LMFDB - Embedded newform 7500.2.d.d.1249.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.6
Root			$-1.82709i$ of defining polynomial
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.d.1249.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -0.547318i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -0.547318i q^{7} -1.00000 q^{9} -1.33826 q^{11} +0.302113i q^{13} -4.04179i q^{17} +5.63282 q^{19} +0.547318 q^{21} +0.245205i q^{23} -1.00000i q^{27} +1.36974 q^{29} -3.53818 q^{31} -1.33826i q^{33} +2.18769i q^{37} -0.302113 q^{39} -9.80284 q^{41} -8.35963i q^{43} +10.4737i q^{47} +6.70044 q^{49} +4.04179 q^{51} -7.69598i q^{53} +5.63282i q^{57} -4.15215 q^{59} +2.07636 q^{61} +0.547318i q^{63} -8.48594i q^{67} -0.245205 q^{69} -9.18323 q^{71} +10.9560i q^{73} +0.732455i q^{77} -15.1829 q^{79} +1.00000 q^{81} -6.65418i q^{83} +1.36974i q^{87} -4.97864 q^{89} +0.165352 q^{91} -3.53818i q^{93} -4.06125i q^{97} +1.33826 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} + 8 q^{29} - 18 q^{31} - 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 4 q^{61} + 18 q^{69} - 40 q^{71} + 6 q^{79} + 8 q^{81} - 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 + 8 * q^29 - 18 * q^31 - 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 4 * q^61 + 18 * q^69 - 40 * q^71 + 6 * q^79 + 8 * q^81 - 30 * q^89 - 20 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.547318i	− 0.206867i	−0.994636	−	0.103433i	$-0.967017\pi$
$7$	− 0.547318i	− 0.206867i	0.994636	−	0.103433i	$-0.0329829\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.33826	−0.403501	−0.201750	−	0.979437i	$-0.564663\pi$
$11$	−1.33826	−0.403501	−0.201750	+	0.979437i	$0.564663\pi$
$12$	0	0
$13$	0.302113i	0.0837912i	0.999122	+	0.0418956i	$0.0133397\pi$
$13$	0.302113i	0.0837912i	−0.999122	+	0.0418956i	$0.986660\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.04179i	− 0.980279i	−0.871644	−	0.490140i	$-0.836946\pi$
$17$	− 4.04179i	− 0.980279i	0.871644	−	0.490140i	$-0.163054\pi$
$18$	0	0
$19$	5.63282	1.29226	0.646128	−	0.763229i	$-0.276387\pi$
$19$	5.63282	1.29226	0.646128	+	0.763229i	$0.276387\pi$
$20$	0	0
$21$	0.547318	0.119435
$22$	0	0
$23$	0.245205i	0.0511287i	0.999673	+	0.0255644i	$0.00813828\pi$
$23$	0.245205i	0.0511287i	−0.999673	+	0.0255644i	$0.991862\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	1.36974	0.254354	0.127177	−	0.991880i	$-0.459408\pi$
$29$	1.36974	0.254354	0.127177	+	0.991880i	$0.459408\pi$
$30$	0	0
$31$	−3.53818	−0.635476	−0.317738	−	0.948179i	$-0.602923\pi$
$31$	−3.53818	−0.635476	−0.317738	+	0.948179i	$0.602923\pi$
$32$	0	0
$33$	− 1.33826i	− 0.232961i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.18769i	0.359654i	0.983698	+	0.179827i	$0.0575539\pi$
$37$	2.18769i	0.359654i	−0.983698	+	0.179827i	$0.942446\pi$
$38$	0	0
$39$	−0.302113	−0.0483769
$40$	0	0
$41$	−9.80284	−1.53095	−0.765473	−	0.643468i	$-0.777495\pi$
$41$	−9.80284	−1.53095	−0.765473	+	0.643468i	$0.777495\pi$
$42$	0	0
$43$	− 8.35963i	− 1.27483i	−0.770520	−	0.637415i	$-0.780004\pi$
$43$	− 8.35963i	− 1.27483i	0.770520	−	0.637415i	$-0.219996\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.4737i	1.52775i	0.645365	+	0.763874i	$0.276705\pi$
$47$	10.4737i	1.52775i	−0.645365	+	0.763874i	$0.723295\pi$
$48$	0	0
$49$	6.70044	0.957206
$50$	0	0
$51$	4.04179	0.565964
$52$	0	0
$53$	− 7.69598i	− 1.05712i	−0.848895	−	0.528562i	$-0.822731\pi$
$53$	− 7.69598i	− 1.05712i	0.848895	−	0.528562i	$-0.177269\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.63282i	0.746085i
$58$	0	0
$59$	−4.15215	−0.540564	−0.270282	−	0.962781i	$-0.587117\pi$
$59$	−4.15215	−0.540564	−0.270282	+	0.962781i	$0.587117\pi$
$60$	0	0
$61$	2.07636	0.265851	0.132926	−	0.991126i	$-0.457563\pi$
$61$	2.07636	0.265851	0.132926	+	0.991126i	$0.457563\pi$
$62$	0	0
$63$	0.547318i	0.0689556i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.48594i	− 1.03672i	−0.855162	−	0.518361i	$-0.826542\pi$
$67$	− 8.48594i	− 1.03672i	0.855162	−	0.518361i	$-0.173458\pi$
$68$	0	0
$69$	−0.245205	−0.0295192
$70$	0	0
$71$	−9.18323	−1.08985	−0.544924	−	0.838485i	$-0.683442\pi$
$71$	−9.18323	−1.08985	−0.544924	+	0.838485i	$0.683442\pi$
$72$	0	0
$73$	10.9560i	1.28230i	0.767416	+	0.641149i	$0.221542\pi$
$73$	10.9560i	1.28230i	−0.767416	+	0.641149i	$0.778458\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.732455i	0.0834710i
$78$	0	0
$79$	−15.1829	−1.70821	−0.854105	−	0.520101i	$-0.825894\pi$
$79$	−15.1829	−1.70821	−0.854105	+	0.520101i	$0.825894\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.65418i	− 0.730391i	−0.930931	−	0.365196i	$-0.881002\pi$
$83$	− 6.65418i	− 0.730391i	0.930931	−	0.365196i	$-0.118998\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.36974i	0.146851i
$88$	0	0
$89$	−4.97864	−0.527734	−0.263867	−	0.964559i	$-0.584998\pi$
$89$	−4.97864	−0.527734	−0.263867	+	0.964559i	$0.584998\pi$
$90$	0	0
$91$	0.165352	0.0173336
$92$	0	0
$93$	− 3.53818i	− 0.366892i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.06125i	− 0.412357i	−0.978514	−	0.206179i	$-0.933897\pi$
$97$	− 4.06125i	− 0.412357i	0.978514	−	0.206179i	$-0.0661028\pi$
$98$	0	0
$99$	1.33826	0.134500
$100$	0	0
$101$	18.4489	1.83573	0.917865	−	0.396892i	$-0.129911\pi$
$101$	18.4489	1.83573	0.917865	+	0.396892i	$0.129911\pi$
$102$	0	0
$103$	− 14.7164i	− 1.45005i	−0.688722	−	0.725025i	$-0.741828\pi$
$103$	− 14.7164i	− 1.45005i	0.688722	−	0.725025i	$-0.258172\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.10528i	0.493546i	0.969073	+	0.246773i	$0.0793703\pi$
$107$	5.10528i	0.493546i	−0.969073	+	0.246773i	$0.920630\pi$
$108$	0	0
$109$	−4.72587	−0.452657	−0.226328	−	0.974051i	$-0.572672\pi$
$109$	−4.72587	−0.452657	−0.226328	+	0.974051i	$0.572672\pi$
$110$	0	0
$111$	−2.18769	−0.207647
$112$	0	0
$113$	4.13834i	0.389302i	0.980873	+	0.194651i	$0.0623575\pi$
$113$	4.13834i	0.389302i	−0.980873	+	0.194651i	$0.937643\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.302113i	− 0.0279304i
$118$	0	0
$119$	−2.21215	−0.202787
$120$	0	0
$121$	−9.20906	−0.837187
$122$	0	0
$123$	− 9.80284i	− 0.883892i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.6781i	1.12500i	0.826797	+	0.562500i	$0.190160\pi$
$127$	12.6781i	1.12500i	−0.826797	+	0.562500i	$0.809840\pi$
$128$	0	0
$129$	8.35963	0.736024
$130$	0	0
$131$	−12.3309	−1.07736	−0.538678	−	0.842512i	$-0.681076\pi$
$131$	−12.3309	−1.07736	−0.538678	+	0.842512i	$0.681076\pi$
$132$	0	0
$133$	− 3.08294i	− 0.267325i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.7230i	− 0.916127i	−0.888920	−	0.458063i	$-0.848543\pi$
$137$	− 10.7230i	− 0.916127i	0.888920	−	0.458063i	$-0.151457\pi$
$138$	0	0
$139$	13.6748	1.15988	0.579941	−	0.814658i	$-0.303075\pi$
$139$	13.6748	1.15988	0.579941	+	0.814658i	$0.303075\pi$
$140$	0	0
$141$	−10.4737	−0.882046
$142$	0	0
$143$	− 0.404307i	− 0.0338098i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.70044i	0.552643i
$148$	0	0
$149$	−8.88176	−0.727622	−0.363811	−	0.931473i	$-0.618525\pi$
$149$	−8.88176	−0.727622	−0.363811	+	0.931473i	$0.618525\pi$
$150$	0	0
$151$	2.68310	0.218348	0.109174	−	0.994023i	$-0.465179\pi$
$151$	2.68310	0.218348	0.109174	+	0.994023i	$0.465179\pi$
$152$	0	0
$153$	4.04179i	0.326760i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.7576i	− 0.938355i	−0.883104	−	0.469178i	$-0.844550\pi$
$157$	− 11.7576i	− 0.938355i	0.883104	−	0.469178i	$-0.155450\pi$
$158$	0	0
$159$	7.69598	0.610331
$160$	0	0
$161$	0.134205	0.0105768
$162$	0	0
$163$	− 18.3127i	− 1.43436i	−0.696887	−	0.717181i	$-0.745432\pi$
$163$	− 18.3127i	− 1.43436i	0.696887	−	0.717181i	$-0.254568\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.40418i	− 0.340806i	−0.985374	−	0.170403i	$-0.945493\pi$
$167$	− 4.40418i	− 0.340806i	0.985374	−	0.170403i	$-0.0545069\pi$
$168$	0	0
$169$	12.9087	0.992979
$170$	0	0
$171$	−5.63282	−0.430752
$172$	0	0
$173$	− 15.9307i	− 1.21119i	−0.795772	−	0.605596i	$-0.792935\pi$
$173$	− 15.9307i	− 1.21119i	0.795772	−	0.605596i	$-0.207065\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.15215i	− 0.312095i
$178$	0	0
$179$	−3.76978	−0.281767	−0.140883	−	0.990026i	$-0.544994\pi$
$179$	−3.76978	−0.281767	−0.140883	+	0.990026i	$0.544994\pi$
$180$	0	0
$181$	−26.6868	−1.98362	−0.991809	−	0.127733i	$-0.959230\pi$
$181$	−26.6868	−1.98362	−0.991809	+	0.127733i	$0.959230\pi$
$182$	0	0
$183$	2.07636i	0.153489i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.40898i	0.395544i
$188$	0	0
$189$	−0.547318	−0.0398115
$190$	0	0
$191$	4.63282	0.335219	0.167609	−	0.985853i	$-0.446395\pi$
$191$	4.63282	0.335219	0.167609	+	0.985853i	$0.446395\pi$
$192$	0	0
$193$	− 20.7440i	− 1.49319i	−0.665280	−	0.746594i	$-0.731688\pi$
$193$	− 20.7440i	− 1.49319i	0.665280	−	0.746594i	$-0.268312\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.5338i	− 1.46297i	−0.681857	−	0.731486i	$-0.738827\pi$
$197$	− 20.5338i	− 1.46297i	0.681857	−	0.731486i	$-0.261173\pi$
$198$	0	0
$199$	−5.91437	−0.419259	−0.209629	−	0.977781i	$-0.567226\pi$
$199$	−5.91437	−0.419259	−0.209629	+	0.977781i	$0.567226\pi$
$200$	0	0
$201$	8.48594	0.598552
$202$	0	0
$203$	− 0.749683i	− 0.0526174i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.245205i	− 0.0170429i
$208$	0	0
$209$	−7.53818	−0.521427
$210$	0	0
$211$	22.9305	1.57860	0.789302	−	0.614006i	$-0.210443\pi$
$211$	22.9305	1.57860	0.789302	+	0.614006i	$0.210443\pi$
$212$	0	0
$213$	− 9.18323i	− 0.629224i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.93651i	0.131459i
$218$	0	0
$219$	−10.9560	−0.740336
$220$	0	0
$221$	1.22108	0.0821387
$222$	0	0
$223$	− 12.7498i	− 0.853789i	−0.904301	−	0.426895i	$-0.859607\pi$
$223$	− 12.7498i	− 0.853789i	0.904301	−	0.426895i	$-0.140393\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.0294i	− 0.732050i	−0.930605	−	0.366025i	$-0.880719\pi$
$227$	− 11.0294i	− 0.732050i	0.930605	−	0.366025i	$-0.119281\pi$
$228$	0	0
$229$	23.0801	1.52518	0.762588	−	0.646885i	$-0.223928\pi$
$229$	23.0801	1.52518	0.762588	+	0.646885i	$0.223928\pi$
$230$	0	0
$231$	−0.732455	−0.0481920
$232$	0	0
$233$	− 29.2882i	− 1.91873i	−0.282164	−	0.959366i	$-0.591052\pi$
$233$	− 29.2882i	− 1.91873i	0.282164	−	0.959366i	$-0.408948\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 15.1829i	− 0.986235i
$238$	0	0
$239$	−10.5986	−0.685565	−0.342783	−	0.939415i	$-0.611369\pi$
$239$	−10.5986	−0.685565	−0.342783	+	0.939415i	$0.611369\pi$
$240$	0	0
$241$	19.0216	1.22529	0.612646	−	0.790358i	$-0.290105\pi$
$241$	19.0216	1.22529	0.612646	+	0.790358i	$0.290105\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.70175i	0.108280i
$248$	0	0
$249$	6.65418	0.421692
$250$	0	0
$251$	0.0370816	0.00234057	0.00117028	−	0.999999i	$-0.499627\pi$
$251$	0.0370816	0.00234057	0.00117028	+	0.999999i	$0.499627\pi$
$252$	0	0
$253$	− 0.328148i	− 0.0206305i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.0237i	0.625262i	0.949875	+	0.312631i	$0.101210\pi$
$257$	10.0237i	0.625262i	−0.949875	+	0.312631i	$0.898790\pi$
$258$	0	0
$259$	1.19736	0.0744006
$260$	0	0
$261$	−1.36974	−0.0847847
$262$	0	0
$263$	− 12.7818i	− 0.788160i	−0.919076	−	0.394080i	$-0.871063\pi$
$263$	− 12.7818i	− 0.788160i	0.919076	−	0.394080i	$-0.128937\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.97864i	− 0.304688i
$268$	0	0
$269$	−21.3115	−1.29939	−0.649693	−	0.760196i	$-0.725103\pi$
$269$	−21.3115	−1.29939	−0.649693	+	0.760196i	$0.725103\pi$
$270$	0	0
$271$	−13.4036	−0.814211	−0.407105	−	0.913381i	$-0.633462\pi$
$271$	−13.4036	−0.814211	−0.407105	+	0.913381i	$0.633462\pi$
$272$	0	0
$273$	0.165352i	0.0100076i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.8720i	1.13391i	0.823748	+	0.566956i	$0.191879\pi$
$277$	18.8720i	1.13391i	−0.823748	+	0.566956i	$0.808121\pi$
$278$	0	0
$279$	3.53818	0.211825
$280$	0	0
$281$	−17.3995	−1.03797	−0.518984	−	0.854784i	$-0.673690\pi$
$281$	−17.3995	−1.03797	−0.518984	+	0.854784i	$0.673690\pi$
$282$	0	0
$283$	23.6677i	1.40690i	0.710747	+	0.703448i	$0.248357\pi$
$283$	23.6677i	1.40690i	−0.710747	+	0.703448i	$0.751643\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.36527i	0.316702i
$288$	0	0
$289$	0.663896	0.0390527
$290$	0	0
$291$	4.06125	0.238075
$292$	0	0
$293$	19.0317i	1.11184i	0.831235	+	0.555921i	$0.187634\pi$
$293$	19.0317i	1.11184i	−0.831235	+	0.555921i	$0.812366\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.33826i	0.0776538i
$298$	0	0
$299$	−0.0740796	−0.00428414
$300$	0	0
$301$	−4.57537	−0.263720
$302$	0	0
$303$	18.4489i	1.05986i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.669899i	0.0382332i	0.999817	+	0.0191166i	$0.00608537\pi$
$307$	0.669899i	0.0382332i	−0.999817	+	0.0191166i	$0.993915\pi$
$308$	0	0
$309$	14.7164	0.837187
$310$	0	0
$311$	31.5857	1.79106	0.895530	−	0.445001i	$-0.146797\pi$
$311$	31.5857	1.79106	0.895530	+	0.445001i	$0.146797\pi$
$312$	0	0
$313$	− 1.45517i	− 0.0822510i	−0.999154	−	0.0411255i	$-0.986906\pi$
$313$	− 1.45517i	− 0.0822510i	0.999154	−	0.0411255i	$-0.0130943\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.0463i	0.901248i	0.892714	+	0.450624i	$0.148798\pi$
$317$	16.0463i	0.901248i	−0.892714	+	0.450624i	$0.851202\pi$
$318$	0	0
$319$	−1.83307	−0.102632
$320$	0	0
$321$	−5.10528	−0.284949
$322$	0	0
$323$	− 22.7667i	− 1.26677i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4.72587i	− 0.261341i
$328$	0	0
$329$	5.73245	0.316040
$330$	0	0
$331$	−26.9002	−1.47857	−0.739284	−	0.673394i	$-0.764836\pi$
$331$	−26.9002	−1.47857	−0.739284	+	0.673394i	$0.764836\pi$
$332$	0	0
$333$	− 2.18769i	− 0.119885i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.36178i	− 0.0741810i	−0.999312	−	0.0370905i	$-0.988191\pi$
$337$	− 1.36178i	− 0.0741810i	0.999312	−	0.0370905i	$-0.0118090\pi$
$338$	0	0
$339$	−4.13834	−0.224764
$340$	0	0
$341$	4.73501	0.256415
$342$	0	0
$343$	− 7.49850i	− 0.404881i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.30678i	0.123835i	0.998081	+	0.0619173i	$0.0197215\pi$
$347$	2.30678i	0.123835i	−0.998081	+	0.0619173i	$0.980278\pi$
$348$	0	0
$349$	−2.35292	−0.125949	−0.0629744	−	0.998015i	$-0.520059\pi$
$349$	−2.35292	−0.125949	−0.0629744	+	0.998015i	$0.520059\pi$
$350$	0	0
$351$	0.302113	0.0161256
$352$	0	0
$353$	− 30.9225i	− 1.64584i	−0.568157	−	0.822920i	$-0.692343\pi$
$353$	− 30.9225i	− 1.64584i	0.568157	−	0.822920i	$-0.307657\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.21215i	− 0.117079i
$358$	0	0
$359$	19.6302	1.03604	0.518021	−	0.855368i	$-0.326669\pi$
$359$	19.6302	1.03604	0.518021	+	0.855368i	$0.326669\pi$
$360$	0	0
$361$	12.7286	0.669928
$362$	0	0
$363$	− 9.20906i	− 0.483350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.1690i	1.67921i	0.543200	+	0.839603i	$0.317212\pi$
$367$	32.1690i	1.67921i	−0.543200	+	0.839603i	$0.682788\pi$
$368$	0	0
$369$	9.80284	0.510315
$370$	0	0
$371$	−4.21215	−0.218684
$372$	0	0
$373$	− 13.6327i	− 0.705874i	−0.935647	−	0.352937i	$-0.885183\pi$
$373$	− 13.6327i	− 0.705874i	0.935647	−	0.352937i	$-0.114817\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.413816i	0.0213126i
$378$	0	0
$379$	−24.6623	−1.26682	−0.633408	−	0.773818i	$-0.718344\pi$
$379$	−24.6623	−1.26682	−0.633408	+	0.773818i	$0.718344\pi$
$380$	0	0
$381$	−12.6781	−0.649519
$382$	0	0
$383$	− 12.2722i	− 0.627081i	−0.949575	−	0.313540i	$-0.898485\pi$
$383$	− 12.2722i	− 0.627081i	0.949575	−	0.313540i	$-0.101515\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.35963i	0.424944i
$388$	0	0
$389$	−11.6169	−0.588998	−0.294499	−	0.955652i	$-0.595153\pi$
$389$	−11.6169	−0.588998	−0.294499	+	0.955652i	$0.595153\pi$
$390$	0	0
$391$	0.991067	0.0501204
$392$	0	0
$393$	− 12.3309i	− 0.622012i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 17.0341i	− 0.854918i	−0.904035	−	0.427459i	$-0.859409\pi$
$397$	− 17.0341i	− 0.854918i	0.904035	−	0.427459i	$-0.140591\pi$
$398$	0	0
$399$	3.08294	0.154340
$400$	0	0
$401$	32.2134	1.60866	0.804330	−	0.594183i	$-0.202524\pi$
$401$	32.2134	1.60866	0.804330	+	0.594183i	$0.202524\pi$
$402$	0	0
$403$	− 1.06893i	− 0.0532473i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.92770i	− 0.145121i
$408$	0	0
$409$	6.97924	0.345101	0.172551	−	0.985001i	$-0.444799\pi$
$409$	6.97924	0.345101	0.172551	+	0.985001i	$0.444799\pi$
$410$	0	0
$411$	10.7230	0.528926
$412$	0	0
$413$	2.27255i	0.111825i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.6748i	0.669659i
$418$	0	0
$419$	−27.8559	−1.36085	−0.680426	−	0.732817i	$-0.738205\pi$
$419$	−27.8559	−1.36085	−0.680426	+	0.732817i	$0.738205\pi$
$420$	0	0
$421$	8.60036	0.419156	0.209578	−	0.977792i	$-0.432791\pi$
$421$	8.60036	0.419156	0.209578	+	0.977792i	$0.432791\pi$
$422$	0	0
$423$	− 10.4737i	− 0.509249i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.13643i	− 0.0549957i
$428$	0	0
$429$	0.404307	0.0195201
$430$	0	0
$431$	0.180908	0.00871403	0.00435702	−	0.999991i	$-0.498613\pi$
$431$	0.180908	0.00871403	0.00435702	+	0.999991i	$0.498613\pi$
$432$	0	0
$433$	− 31.4400i	− 1.51091i	−0.655201	−	0.755455i	$-0.727416\pi$
$433$	− 31.4400i	− 1.51091i	0.655201	−	0.755455i	$-0.272584\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.38119i	0.0660715i
$438$	0	0
$439$	16.8128	0.802432	0.401216	−	0.915983i	$-0.368588\pi$
$439$	16.8128	0.802432	0.401216	+	0.915983i	$0.368588\pi$
$440$	0	0
$441$	−6.70044	−0.319069
$442$	0	0
$443$	36.1952i	1.71969i	0.510559	+	0.859843i	$0.329439\pi$
$443$	36.1952i	1.71969i	−0.510559	+	0.859843i	$0.670561\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.88176i	− 0.420093i
$448$	0	0
$449$	−28.3299	−1.33697	−0.668486	−	0.743725i	$-0.733057\pi$
$449$	−28.3299	−1.33697	−0.668486	+	0.743725i	$0.733057\pi$
$450$	0	0
$451$	13.1188	0.617738
$452$	0	0
$453$	2.68310i	0.126063i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16.5396i	− 0.773691i	−0.922144	−	0.386846i	$-0.873565\pi$
$457$	− 16.5396i	− 0.773691i	0.922144	−	0.386846i	$-0.126435\pi$
$458$	0	0
$459$	−4.04179	−0.188655
$460$	0	0
$461$	0.434035	0.0202150	0.0101075	−	0.999949i	$-0.496783\pi$
$461$	0.434035	0.0202150	0.0101075	+	0.999949i	$0.496783\pi$
$462$	0	0
$463$	− 19.9625i	− 0.927738i	−0.885904	−	0.463869i	$-0.846461\pi$
$463$	− 19.9625i	− 0.927738i	0.885904	−	0.463869i	$-0.153539\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.20727i	0.287238i	0.989633	+	0.143619i	$0.0458740\pi$
$467$	6.20727i	0.287238i	−0.989633	+	0.143619i	$0.954126\pi$
$468$	0	0
$469$	−4.64451	−0.214464
$470$	0	0
$471$	11.7576	0.541760
$472$	0	0
$473$	11.1874i	0.514395i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.69598i	0.352375i
$478$	0	0
$479$	−23.4677	−1.07227	−0.536133	−	0.844134i	$-0.680115\pi$
$479$	−23.4677	−1.07227	−0.536133	+	0.844134i	$0.680115\pi$
$480$	0	0
$481$	−0.660931	−0.0301359
$482$	0	0
$483$	0.134205i	0.00610654i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.34615i	0.151628i	0.997122	+	0.0758142i	$0.0241556\pi$
$487$	3.34615i	0.151628i	−0.997122	+	0.0758142i	$0.975844\pi$
$488$	0	0
$489$	18.3127	0.828129
$490$	0	0
$491$	−0.546988	−0.0246852	−0.0123426	−	0.999924i	$-0.503929\pi$
$491$	−0.546988	−0.0246852	−0.0123426	+	0.999924i	$0.503929\pi$
$492$	0	0
$493$	− 5.53620i	− 0.249338i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.02615i	0.225453i
$498$	0	0
$499$	8.08164	0.361784	0.180892	−	0.983503i	$-0.442102\pi$
$499$	8.08164	0.361784	0.180892	+	0.983503i	$0.442102\pi$
$500$	0	0
$501$	4.40418	0.196764
$502$	0	0
$503$	− 28.3131i	− 1.26242i	−0.775613	−	0.631209i	$-0.782559\pi$
$503$	− 28.3131i	− 1.26242i	0.775613	−	0.631209i	$-0.217441\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.9087i	0.573297i
$508$	0	0
$509$	−5.71024	−0.253102	−0.126551	−	0.991960i	$-0.540391\pi$
$509$	−5.71024	−0.253102	−0.126551	+	0.991960i	$0.540391\pi$
$510$	0	0
$511$	5.99640	0.265265
$512$	0	0
$513$	− 5.63282i	− 0.248695i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 14.0166i	− 0.616448i
$518$	0	0
$519$	15.9307	0.699282
$520$	0	0
$521$	−25.4900	−1.11674	−0.558368	−	0.829594i	$-0.688572\pi$
$521$	−25.4900	−1.11674	−0.558368	+	0.829594i	$0.688572\pi$
$522$	0	0
$523$	2.83180i	0.123826i	0.998082	+	0.0619131i	$0.0197202\pi$
$523$	2.83180i	0.123826i	−0.998082	+	0.0619131i	$0.980280\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.3006i	0.622944i
$528$	0	0
$529$	22.9399	0.997386
$530$	0	0
$531$	4.15215	0.180188
$532$	0	0
$533$	− 2.96157i	− 0.128280i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3.76978i	− 0.162678i
$538$	0	0
$539$	−8.96694	−0.386234
$540$	0	0
$541$	4.50090	0.193509	0.0967543	−	0.995308i	$-0.469154\pi$
$541$	4.50090	0.193509	0.0967543	+	0.995308i	$0.469154\pi$
$542$	0	0
$543$	− 26.6868i	− 1.14524i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 9.51551i	− 0.406854i	−0.979090	−	0.203427i	$-0.934792\pi$
$547$	− 9.51551i	− 0.406854i	0.979090	−	0.203427i	$-0.0652079\pi$
$548$	0	0
$549$	−2.07636	−0.0886170
$550$	0	0
$551$	7.71549	0.328691
$552$	0	0
$553$	8.30987i	0.353372i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 25.5173i	− 1.08120i	−0.841279	−	0.540601i	$-0.818197\pi$
$557$	− 25.5173i	− 1.08120i	0.841279	−	0.540601i	$-0.181803\pi$
$558$	0	0
$559$	2.52555	0.106820
$560$	0	0
$561$	−5.40898	−0.228367
$562$	0	0
$563$	13.7426i	0.579183i	0.957150	+	0.289592i	$0.0935195\pi$
$563$	13.7426i	0.579183i	−0.957150	+	0.289592i	$0.906480\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 0.547318i	− 0.0229852i
$568$	0	0
$569$	28.3439	1.18824	0.594118	−	0.804378i	$-0.297501\pi$
$569$	28.3439	1.18824	0.594118	+	0.804378i	$0.297501\pi$
$570$	0	0
$571$	−1.95314	−0.0817362	−0.0408681	−	0.999165i	$-0.513012\pi$
$571$	−1.95314	−0.0817362	−0.0408681	+	0.999165i	$0.513012\pi$
$572$	0	0
$573$	4.63282i	0.193539i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 42.8387i	− 1.78340i	−0.452630	−	0.891698i	$-0.649514\pi$
$577$	− 42.8387i	− 1.78340i	0.452630	−	0.891698i	$-0.350486\pi$
$578$	0	0
$579$	20.7440	0.862092
$580$	0	0
$581$	−3.64195	−0.151094
$582$	0	0
$583$	10.2992i	0.426550i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.97579i	0.0815497i	0.999168	+	0.0407748i	$0.0129826\pi$
$587$	1.97579i	0.0815497i	−0.999168	+	0.0407748i	$0.987017\pi$
$588$	0	0
$589$	−19.9299	−0.821198
$590$	0	0
$591$	20.5338	0.844647
$592$	0	0
$593$	− 17.3986i	− 0.714475i	−0.934014	−	0.357237i	$-0.883719\pi$
$593$	− 17.3986i	− 0.714475i	0.934014	−	0.357237i	$-0.116281\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.91437i	− 0.242059i
$598$	0	0
$599$	3.17298	0.129644	0.0648222	−	0.997897i	$-0.479352\pi$
$599$	3.17298	0.129644	0.0648222	+	0.997897i	$0.479352\pi$
$600$	0	0
$601$	32.1659	1.31207	0.656037	−	0.754729i	$-0.272232\pi$
$601$	32.1659	1.31207	0.656037	+	0.754729i	$0.272232\pi$
$602$	0	0
$603$	8.48594i	0.345574i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.4215i	1.15359i	0.816888	+	0.576797i	$0.195698\pi$
$607$	28.4215i	1.15359i	−0.816888	+	0.576797i	$0.804302\pi$
$608$	0	0
$609$	0.749683	0.0303787
$610$	0	0
$611$	−3.16425	−0.128012
$612$	0	0
$613$	− 2.48351i	− 0.100308i	−0.998741	−	0.0501541i	$-0.984029\pi$
$613$	− 2.48351i	− 0.100308i	0.998741	−	0.0501541i	$-0.0159712\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.83668i	− 0.396010i	−0.980201	−	0.198005i	$-0.936554\pi$
$617$	− 9.83668i	− 0.396010i	0.980201	−	0.198005i	$-0.0634462\pi$
$618$	0	0
$619$	13.6885	0.550186	0.275093	−	0.961418i	$-0.411291\pi$
$619$	13.6885	0.550186	0.275093	+	0.961418i	$0.411291\pi$
$620$	0	0
$621$	0.245205	0.00983973
$622$	0	0
$623$	2.72490i	0.109171i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 7.53818i	− 0.301046i
$628$	0	0
$629$	8.84220	0.352562
$630$	0	0
$631$	19.9161	0.792847	0.396423	−	0.918068i	$-0.370251\pi$
$631$	19.9161	0.792847	0.396423	+	0.918068i	$0.370251\pi$
$632$	0	0
$633$	22.9305i	0.911407i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.02429i	0.0802054i
$638$	0	0
$639$	9.18323	0.363283
$640$	0	0
$641$	−13.1526	−0.519496	−0.259748	−	0.965676i	$-0.583640\pi$
$641$	−13.1526	−0.519496	−0.259748	+	0.965676i	$0.583640\pi$
$642$	0	0
$643$	− 34.4841i	− 1.35992i	−0.733249	−	0.679960i	$-0.761997\pi$
$643$	− 34.4841i	− 1.35992i	0.733249	−	0.679960i	$-0.238003\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.1515i	1.49989i	0.661500	+	0.749945i	$0.269920\pi$
$647$	38.1515i	1.49989i	−0.661500	+	0.749945i	$0.730080\pi$
$648$	0	0
$649$	5.55666	0.218118
$650$	0	0
$651$	−1.93651	−0.0758978
$652$	0	0
$653$	− 34.6440i	− 1.35572i	−0.735189	−	0.677862i	$-0.762907\pi$
$653$	− 34.6440i	− 1.35572i	0.735189	−	0.677862i	$-0.237093\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.9560i	− 0.427433i
$658$	0	0
$659$	−50.2493	−1.95743	−0.978717	−	0.205213i	$-0.934211\pi$
$659$	−50.2493	−1.95743	−0.978717	+	0.205213i	$0.934211\pi$
$660$	0	0
$661$	10.7839	0.419445	0.209723	−	0.977761i	$-0.432744\pi$
$661$	10.7839	0.419445	0.209723	+	0.977761i	$0.432744\pi$
$662$	0	0
$663$	1.22108i	0.0474228i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.335866i	0.0130048i
$668$	0	0
$669$	12.7498	0.492936
$670$	0	0
$671$	−2.77872	−0.107271
$672$	0	0
$673$	− 16.8119i	− 0.648051i	−0.946048	−	0.324026i	$-0.894964\pi$
$673$	− 16.8119i	− 0.648051i	0.946048	−	0.324026i	$-0.105036\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 37.2375i	− 1.43115i	−0.698534	−	0.715577i	$-0.746164\pi$
$677$	− 37.2375i	− 1.43115i	0.698534	−	0.715577i	$-0.253836\pi$
$678$	0	0
$679$	−2.22280	−0.0853030
$680$	0	0
$681$	11.0294	0.422649
$682$	0	0
$683$	− 31.8309i	− 1.21798i	−0.793180	−	0.608988i	$-0.791576\pi$
$683$	− 31.8309i	− 1.21798i	0.793180	−	0.608988i	$-0.208424\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	23.0801i	0.880561i
$688$	0	0
$689$	2.32506	0.0885776
$690$	0	0
$691$	−18.9658	−0.721492	−0.360746	−	0.932664i	$-0.617478\pi$
$691$	−18.9658	−0.721492	−0.360746	+	0.932664i	$0.617478\pi$
$692$	0	0
$693$	− 0.732455i	− 0.0278237i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	39.6211i	1.50075i
$698$	0	0
$699$	29.2882	1.10778
$700$	0	0
$701$	−19.2027	−0.725275	−0.362638	−	0.931930i	$-0.618124\pi$
$701$	−19.2027	−0.725275	−0.362638	+	0.931930i	$0.618124\pi$
$702$	0	0
$703$	12.3229i	0.464766i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.0974i	− 0.379752i
$708$	0	0
$709$	11.0244	0.414032	0.207016	−	0.978338i	$-0.433625\pi$
$709$	11.0244	0.414032	0.207016	+	0.978338i	$0.433625\pi$
$710$	0	0
$711$	15.1829	0.569403
$712$	0	0
$713$	− 0.867579i	− 0.0324911i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10.5986i	− 0.395811i
$718$	0	0
$719$	−26.5680	−0.990818	−0.495409	−	0.868660i	$-0.664982\pi$
$719$	−26.5680	−0.990818	−0.495409	+	0.868660i	$0.664982\pi$
$720$	0	0
$721$	−8.05456	−0.299967
$722$	0	0
$723$	19.0216i	0.707422i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 27.3448i	− 1.01416i	−0.861898	−	0.507081i	$-0.830725\pi$
$727$	− 27.3448i	− 1.01416i	0.861898	−	0.507081i	$-0.169275\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−33.7879	−1.24969
$732$	0	0
$733$	8.30460i	0.306737i	0.988169	+	0.153369i	$0.0490122\pi$
$733$	8.30460i	0.306737i	−0.988169	+	0.153369i	$0.950988\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.3564i	0.418319i
$738$	0	0
$739$	−1.61459	−0.0593935	−0.0296968	−	0.999559i	$-0.509454\pi$
$739$	−1.61459	−0.0593935	−0.0296968	+	0.999559i	$0.509454\pi$
$740$	0	0
$741$	−1.70175	−0.0625153
$742$	0	0
$743$	− 3.77367i	− 0.138443i	−0.997601	−	0.0692213i	$-0.977949\pi$
$743$	− 3.77367i	− 0.138443i	0.997601	−	0.0692213i	$-0.0220515\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.65418i	0.243464i
$748$	0	0
$749$	2.79421	0.102098
$750$	0	0
$751$	53.4032	1.94871	0.974355	−	0.225016i	$-0.0722435\pi$
$751$	53.4032	1.94871	0.974355	+	0.225016i	$0.0722435\pi$
$752$	0	0
$753$	0.0370816i	0.00135133i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 11.6322i	− 0.422781i	−0.977402	−	0.211390i	$-0.932201\pi$
$757$	− 11.6322i	− 0.422781i	0.977402	−	0.211390i	$-0.0677992\pi$
$758$	0	0
$759$	0.328148	0.0119110
$760$	0	0
$761$	4.66772	0.169205	0.0846023	−	0.996415i	$-0.473038\pi$
$761$	4.66772	0.169205	0.0846023	+	0.996415i	$0.473038\pi$
$762$	0	0
$763$	2.58656i	0.0936396i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.25442i	− 0.0452945i
$768$	0	0
$769$	22.6044	0.815137	0.407568	−	0.913175i	$-0.366377\pi$
$769$	22.6044	0.815137	0.407568	+	0.913175i	$0.366377\pi$
$770$	0	0
$771$	−10.0237	−0.360995
$772$	0	0
$773$	39.6786i	1.42714i	0.700584	+	0.713570i	$0.252923\pi$
$773$	39.6786i	1.42714i	−0.700584	+	0.713570i	$0.747077\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.19736i	0.0429552i
$778$	0	0
$779$	−55.2176	−1.97838
$780$	0	0
$781$	12.2896	0.439755
$782$	0	0
$783$	− 1.36974i	− 0.0489505i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.73737i	− 0.168869i	−0.996429	−	0.0844345i	$-0.973092\pi$
$787$	− 4.73737i	− 0.168869i	0.996429	−	0.0844345i	$-0.0269084\pi$
$788$	0	0
$789$	12.7818	0.455044
$790$	0	0
$791$	2.26499	0.0805337
$792$	0	0
$793$	0.627297i	0.0222760i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 28.9009i	− 1.02372i	−0.859068	−	0.511862i	$-0.828956\pi$
$797$	− 28.9009i	− 1.02372i	0.859068	−	0.511862i	$-0.171044\pi$
$798$	0	0
$799$	42.3326	1.49762
$800$	0	0
$801$	4.97864	0.175911
$802$	0	0
$803$	− 14.6619i	− 0.517409i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 21.3115i	− 0.750201i
$808$	0	0
$809$	−44.6499	−1.56981	−0.784903	−	0.619619i	$-0.787287\pi$
$809$	−44.6499	−1.56981	−0.784903	+	0.619619i	$0.787287\pi$
$810$	0	0
$811$	−45.1603	−1.58579	−0.792896	−	0.609357i	$-0.791428\pi$
$811$	−45.1603	−1.58579	−0.792896	+	0.609357i	$0.791428\pi$
$812$	0	0
$813$	− 13.4036i	− 0.470085i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 47.0882i	− 1.64741i
$818$	0	0
$819$	−0.165352	−0.00577787
$820$	0	0
$821$	47.7380	1.66607	0.833035	−	0.553221i	$-0.186601\pi$
$821$	47.7380	1.66607	0.833035	+	0.553221i	$0.186601\pi$
$822$	0	0
$823$	− 6.46344i	− 0.225301i	−0.993635	−	0.112651i	$-0.964066\pi$
$823$	− 6.46344i	− 0.225301i	0.993635	−	0.112651i	$-0.0359341\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	50.1850i	1.74510i	0.488522	+	0.872552i	$0.337536\pi$
$827$	50.1850i	1.74510i	−0.488522	+	0.872552i	$0.662464\pi$
$828$	0	0
$829$	37.2464	1.29362	0.646810	−	0.762651i	$-0.276103\pi$
$829$	37.2464	1.29362	0.646810	+	0.762651i	$0.276103\pi$
$830$	0	0
$831$	−18.8720	−0.654664
$832$	0	0
$833$	− 27.0818i	− 0.938329i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.53818i	0.122297i
$838$	0	0
$839$	−13.2471	−0.457339	−0.228670	−	0.973504i	$-0.573438\pi$
$839$	−13.2471	−0.457339	−0.228670	+	0.973504i	$0.573438\pi$
$840$	0	0
$841$	−27.1238	−0.935304
$842$	0	0
$843$	− 17.3995i	− 0.599271i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.04028i	0.173186i
$848$	0	0
$849$	−23.6677	−0.812272
$850$	0	0
$851$	−0.536433	−0.0183887
$852$	0	0
$853$	44.6784i	1.52976i	0.644173	+	0.764880i	$0.277202\pi$
$853$	44.6784i	1.52976i	−0.644173	+	0.764880i	$0.722798\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.6798i	1.42375i	0.702304	+	0.711877i	$0.252155\pi$
$857$	41.6798i	1.42375i	−0.702304	+	0.711877i	$0.747845\pi$
$858$	0	0
$859$	37.5766	1.28210	0.641048	−	0.767501i	$-0.278500\pi$
$859$	37.5766	1.28210	0.641048	+	0.767501i	$0.278500\pi$
$860$	0	0
$861$	−5.36527	−0.182848
$862$	0	0
$863$	56.2069i	1.91331i	0.291230	+	0.956653i	$0.405936\pi$
$863$	56.2069i	1.91331i	−0.291230	+	0.956653i	$0.594064\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.663896i	0.0225471i
$868$	0	0
$869$	20.3187	0.689264
$870$	0	0
$871$	2.56372	0.0868682
$872$	0	0
$873$	4.06125i	0.137452i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.36231i	− 0.0797695i	−0.999204	−	0.0398847i	$-0.987301\pi$
$877$	− 2.36231i	− 0.0797695i	0.999204	−	0.0398847i	$-0.0126991\pi$
$878$	0	0
$879$	−19.0317	−0.641923
$880$	0	0
$881$	47.6703	1.60605	0.803027	−	0.595942i	$-0.203221\pi$
$881$	47.6703	1.60605	0.803027	+	0.595942i	$0.203221\pi$
$882$	0	0
$883$	9.95268i	0.334934i	0.985878	+	0.167467i	$0.0535588\pi$
$883$	9.95268i	0.334934i	−0.985878	+	0.167467i	$0.946441\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.3636i	0.448707i	0.974508	+	0.224353i	$0.0720270\pi$
$887$	13.3636i	0.448707i	−0.974508	+	0.224353i	$0.927973\pi$
$888$	0	0
$889$	6.93896	0.232725
$890$	0	0
$891$	−1.33826	−0.0448334
$892$	0	0
$893$	58.9965i	1.97424i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 0.0740796i	− 0.00247345i
$898$	0	0
$899$	−4.84638	−0.161636
$900$	0	0
$901$	−31.1056	−1.03628
$902$	0	0
$903$	− 4.57537i	− 0.152259i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.4136i	1.07627i	0.842857	+	0.538137i	$0.180872\pi$
$907$	32.4136i	1.07627i	−0.842857	+	0.538137i	$0.819128\pi$
$908$	0	0
$909$	−18.4489	−0.611910
$910$	0	0
$911$	−40.1655	−1.33074	−0.665371	−	0.746513i	$-0.731727\pi$
$911$	−40.1655	−1.33074	−0.665371	+	0.746513i	$0.731727\pi$
$912$	0	0
$913$	8.90503i	0.294714i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.74893i	0.222869i
$918$	0	0
$919$	−32.3349	−1.06663	−0.533315	−	0.845917i	$-0.679054\pi$
$919$	−32.3349	−1.06663	−0.533315	+	0.845917i	$0.679054\pi$
$920$	0	0
$921$	−0.669899	−0.0220739
$922$	0	0
$923$	− 2.77438i	− 0.0913197i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.7164i	0.483350i
$928$	0	0
$929$	−53.3773	−1.75125	−0.875626	−	0.482990i	$-0.839551\pi$
$929$	−53.3773	−1.75125	−0.875626	+	0.482990i	$0.839551\pi$
$930$	0	0
$931$	37.7424	1.23696
$932$	0	0
$933$	31.5857i	1.03407i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.55999i	0.279643i	0.990177	+	0.139821i	$0.0446528\pi$
$937$	8.55999i	0.279643i	−0.990177	+	0.139821i	$0.955347\pi$
$938$	0	0
$939$	1.45517	0.0474876
$940$	0	0
$941$	33.8168	1.10239	0.551197	−	0.834375i	$-0.314171\pi$
$941$	33.8168	1.10239	0.551197	+	0.834375i	$0.314171\pi$
$942$	0	0
$943$	− 2.40370i	− 0.0782753i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.0492i	0.554025i	0.960866	+	0.277013i	$0.0893444\pi$
$947$	17.0492i	0.554025i	−0.960866	+	0.277013i	$0.910656\pi$
$948$	0	0
$949$	−3.30994	−0.107445
$950$	0	0
$951$	−16.0463	−0.520336
$952$	0	0
$953$	− 37.6100i	− 1.21831i	−0.793052	−	0.609154i	$-0.791509\pi$
$953$	− 37.6100i	− 1.21831i	0.793052	−	0.609154i	$-0.208491\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.83307i	− 0.0592547i
$958$	0	0
$959$	−5.86889	−0.189516
$960$	0	0
$961$	−18.4813	−0.596170
$962$	0	0
$963$	− 5.10528i	− 0.164515i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.7432i	0.538427i	0.963081	+	0.269213i	$0.0867637\pi$
$967$	16.7432i	0.538427i	−0.963081	+	0.269213i	$0.913236\pi$
$968$	0	0
$969$	22.7667	0.731372
$970$	0	0
$971$	−6.17818	−0.198267	−0.0991337	−	0.995074i	$-0.531607\pi$
$971$	−6.17818	−0.198267	−0.0991337	+	0.995074i	$0.531607\pi$
$972$	0	0
$973$	− 7.48448i	− 0.239941i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 13.1652i	− 0.421191i	−0.977573	−	0.210595i	$-0.932460\pi$
$977$	− 13.1652i	− 0.421191i	0.977573	−	0.210595i	$-0.0675402\pi$
$978$	0	0
$979$	6.66272	0.212941
$980$	0	0
$981$	4.72587	0.150886
$982$	0	0
$983$	13.7222i	0.437670i	0.975762	+	0.218835i	$0.0702257\pi$
$983$	13.7222i	0.437670i	−0.975762	+	0.218835i	$0.929774\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.73245i	0.182466i
$988$	0	0
$989$	2.04982	0.0651805
$990$	0	0
$991$	−47.6522	−1.51372	−0.756861	−	0.653576i	$-0.773268\pi$
$991$	−47.6522	−1.51372	−0.756861	+	0.653576i	$0.773268\pi$
$992$	0	0
$993$	− 26.9002i	− 0.853652i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.8494i	0.628636i	0.949318	+	0.314318i	$0.101776\pi$
$997$	19.8494i	0.628636i	−0.949318	+	0.314318i	$0.898224\pi$
$998$	0	0
$999$	2.18769	0.0692155

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.d.1249.6		8
5.2	odd	4		7500.2.a.g.1.3		4
5.3	odd	4		7500.2.a.d.1.2		4
5.4	even	2	inner	7500.2.d.d.1249.3		8
25.3	odd	20		1500.2.m.b.1201.1		8
25.4	even	10		1500.2.o.a.49.4		16
25.6	even	5		1500.2.o.a.949.3		16
25.8	odd	20		1500.2.m.b.301.1		8
25.17	odd	20		300.2.m.a.61.2	✓	8
25.19	even	10		1500.2.o.a.949.2		16
25.21	even	5		1500.2.o.a.49.1		16
25.22	odd	20		300.2.m.a.241.2	yes	8
75.17	even	20		900.2.n.a.361.1		8
75.47	even	20		900.2.n.a.541.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.a.61.2	✓	8	25.17	odd	20
300.2.m.a.241.2	yes	8	25.22	odd	20
900.2.n.a.361.1		8	75.17	even	20
900.2.n.a.541.1		8	75.47	even	20
1500.2.m.b.301.1		8	25.8	odd	20
1500.2.m.b.1201.1		8	25.3	odd	20
1500.2.o.a.49.1		16	25.21	even	5
1500.2.o.a.49.4		16	25.4	even	10
1500.2.o.a.949.2		16	25.19	even	10
1500.2.o.a.949.3		16	25.6	even	5
7500.2.a.d.1.2		4	5.3	odd	4
7500.2.a.g.1.3		4	5.2	odd	4
7500.2.d.d.1249.3		8	5.4	even	2	inner
7500.2.d.d.1249.6		8	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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -0.547318i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -0.547318i q^{7} -1.00000 q^{9} -1.33826 q^{11} +0.302113i q^{13} -4.04179i q^{17} +5.63282 q^{19} +0.547318 q^{21} +0.245205i q^{23} -1.00000i q^{27} +1.36974 q^{29} -3.53818 q^{31} -1.33826i q^{33} +2.18769i q^{37} -0.302113 q^{39} -9.80284 q^{41} -8.35963i q^{43} +10.4737i q^{47} +6.70044 q^{49} +4.04179 q^{51} -7.69598i q^{53} +5.63282i q^{57} -4.15215 q^{59} +2.07636 q^{61} +0.547318i q^{63} -8.48594i q^{67} -0.245205 q^{69} -9.18323 q^{71} +10.9560i q^{73} +0.732455i q^{77} -15.1829 q^{79} +1.00000 q^{81} -6.65418i q^{83} +1.36974i q^{87} -4.97864 q^{89} +0.165352 q^{91} -3.53818i q^{93} -4.06125i q^{97} +1.33826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} + 8 q^{29} - 18 q^{31} - 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 4 q^{61} + 18 q^{69} - 40 q^{71} + 6 q^{79} + 8 q^{81} - 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 + 8 * q^29 - 18 * q^31 - 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 4 * q^61 + 18 * q^69 - 40 * q^71 + 6 * q^79 + 8 * q^81 - 30 * q^89 - 20 * q^91 + 2 * q^99

Introduction

L-functions

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