$	34.3431	1.33579	0.667897	−	0.744254i	$-0.267195\pi$
$661$	34.3431	1.33579	0.667897	+	0.744254i	$0.267195\pi$
$662$	24.0711	0.935549
$663$	17.4853	0.679072
$664$	0.0710678	0.00275797
$665$	0	0
$666$	−7.07107	−0.273998
$667$	−0.414214	−0.0160384
$668$	−8.82843	−0.341582
$669$	24.5563	0.949403
$670$	0	0
$671$	−0.585786	−0.0226140
$672$	0	0
$673$	−0.798990	−0.0307988	−0.0153994	−	0.999881i	$-0.504902\pi$
$673$	−0.798990	−0.0307988	−0.0153994	+	0.999881i	$0.504902\pi$
$674$	−4.79899	−0.184850
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−4.38478	−0.168521	−0.0842603	−	0.996444i	$-0.526853\pi$
$677$	−4.38478	−0.168521	−0.0842603	+	0.996444i	$0.526853\pi$
$678$	−6.58579	−0.252926
$679$	0	0
$680$	0	0
$681$	25.3848	0.972747
$682$	8.24264	0.315627
$683$	25.0711	0.959318	0.479659	−	0.877455i	$-0.340760\pi$
$683$	25.0711	0.959318	0.479659	+	0.877455i	$0.340760\pi$
$684$	1.41421	0.0540738
$685$	0	0
$686$	0	0
$687$	29.3137	1.11839
$688$	1.58579	0.0604575
$689$	37.9706	1.44656
$690$	0	0
$691$	33.1127	1.25967	0.629833	−	0.776730i	$-0.283123\pi$
$691$	33.1127	1.25967	0.629833	+	0.776730i	$0.283123\pi$
$692$	14.2426	0.541424
$693$	0	0
$694$	9.79899	0.371965
$695$	0	0
$696$	−1.00000	−0.0379049
$697$	−33.9706	−1.28673
$698$	−24.6569	−0.933276
$699$	14.2426	0.538706
$700$	0	0
$701$	36.3137	1.37155	0.685775	−	0.727814i	$-0.259463\pi$
$701$	36.3137	1.37155	0.685775	+	0.727814i	$0.259463\pi$
$702$	−3.00000	−0.113228
$703$	10.0000	0.377157
$704$	3.41421	0.128678
$705$	0	0
$706$	−20.4853	−0.770974
$707$	0	0
$708$	7.24264	0.272195
$709$	19.8995	0.747341	0.373671	−	0.927561i	$-0.378099\pi$
$709$	19.8995	0.747341	0.373671	+	0.927561i	$0.378099\pi$
$710$	0	0
$711$	1.07107	0.0401682
$712$	−4.00000	−0.149906
$713$	1.00000	0.0374503
$714$	0	0
$715$	0	0
$716$	−5.07107	−0.189515
$717$	−11.3137	−0.422518
$718$	−33.8701	−1.26402
$719$	18.9289	0.705930	0.352965	−	0.935637i	$-0.385173\pi$
$719$	18.9289	0.705930	0.352965	+	0.935637i	$0.385173\pi$
$720$	0	0
$721$	0	0
$722$	17.0000	0.632674
$723$	−6.00000	−0.223142
$724$	−23.3137	−0.866447
$725$	0	0
$726$	−0.656854	−0.0243781
$727$	−37.8701	−1.40452	−0.702261	−	0.711919i	$-0.747826\pi$
$727$	−37.8701	−1.40452	−0.702261	+	0.711919i	$0.747826\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.24264	0.341851
$732$	−0.171573	−0.00634152
$733$	−0.514719	−0.0190116	−0.00950578	−	0.999955i	$-0.503026\pi$
$733$	−0.514719	−0.0190116	−0.00950578	+	0.999955i	$0.503026\pi$
$734$	−0.757359	−0.0279546
$735$	0	0
$736$	0.414214	0.0152681
$737$	−52.2843	−1.92592
$738$	5.82843	0.214547
$739$	53.8701	1.98164	0.990821	−	0.135180i	$-0.0431613\pi$
$739$	53.8701	1.98164	0.990821	+	0.135180i	$0.0431613\pi$
$740$	0	0
$741$	4.24264	0.155857
$742$	0	0
$743$	−36.5563	−1.34112	−0.670561	−	0.741854i	$-0.733947\pi$
$743$	−36.5563	−1.34112	−0.670561	+	0.741854i	$0.733947\pi$
$744$	2.41421	0.0885094
$745$	0	0
$746$	−2.34315	−0.0857887
$747$	−0.0710678	−0.00260024
$748$	19.8995	0.727598
$749$	0	0
$750$	0	0
$751$	−12.9706	−0.473303	−0.236651	−	0.971595i	$-0.576050\pi$
$751$	−12.9706	−0.473303	−0.236651	+	0.971595i	$0.576050\pi$
$752$	−1.65685	−0.0604193
$753$	−1.92893	−0.0702942
$754$	−3.00000	−0.109254
$755$	0	0
$756$	0	0
$757$	−44.2426	−1.60803	−0.804013	−	0.594612i	$-0.797306\pi$
$757$	−44.2426	−1.60803	−0.804013	+	0.594612i	$0.797306\pi$
$758$	34.2132	1.24268
$759$	−1.41421	−0.0513327
$760$	0	0
$761$	15.0294	0.544817	0.272408	−	0.962182i	$-0.412180\pi$
$761$	15.0294	0.544817	0.272408	+	0.962182i	$0.412180\pi$
$762$	3.17157	0.114894
$763$	0	0
$764$	16.0711	0.581431
$765$	0	0
$766$	−27.5563	−0.995651
$767$	21.7279	0.784550
$768$	1.00000	0.0360844
$769$	−15.3137	−0.552226	−0.276113	−	0.961125i	$-0.589046\pi$
$769$	−15.3137	−0.552226	−0.276113	+	0.961125i	$0.589046\pi$
$770$	0	0
$771$	13.3431	0.480542
$772$	−6.82843	−0.245760
$773$	−19.3137	−0.694666	−0.347333	−	0.937742i	$-0.612913\pi$
$773$	−19.3137	−0.694666	−0.347333	+	0.937742i	$0.612913\pi$
$774$	−1.58579	−0.0569999
$775$	0	0
$776$	0.343146	0.0123182
$777$	0	0
$778$	10.3431	0.370820
$779$	−8.24264	−0.295323
$780$	0	0
$781$	−31.7990	−1.13786
$782$	2.41421	0.0863321
$783$	1.00000	0.0357371
$784$	0	0
$785$	0	0
$786$	18.8284	0.671588
$787$	−43.4558	−1.54903	−0.774517	−	0.632553i	$-0.782007\pi$
$787$	−43.4558	−1.54903	−0.774517	+	0.632553i	$0.782007\pi$
$788$	23.1421	0.824404
$789$	−10.5563	−0.375816
$790$	0	0
$791$	0	0
$792$	−3.41421	−0.121319
$793$	−0.514719	−0.0182782
$794$	−30.3137	−1.07579
$795$	0	0
$796$	19.6569	0.696719
$797$	16.2426	0.575344	0.287672	−	0.957729i	$-0.407119\pi$
$797$	16.2426	0.575344	0.287672	+	0.957729i	$0.407119\pi$
$798$	0	0
$799$	−9.65685	−0.341635
$800$	0	0
$801$	4.00000	0.141333
$802$	−12.1005	−0.427284
$803$	51.4558	1.81584
$804$	−15.3137	−0.540073
$805$	0	0
$806$	7.24264	0.255111
$807$	0.443651	0.0156172
$808$	−5.65685	−0.199007
$809$	−34.6274	−1.21744	−0.608718	−	0.793387i	$-0.708316\pi$
$809$	−34.6274	−1.21744	−0.608718	+	0.793387i	$0.708316\pi$
$810$	0	0
$811$	33.7574	1.18538	0.592691	−	0.805430i	$-0.298066\pi$
$811$	33.7574	1.18538	0.592691	+	0.805430i	$0.298066\pi$
$812$	0	0
$813$	−11.5147	−0.403839
$814$	−24.1421	−0.846181
$815$	0	0
$816$	5.82843	0.204036
$817$	2.24264	0.0784601
$818$	−22.5858	−0.789694
$819$	0	0
$820$	0	0
$821$	13.1716	0.459691	0.229846	−	0.973227i	$-0.426178\pi$
$821$	13.1716	0.459691	0.229846	+	0.973227i	$0.426178\pi$
$822$	23.2132	0.809653
$823$	28.5858	0.996438	0.498219	−	0.867051i	$-0.333988\pi$
$823$	28.5858	0.996438	0.498219	+	0.867051i	$0.333988\pi$
$824$	0.0710678	0.00247576
$825$	0	0
$826$	0	0
$827$	−11.8995	−0.413786	−0.206893	−	0.978364i	$-0.566335\pi$
$827$	−11.8995	−0.413786	−0.206893	+	0.978364i	$0.566335\pi$
$828$	−0.414214	−0.0143949
$829$	−0.798990	−0.0277501	−0.0138750	−	0.999904i	$-0.504417\pi$
$829$	−0.798990	−0.0277501	−0.0138750	+	0.999904i	$0.504417\pi$
$830$	0	0
$831$	−24.0416	−0.833995
$832$	3.00000	0.104006
$833$	0	0
$834$	15.5563	0.538672
$835$	0	0
$836$	4.82843	0.166995
$837$	−2.41421	−0.0834474
$838$	−24.5563	−0.848285
$839$	17.4142	0.601205	0.300603	−	0.953749i	$-0.402812\pi$
$839$	17.4142	0.601205	0.300603	+	0.953749i	$0.402812\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	35.6569	1.22882
$843$	20.0000	0.688837
$844$	−26.5563	−0.914107
$845$	0	0
$846$	1.65685	0.0569638
$847$	0	0
$848$	12.6569	0.434638
$849$	−0.242641	−0.00832741
$850$	0	0
$851$	−2.92893	−0.100403
$852$	−9.31371	−0.319082
$853$	14.3137	0.490092	0.245046	−	0.969511i	$-0.421197\pi$
$853$	14.3137	0.490092	0.245046	+	0.969511i	$0.421197\pi$
$854$	0	0
$855$	0	0
$856$	−6.48528	−0.221662
$857$	29.4558	1.00619	0.503096	−	0.864230i	$-0.332194\pi$
$857$	29.4558	1.00619	0.503096	+	0.864230i	$0.332194\pi$
$858$	−10.2426	−0.349678
$859$	21.9411	0.748622	0.374311	−	0.927303i	$-0.377879\pi$
$859$	21.9411	0.748622	0.374311	+	0.927303i	$0.377879\pi$
$860$	0	0
$861$	0	0
$862$	−33.0416	−1.12540
$863$	24.9706	0.850008	0.425004	−	0.905192i	$-0.360273\pi$
$863$	24.9706	0.850008	0.425004	+	0.905192i	$0.360273\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−28.6274	−0.972799
$867$	16.9706	0.576351
$868$	0	0
$869$	3.65685	0.124050
$870$	0	0
$871$	−45.9411	−1.55666
$872$	11.3137	0.383131
$873$	−0.343146	−0.0116137
$874$	0.585786	0.0198145
$875$	0	0
$876$	15.0711	0.509204
$877$	−26.3431	−0.889545	−0.444772	−	0.895644i	$-0.646715\pi$
$877$	−26.3431	−0.889545	−0.444772	+	0.895644i	$0.646715\pi$
$878$	−1.58579	−0.0535177
$879$	−11.3137	−0.381602
$880$	0	0
$881$	41.6274	1.40246	0.701232	−	0.712933i	$-0.252634\pi$
$881$	41.6274	1.40246	0.701232	+	0.712933i	$0.252634\pi$
$882$	0	0
$883$	−31.5269	−1.06097	−0.530483	−	0.847696i	$-0.677989\pi$
$883$	−31.5269	−1.06097	−0.530483	+	0.847696i	$0.677989\pi$
$884$	17.4853	0.588094
$885$	0	0
$886$	14.3431	0.481867
$887$	−25.7574	−0.864847	−0.432424	−	0.901671i	$-0.642342\pi$
$887$	−25.7574	−0.864847	−0.432424	+	0.901671i	$0.642342\pi$
$888$	−7.07107	−0.237289
$889$	0	0
$890$	0	0
$891$	3.41421	0.114380
$892$	24.5563	0.822207
$893$	−2.34315	−0.0784104
$894$	−4.65685	−0.155749
$895$	0	0
$896$	0	0
$897$	−1.24264	−0.0414906
$898$	25.0711	0.836632
$899$	−2.41421	−0.0805185
$900$	0	0
$901$	73.7696	2.45762
$902$	19.8995	0.662581
$903$	0	0
$904$	−6.58579	−0.219040
$905$	0	0
$906$	−7.89949	−0.262443
$907$	40.6985	1.35137	0.675686	−	0.737190i	$-0.263848\pi$
$907$	40.6985	1.35137	0.675686	+	0.737190i	$0.263848\pi$
$908$	25.3848	0.842423
$909$	5.65685	0.187626
$910$	0	0
$911$	22.0711	0.731247	0.365624	−	0.930763i	$-0.380856\pi$
$911$	22.0711	0.731247	0.365624	+	0.930763i	$0.380856\pi$
$912$	1.41421	0.0468293
$913$	−0.242641	−0.00803023
$914$	3.82843	0.126633
$915$	0	0
$916$	29.3137	0.968552
$917$	0	0
$918$	−5.82843	−0.192367
$919$	4.44365	0.146583	0.0732913	−	0.997311i	$-0.476650\pi$
$919$	4.44365	0.146583	0.0732913	+	0.997311i	$0.476650\pi$
$920$	0	0
$921$	8.58579	0.282911
$922$	−20.0000	−0.658665
$923$	−27.9411	−0.919693
$924$	0	0
$925$	0	0
$926$	−29.8995	−0.982558
$927$	−0.0710678	−0.00233417
$928$	−1.00000	−0.0328266
$929$	−8.65685	−0.284022	−0.142011	−	0.989865i	$-0.545357\pi$
$929$	−8.65685	−0.284022	−0.142011	+	0.989865i	$0.545357\pi$
$930$	0	0
$931$	0	0
$932$	14.2426	0.466533
$933$	−11.4142	−0.373685
$934$	1.24264	0.0406604
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	0.242641	0.00792673	0.00396336	−	0.999992i	$-0.498738\pi$
$937$	0.242641	0.00792673	0.00396336	+	0.999992i	$0.498738\pi$
$938$	0	0
$939$	−16.0416	−0.523499
$940$	0	0
$941$	51.3137	1.67278	0.836390	−	0.548136i	$-0.184662\pi$
$941$	51.3137	1.67278	0.836390	+	0.548136i	$0.184662\pi$
$942$	19.7990	0.645086
$943$	2.41421	0.0786176
$944$	7.24264	0.235728
$945$	0	0
$946$	−5.41421	−0.176031
$947$	−25.2132	−0.819319	−0.409660	−	0.912239i	$-0.634352\pi$
$947$	−25.2132	−0.819319	−0.409660	+	0.912239i	$0.634352\pi$
$948$	1.07107	0.0347867
$949$	45.2132	1.46768
$950$	0	0
$951$	19.9706	0.647590
$952$	0	0
$953$	−31.5980	−1.02356	−0.511779	−	0.859117i	$-0.671014\pi$
$953$	−31.5980	−1.02356	−0.511779	+	0.859117i	$0.671014\pi$
$954$	−12.6569	−0.409781
$955$	0	0
$956$	−11.3137	−0.365911
$957$	3.41421	0.110366
$958$	−20.2426	−0.654010
$959$	0	0
$960$	0	0
$961$	−25.1716	−0.811986
$962$	−21.2132	−0.683941
$963$	6.48528	0.208985
$964$	−6.00000	−0.193247
$965$	0	0
$966$	0	0
$967$	−44.0416	−1.41628	−0.708142	−	0.706070i	$-0.750466\pi$
$967$	−44.0416	−1.41628	−0.708142	+	0.706070i	$0.750466\pi$
$968$	−0.656854	−0.0211121
$969$	8.24264	0.264792
$970$	0	0
$971$	−44.9706	−1.44317	−0.721587	−	0.692324i	$-0.756587\pi$
$971$	−44.9706	−1.44317	−0.721587	+	0.692324i	$0.756587\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−5.51472	−0.176703
$975$	0	0
$976$	−0.171573	−0.00549191
$977$	−8.44365	−0.270136	−0.135068	−	0.990836i	$-0.543125\pi$
$977$	−8.44365	−0.270136	−0.135068	+	0.990836i	$0.543125\pi$
$978$	−5.24264	−0.167641
$979$	13.6569	0.436475
$980$	0	0
$981$	−11.3137	−0.361219
$982$	0.142136	0.00453573
$983$	28.6863	0.914951	0.457475	−	0.889222i	$-0.348754\pi$
$983$	28.6863	0.914951	0.457475	+	0.889222i	$0.348754\pi$
$984$	5.82843	0.185803
$985$	0	0
$986$	−5.82843	−0.185615
$987$	0	0
$988$	4.24264	0.134976
$989$	−0.656854	−0.0208867
$990$	0	0
$991$	−18.3431	−0.582689	−0.291345	−	0.956618i	$-0.594103\pi$
$991$	−18.3431	−0.582689	−0.291345	+	0.956618i	$0.594103\pi$
$992$	2.41421	0.0766514
$993$	−24.0711	−0.763872
$994$	0	0
$995$	0	0
$996$	−0.0710678	−0.00225187
$997$	−25.3137	−0.801693	−0.400847	−	0.916145i	$-0.631284\pi$
$997$	−25.3137	−0.801693	−0.400847	+	0.916145i	$0.631284\pi$
$998$	26.0711	0.825265
$999$	7.07107	0.223719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dg.1.2	yes	2
5.4	even	2		7350.2.a.dj.1.2	yes	2
7.6	odd	2		7350.2.a.dc.1.2	✓	2
35.34	odd	2		7350.2.a.dm.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7350.2.a.dc.1.2	✓	2	7.6	odd	2
7350.2.a.dg.1.2	yes	2	1.1	even	1	trivial
7350.2.a.dj.1.2	yes	2	5.4	even	2
7350.2.a.dm.1.2	yes	2	35.34	odd	2

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:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.41421 q^{11} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{16} +5.82843 q^{17} -1.00000 q^{18} +1.41421 q^{19} -3.41421 q^{22} -0.414214 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} +1.00000 q^{29} -2.41421 q^{31} -1.00000 q^{32} +3.41421 q^{33} -5.82843 q^{34} +1.00000 q^{36} +7.07107 q^{37} -1.41421 q^{38} +3.00000 q^{39} -5.82843 q^{41} +1.58579 q^{43} +3.41421 q^{44} +0.414214 q^{46} -1.65685 q^{47} +1.00000 q^{48} +5.82843 q^{51} +3.00000 q^{52} +12.6569 q^{53} -1.00000 q^{54} +1.41421 q^{57} -1.00000 q^{58} +7.24264 q^{59} -0.171573 q^{61} +2.41421 q^{62} +1.00000 q^{64} -3.41421 q^{66} -15.3137 q^{67} +5.82843 q^{68} -0.414214 q^{69} -9.31371 q^{71} -1.00000 q^{72} +15.0711 q^{73} -7.07107 q^{74} +1.41421 q^{76} -3.00000 q^{78} +1.07107 q^{79} +1.00000 q^{81} +5.82843 q^{82} -0.0710678 q^{83} -1.58579 q^{86} +1.00000 q^{87} -3.41421 q^{88} +4.00000 q^{89} -0.414214 q^{92} -2.41421 q^{93} +1.65685 q^{94} -1.00000 q^{96} -0.343146 q^{97} +3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 4 q^{22} + 2 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{36} + 6 q^{39} - 6 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} + 8 q^{47} + 2 q^{48} + 6 q^{51} + 6 q^{52} + 14 q^{53} - 2 q^{54} - 2 q^{58} + 6 q^{59} - 6 q^{61} + 2 q^{62} + 2 q^{64} - 4 q^{66} - 8 q^{67} + 6 q^{68} + 2 q^{69} + 4 q^{71} - 2 q^{72} + 16 q^{73} - 6 q^{78} - 12 q^{79} + 2 q^{81} + 6 q^{82} + 14 q^{83} - 6 q^{86} + 2 q^{87} - 4 q^{88} + 8 q^{89} + 2 q^{92} - 2 q^{93} - 8 q^{94} - 2 q^{96} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 6 * q^13 + 2 * q^16 + 6 * q^17 - 2 * q^18 - 4 * q^22 + 2 * q^23 - 2 * q^24 - 6 * q^26 + 2 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^36 + 6 * q^39 - 6 * q^41 + 6 * q^43 + 4 * q^44 - 2 * q^46 + 8 * q^47 + 2 * q^48 + 6 * q^51 + 6 * q^52 + 14 * q^53 - 2 * q^54 - 2 * q^58 + 6 * q^59 - 6 * q^61 + 2 * q^62 + 2 * q^64 - 4 * q^66 - 8 * q^67 + 6 * q^68 + 2 * q^69 + 4 * q^71 - 2 * q^72 + 16 * q^73 - 6 * q^78 - 12 * q^79 + 2 * q^81 + 6 * q^82 + 14 * q^83 - 6 * q^86 + 2 * q^87 - 4 * q^88 + 8 * q^89 + 2 * q^92 - 2 * q^93 - 8 * q^94 - 2 * q^96 - 12 * q^97 + 4 * q^99

Introduction

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