LMFDB - Embedded newform 2300.2.c.h.1749.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.c (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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.2.c.h.1749.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{3} -1.56155i q^{7} -3.56155 q^{9} +O(q^{10})$	$q-2.56155i q^{3} -1.56155i q^{7} -3.56155 q^{9} +2.00000 q^{11} -0.561553i q^{13} -1.56155i q^{17} -6.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} +1.43845i q^{27} +2.12311 q^{29} -9.24621 q^{31} -5.12311i q^{33} -0.438447i q^{37} -1.43845 q^{39} -4.12311 q^{41} -7.68466i q^{47} +4.56155 q^{49} -4.00000 q^{51} +0.438447i q^{53} +15.3693i q^{57} -8.68466 q^{59} +1.12311 q^{61} +5.56155i q^{63} -4.43845i q^{67} -2.56155 q^{69} +1.87689 q^{71} +8.56155i q^{73} -3.12311i q^{77} -13.1231 q^{79} -7.00000 q^{81} -14.9309i q^{83} -5.43845i q^{87} +2.24621 q^{89} -0.876894 q^{91} +23.6847i q^{93} -4.87689i q^{97} -7.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} + 8 q^{11} - 24 q^{19} - 16 q^{21} - 8 q^{29} - 4 q^{31} - 14 q^{39} + 10 q^{49} - 16 q^{51} - 10 q^{59} - 12 q^{61} - 2 q^{69} + 24 q^{71} - 36 q^{79} - 28 q^{81} - 24 q^{89} - 20 q^{91} - 12 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 + 8 * q^11 - 24 * q^19 - 16 * q^21 - 8 * q^29 - 4 * q^31 - 14 * q^39 + 10 * q^49 - 16 * q^51 - 10 * q^59 - 12 * q^61 - 2 * q^69 + 24 * q^71 - 36 * q^79 - 28 * q^81 - 24 * q^89 - 20 * q^91 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.56155i	− 1.47891i	−0.673204	−	0.739457i	$-0.735083\pi$
$3$	− 2.56155i	− 1.47891i	0.673204	−	0.739457i	$-0.264917\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.56155i	− 0.590211i	−0.955465	−	0.295106i	$-0.904645\pi$
$7$	− 1.56155i	− 0.590211i	0.955465	−	0.295106i	$-0.0953549\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	− 0.561553i	− 0.155747i	−0.996963	−	0.0778734i	$-0.975187\pi$
$13$	− 0.561553i	− 0.155747i	0.996963	−	0.0778734i	$-0.0248130\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.56155i	− 0.378732i	−0.981907	−	0.189366i	$-0.939357\pi$
$17$	− 1.56155i	− 0.378732i	0.981907	−	0.189366i	$-0.0606433\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.43845i	0.276829i
$28$	0	0
$29$	2.12311	0.394251	0.197125	−	0.980378i	$-0.436839\pi$
$29$	2.12311	0.394251	0.197125	+	0.980378i	$0.436839\pi$
$30$	0	0
$31$	−9.24621	−1.66067	−0.830334	−	0.557266i	$-0.811851\pi$
$31$	−9.24621	−1.66067	−0.830334	+	0.557266i	$0.811851\pi$
$32$	0	0
$33$	− 5.12311i	− 0.891818i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.438447i	− 0.0720803i	−0.999350	−	0.0360401i	$-0.988526\pi$
$37$	− 0.438447i	− 0.0720803i	0.999350	−	0.0360401i	$-0.0114744\pi$
$38$	0	0
$39$	−1.43845	−0.230336
$40$	0	0
$41$	−4.12311	−0.643921	−0.321960	−	0.946753i	$-0.604342\pi$
$41$	−4.12311	−0.643921	−0.321960	+	0.946753i	$0.604342\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.68466i	− 1.12092i	−0.828181	−	0.560461i	$-0.810624\pi$
$47$	− 7.68466i	− 1.12092i	0.828181	−	0.560461i	$-0.189376\pi$
$48$	0	0
$49$	4.56155	0.651650
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	0.438447i	0.0602254i	0.999547	+	0.0301127i	$0.00958661\pi$
$53$	0.438447i	0.0602254i	−0.999547	+	0.0301127i	$0.990413\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15.3693i	2.03572i
$58$	0	0
$59$	−8.68466	−1.13065	−0.565323	−	0.824870i	$-0.691249\pi$
$59$	−8.68466	−1.13065	−0.565323	+	0.824870i	$0.691249\pi$
$60$	0	0
$61$	1.12311	0.143799	0.0718995	−	0.997412i	$-0.477094\pi$
$61$	1.12311	0.143799	0.0718995	+	0.997412i	$0.477094\pi$
$62$	0	0
$63$	5.56155i	0.700690i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.43845i	− 0.542243i	−0.962545	−	0.271121i	$-0.912606\pi$
$67$	− 4.43845i	− 0.542243i	0.962545	−	0.271121i	$-0.0873945\pi$
$68$	0	0
$69$	−2.56155	−0.308375
$70$	0	0
$71$	1.87689	0.222746	0.111373	−	0.993779i	$-0.464475\pi$
$71$	1.87689	0.222746	0.111373	+	0.993779i	$0.464475\pi$
$72$	0	0
$73$	8.56155i	1.00205i	0.865432	+	0.501027i	$0.167044\pi$
$73$	8.56155i	1.00205i	−0.865432	+	0.501027i	$0.832956\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.12311i	− 0.355911i
$78$	0	0
$79$	−13.1231	−1.47646	−0.738232	−	0.674546i	$-0.764339\pi$
$79$	−13.1231	−1.47646	−0.738232	+	0.674546i	$0.764339\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 14.9309i	− 1.63888i	−0.573168	−	0.819438i	$-0.694286\pi$
$83$	− 14.9309i	− 1.63888i	0.573168	−	0.819438i	$-0.305714\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.43845i	− 0.583063i
$88$	0	0
$89$	2.24621	0.238098	0.119049	−	0.992888i	$-0.462015\pi$
$89$	2.24621	0.238098	0.119049	+	0.992888i	$0.462015\pi$
$90$	0	0
$91$	−0.876894	−0.0919235
$92$	0	0
$93$	23.6847i	2.45598i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.87689i	− 0.495174i	−0.968866	−	0.247587i	$-0.920362\pi$
$97$	− 4.87689i	− 0.495174i	0.968866	−	0.247587i	$-0.0796375\pi$
$98$	0	0
$99$	−7.12311	−0.715899
$100$	0	0
$101$	−5.31534	−0.528896	−0.264448	−	0.964400i	$-0.585190\pi$
$101$	−5.31534	−0.528896	−0.264448	+	0.964400i	$0.585190\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.56155i	0.731003i	0.930811	+	0.365501i	$0.119102\pi$
$107$	7.56155i	0.731003i	−0.930811	+	0.365501i	$0.880898\pi$
$108$	0	0
$109$	9.36932	0.897418	0.448709	−	0.893678i	$-0.351884\pi$
$109$	9.36932	0.897418	0.448709	+	0.893678i	$0.351884\pi$
$110$	0	0
$111$	−1.12311	−0.106600
$112$	0	0
$113$	7.80776i	0.734493i	0.930124	+	0.367246i	$0.119699\pi$
$113$	7.80776i	0.734493i	−0.930124	+	0.367246i	$0.880301\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−2.43845	−0.223532
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	10.5616i	0.952303i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 12.8078i	− 1.13651i	−0.822854	−	0.568253i	$-0.807620\pi$
$127$	− 12.8078i	− 1.13651i	0.822854	−	0.568253i	$-0.192380\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.8078	1.81798	0.908991	−	0.416815i	$-0.136854\pi$
$131$	20.8078	1.81798	0.908991	+	0.416815i	$0.136854\pi$
$132$	0	0
$133$	9.36932i	0.812423i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.1231i	1.29205i	0.763315	+	0.646027i	$0.223571\pi$
$137$	15.1231i	1.29205i	−0.763315	+	0.646027i	$0.776429\pi$
$138$	0	0
$139$	−5.24621	−0.444978	−0.222489	−	0.974935i	$-0.571418\pi$
$139$	−5.24621	−0.444978	−0.222489	+	0.974935i	$0.571418\pi$
$140$	0	0
$141$	−19.6847	−1.65775
$142$	0	0
$143$	− 1.12311i	− 0.0939188i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.6847i	− 0.963734i
$148$	0	0
$149$	−15.3693	−1.25910	−0.629552	−	0.776959i	$-0.716761\pi$
$149$	−15.3693	−1.25910	−0.629552	+	0.776959i	$0.716761\pi$
$150$	0	0
$151$	23.0540	1.87611	0.938053	−	0.346492i	$-0.112627\pi$
$151$	23.0540	1.87611	0.938053	+	0.346492i	$0.112627\pi$
$152$	0	0
$153$	5.56155i	0.449625i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.6847i	− 1.33158i	−0.746139	−	0.665790i	$-0.768095\pi$
$157$	− 16.6847i	− 1.33158i	0.746139	−	0.665790i	$-0.231905\pi$
$158$	0	0
$159$	1.12311	0.0890681
$160$	0	0
$161$	−1.56155	−0.123068
$162$	0	0
$163$	11.9309i	0.934498i	0.884126	+	0.467249i	$0.154755\pi$
$163$	11.9309i	0.934498i	−0.884126	+	0.467249i	$0.845245\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	12.6847	0.975743
$170$	0	0
$171$	21.3693	1.63415
$172$	0	0
$173$	19.3693i	1.47262i	0.676643	+	0.736311i	$0.263434\pi$
$173$	19.3693i	1.47262i	−0.676643	+	0.736311i	$0.736566\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	22.2462i	1.67213i
$178$	0	0
$179$	−21.9309	−1.63919	−0.819595	−	0.572943i	$-0.805802\pi$
$179$	−21.9309	−1.63919	−0.819595	+	0.572943i	$0.805802\pi$
$180$	0	0
$181$	19.3693	1.43971	0.719855	−	0.694124i	$-0.244208\pi$
$181$	19.3693	1.43971	0.719855	+	0.694124i	$0.244208\pi$
$182$	0	0
$183$	− 2.87689i	− 0.212666i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.12311i	− 0.228384i
$188$	0	0
$189$	2.24621	0.163388
$190$	0	0
$191$	−7.36932	−0.533225	−0.266613	−	0.963804i	$-0.585904\pi$
$191$	−7.36932	−0.533225	−0.266613	+	0.963804i	$0.585904\pi$
$192$	0	0
$193$	− 6.56155i	− 0.472311i	−0.971715	−	0.236155i	$-0.924113\pi$
$193$	− 6.56155i	− 0.472311i	0.971715	−	0.236155i	$-0.0758875\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.31534i	− 0.449949i	−0.974365	−	0.224975i	$-0.927770\pi$
$197$	− 6.31534i	− 0.449949i	0.974365	−	0.224975i	$-0.0722300\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−11.3693	−0.801930
$202$	0	0
$203$	− 3.31534i	− 0.232691i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.56155i	0.247545i
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	12.6847	0.873248	0.436624	−	0.899644i	$-0.356174\pi$
$211$	12.6847	0.873248	0.436624	+	0.899644i	$0.356174\pi$
$212$	0	0
$213$	− 4.80776i	− 0.329423i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.4384i	0.980146i
$218$	0	0
$219$	21.9309	1.48195
$220$	0	0
$221$	−0.876894	−0.0589863
$222$	0	0
$223$	− 23.6155i	− 1.58141i	−0.612196	−	0.790706i	$-0.709713\pi$
$223$	− 23.6155i	− 1.58141i	0.612196	−	0.790706i	$-0.290287\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.75379i	0.647382i	0.946163	+	0.323691i	$0.104924\pi$
$227$	9.75379i	0.647382i	−0.946163	+	0.323691i	$0.895076\pi$
$228$	0	0
$229$	−22.7386	−1.50261	−0.751306	−	0.659954i	$-0.770576\pi$
$229$	−22.7386	−1.50261	−0.751306	+	0.659954i	$0.770576\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	− 11.6847i	− 0.765487i	−0.923855	−	0.382744i	$-0.874979\pi$
$233$	− 11.6847i	− 0.765487i	0.923855	−	0.382744i	$-0.125021\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	33.6155i	2.18356i
$238$	0	0
$239$	−19.2462	−1.24493	−0.622467	−	0.782646i	$-0.713869\pi$
$239$	−19.2462	−1.24493	−0.622467	+	0.782646i	$0.713869\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	22.2462i	1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.36932i	0.214384i
$248$	0	0
$249$	−38.2462	−2.42376
$250$	0	0
$251$	17.3693	1.09634	0.548171	−	0.836366i	$-0.315324\pi$
$251$	17.3693	1.09634	0.548171	+	0.836366i	$0.315324\pi$
$252$	0	0
$253$	− 2.00000i	− 0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.1922i	0.698152i	0.937094	+	0.349076i	$0.113505\pi$
$257$	11.1922i	0.698152i	−0.937094	+	0.349076i	$0.886495\pi$
$258$	0	0
$259$	−0.684658	−0.0425426
$260$	0	0
$261$	−7.56155	−0.468048
$262$	0	0
$263$	− 28.9309i	− 1.78395i	−0.452081	−	0.891977i	$-0.649318\pi$
$263$	− 28.9309i	− 1.78395i	0.452081	−	0.891977i	$-0.350682\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.75379i	− 0.352126i
$268$	0	0
$269$	−6.75379	−0.411786	−0.205893	−	0.978575i	$-0.566010\pi$
$269$	−6.75379	−0.411786	−0.205893	+	0.978575i	$0.566010\pi$
$270$	0	0
$271$	−6.93087	−0.421020	−0.210510	−	0.977592i	$-0.567513\pi$
$271$	−6.93087	−0.421020	−0.210510	+	0.977592i	$0.567513\pi$
$272$	0	0
$273$	2.24621i	0.135947i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.68466i	0.341558i	0.985309	+	0.170779i	$0.0546284\pi$
$277$	5.68466i	0.341558i	−0.985309	+	0.170779i	$0.945372\pi$
$278$	0	0
$279$	32.9309	1.97152
$280$	0	0
$281$	−15.1231	−0.902169	−0.451084	−	0.892481i	$-0.648963\pi$
$281$	−15.1231	−0.902169	−0.451084	+	0.892481i	$0.648963\pi$
$282$	0	0
$283$	5.80776i	0.345236i	0.984989	+	0.172618i	$0.0552226\pi$
$283$	5.80776i	0.345236i	−0.984989	+	0.172618i	$0.944777\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.43845i	0.380050i
$288$	0	0
$289$	14.5616	0.856562
$290$	0	0
$291$	−12.4924	−0.732319
$292$	0	0
$293$	− 3.80776i	− 0.222452i	−0.993795	−	0.111226i	$-0.964522\pi$
$293$	− 3.80776i	− 0.222452i	0.993795	−	0.111226i	$-0.0354777\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.87689i	0.166934i
$298$	0	0
$299$	−0.561553	−0.0324754
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.6155i	0.782192i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.8769i	− 0.734923i	−0.930039	−	0.367462i	$-0.880227\pi$
$307$	− 12.8769i	− 0.734923i	0.930039	−	0.367462i	$-0.119773\pi$
$308$	0	0
$309$	40.9848	2.33155
$310$	0	0
$311$	−3.68466	−0.208938	−0.104469	−	0.994528i	$-0.533314\pi$
$311$	−3.68466	−0.208938	−0.104469	+	0.994528i	$0.533314\pi$
$312$	0	0
$313$	− 19.8078i	− 1.11960i	−0.828627	−	0.559801i	$-0.810878\pi$
$313$	− 19.8078i	− 1.11960i	0.828627	−	0.559801i	$-0.189122\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.87689i	0.386245i	0.981175	+	0.193122i	$0.0618615\pi$
$317$	6.87689i	0.386245i	−0.981175	+	0.193122i	$0.938139\pi$
$318$	0	0
$319$	4.24621	0.237742
$320$	0	0
$321$	19.3693	1.08109
$322$	0	0
$323$	9.36932i	0.521323i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 24.0000i	− 1.32720i
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−12.6155	−0.693412	−0.346706	−	0.937974i	$-0.612700\pi$
$331$	−12.6155	−0.693412	−0.346706	+	0.937974i	$0.612700\pi$
$332$	0	0
$333$	1.56155i	0.0855726i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 27.6155i	− 1.50431i	−0.658984	−	0.752157i	$-0.729014\pi$
$337$	− 27.6155i	− 1.50431i	0.658984	−	0.752157i	$-0.270986\pi$
$338$	0	0
$339$	20.0000	1.08625
$340$	0	0
$341$	−18.4924	−1.00142
$342$	0	0
$343$	− 18.0540i	− 0.974823i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.4924i	− 0.885360i	−0.896680	−	0.442680i	$-0.854028\pi$
$347$	− 16.4924i	− 0.885360i	0.896680	−	0.442680i	$-0.145972\pi$
$348$	0	0
$349$	1.63068	0.0872885	0.0436442	−	0.999047i	$-0.486103\pi$
$349$	1.63068	0.0872885	0.0436442	+	0.999047i	$0.486103\pi$
$350$	0	0
$351$	0.807764	0.0431153
$352$	0	0
$353$	− 35.0540i	− 1.86573i	−0.360220	−	0.932867i	$-0.617298\pi$
$353$	− 35.0540i	− 1.86573i	0.360220	−	0.932867i	$-0.382702\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.24621i	0.330585i
$358$	0	0
$359$	25.6155	1.35194	0.675968	−	0.736931i	$-0.263726\pi$
$359$	25.6155	1.35194	0.675968	+	0.736931i	$0.263726\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	17.9309i	0.941127i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 18.4384i	− 0.962479i	−0.876589	−	0.481240i	$-0.840187\pi$
$367$	− 18.4384i	− 0.962479i	0.876589	−	0.481240i	$-0.159813\pi$
$368$	0	0
$369$	14.6847	0.764453
$370$	0	0
$371$	0.684658	0.0355457
$372$	0	0
$373$	3.75379i	0.194364i	0.995267	+	0.0971819i	$0.0309829\pi$
$373$	3.75379i	0.194364i	−0.995267	+	0.0971819i	$0.969017\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.19224i	− 0.0614033i
$378$	0	0
$379$	3.50758	0.180172	0.0900861	−	0.995934i	$-0.471286\pi$
$379$	3.50758	0.180172	0.0900861	+	0.995934i	$0.471286\pi$
$380$	0	0
$381$	−32.8078	−1.68079
$382$	0	0
$383$	− 23.8078i	− 1.21652i	−0.793738	−	0.608260i	$-0.791868\pi$
$383$	− 23.8078i	− 1.21652i	0.793738	−	0.608260i	$-0.208132\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.1231	0.969580	0.484790	−	0.874631i	$-0.338896\pi$
$389$	19.1231	0.969580	0.484790	+	0.874631i	$0.338896\pi$
$390$	0	0
$391$	−1.56155	−0.0789711
$392$	0	0
$393$	− 53.3002i	− 2.68864i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 35.5464i	− 1.78402i	−0.452013	−	0.892011i	$-0.649294\pi$
$397$	− 35.5464i	− 1.78402i	0.452013	−	0.892011i	$-0.350706\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	5.19224i	0.258644i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 0.876894i	− 0.0434660i
$408$	0	0
$409$	−29.9848	−1.48266	−0.741328	−	0.671143i	$-0.765803\pi$
$409$	−29.9848	−1.48266	−0.741328	+	0.671143i	$0.765803\pi$
$410$	0	0
$411$	38.7386	1.91084
$412$	0	0
$413$	13.5616i	0.667320i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.4384i	0.658084i
$418$	0	0
$419$	27.1231	1.32505	0.662525	−	0.749040i	$-0.269485\pi$
$419$	27.1231	1.32505	0.662525	+	0.749040i	$0.269485\pi$
$420$	0	0
$421$	16.8769	0.822530	0.411265	−	0.911516i	$-0.365087\pi$
$421$	16.8769	0.822530	0.411265	+	0.911516i	$0.365087\pi$
$422$	0	0
$423$	27.3693i	1.33074i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.75379i	− 0.0848718i
$428$	0	0
$429$	−2.87689	−0.138898
$430$	0	0
$431$	6.24621	0.300869	0.150435	−	0.988620i	$-0.451933\pi$
$431$	6.24621	0.300869	0.150435	+	0.988620i	$0.451933\pi$
$432$	0	0
$433$	− 11.0691i	− 0.531948i	−0.963980	−	0.265974i	$-0.914306\pi$
$433$	− 11.0691i	− 0.531948i	0.963980	−	0.265974i	$-0.0856936\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000i	0.287019i
$438$	0	0
$439$	0.807764	0.0385525	0.0192762	−	0.999814i	$-0.493864\pi$
$439$	0.807764	0.0385525	0.0192762	+	0.999814i	$0.493864\pi$
$440$	0	0
$441$	−16.2462	−0.773629
$442$	0	0
$443$	− 8.80776i	− 0.418469i	−0.977865	−	0.209235i	$-0.932903\pi$
$443$	− 8.80776i	− 0.418469i	0.977865	−	0.209235i	$-0.0670973\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	39.3693i	1.86210i
$448$	0	0
$449$	9.31534	0.439618	0.219809	−	0.975543i	$-0.429457\pi$
$449$	9.31534	0.439618	0.219809	+	0.975543i	$0.429457\pi$
$450$	0	0
$451$	−8.24621	−0.388299
$452$	0	0
$453$	− 59.0540i	− 2.77460i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.5616i	0.634383i	0.948362	+	0.317191i	$0.102740\pi$
$457$	13.5616i	0.634383i	−0.948362	+	0.317191i	$0.897260\pi$
$458$	0	0
$459$	2.24621	0.104844
$460$	0	0
$461$	−12.0691	−0.562115	−0.281058	−	0.959691i	$-0.590685\pi$
$461$	−12.0691	−0.562115	−0.281058	+	0.959691i	$0.590685\pi$
$462$	0	0
$463$	6.63068i	0.308154i	0.988059	+	0.154077i	$0.0492404\pi$
$463$	6.63068i	0.308154i	−0.988059	+	0.154077i	$0.950760\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 34.6847i	− 1.60501i	−0.596642	−	0.802507i	$-0.703499\pi$
$467$	− 34.6847i	− 1.60501i	0.596642	−	0.802507i	$-0.296501\pi$
$468$	0	0
$469$	−6.93087	−0.320038
$470$	0	0
$471$	−42.7386	−1.96929
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.56155i	− 0.0714986i
$478$	0	0
$479$	−39.2311	−1.79251	−0.896256	−	0.443536i	$-0.853724\pi$
$479$	−39.2311	−1.79251	−0.896256	+	0.443536i	$0.853724\pi$
$480$	0	0
$481$	−0.246211	−0.0112263
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.8078i	− 0.580375i	−0.956970	−	0.290188i	$-0.906282\pi$
$487$	− 12.8078i	− 0.580375i	0.956970	−	0.290188i	$-0.0937177\pi$
$488$	0	0
$489$	30.5616	1.38204
$490$	0	0
$491$	−21.4924	−0.969939	−0.484970	−	0.874531i	$-0.661169\pi$
$491$	−21.4924	−0.969939	−0.484970	+	0.874531i	$0.661169\pi$
$492$	0	0
$493$	− 3.31534i	− 0.149315i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.93087i	− 0.131467i
$498$	0	0
$499$	18.6155	0.833345	0.416673	−	0.909057i	$-0.363196\pi$
$499$	18.6155	0.833345	0.416673	+	0.909057i	$0.363196\pi$
$500$	0	0
$501$	−20.4924	−0.915534
$502$	0	0
$503$	− 24.9309i	− 1.11161i	−0.831312	−	0.555806i	$-0.812410\pi$
$503$	− 24.9309i	− 1.11161i	0.831312	−	0.555806i	$-0.187590\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 32.4924i	− 1.44304i
$508$	0	0
$509$	−18.1771	−0.805685	−0.402842	−	0.915269i	$-0.631978\pi$
$509$	−18.1771	−0.805685	−0.402842	+	0.915269i	$0.631978\pi$
$510$	0	0
$511$	13.3693	0.591424
$512$	0	0
$513$	− 8.63068i	− 0.381054i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 15.3693i	− 0.675942i
$518$	0	0
$519$	49.6155	2.17788
$520$	0	0
$521$	27.6155	1.20986	0.604929	−	0.796279i	$-0.293201\pi$
$521$	27.6155	1.20986	0.604929	+	0.796279i	$0.293201\pi$
$522$	0	0
$523$	− 34.7386i	− 1.51901i	−0.650499	−	0.759507i	$-0.725440\pi$
$523$	− 34.7386i	− 1.51901i	0.650499	−	0.759507i	$-0.274560\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.4384i	0.628949i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	30.9309	1.34229
$532$	0	0
$533$	2.31534i	0.100289i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	56.1771i	2.42422i
$538$	0	0
$539$	9.12311	0.392960
$540$	0	0
$541$	9.68466	0.416376	0.208188	−	0.978089i	$-0.433243\pi$
$541$	9.68466	0.416376	0.208188	+	0.978089i	$0.433243\pi$
$542$	0	0
$543$	− 49.6155i	− 2.12921i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.1771i	1.03374i	0.856065	+	0.516869i	$0.172902\pi$
$547$	24.1771i	1.03374i	−0.856065	+	0.516869i	$0.827098\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	−12.7386	−0.542684
$552$	0	0
$553$	20.4924i	0.871426i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.31534i	− 0.140476i	−0.997530	−	0.0702378i	$-0.977624\pi$
$557$	− 3.31534i	− 0.140476i	0.997530	−	0.0702378i	$-0.0223758\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	45.6695i	1.92474i	0.271742	+	0.962370i	$0.412400\pi$
$563$	45.6695i	1.92474i	−0.271742	+	0.962370i	$0.587600\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.9309i	0.459053i
$568$	0	0
$569$	32.7386	1.37247	0.686237	−	0.727378i	$-0.259261\pi$
$569$	32.7386	1.37247	0.686237	+	0.727378i	$0.259261\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	18.8769i	0.788594i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 25.4384i	− 1.05902i	−0.848305	−	0.529508i	$-0.822376\pi$
$577$	− 25.4384i	− 1.05902i	0.848305	−	0.529508i	$-0.177624\pi$
$578$	0	0
$579$	−16.8078	−0.698507
$580$	0	0
$581$	−23.3153	−0.967283
$582$	0	0
$583$	0.876894i	0.0363173i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.9309i	1.48303i	0.670939	+	0.741513i	$0.265891\pi$
$587$	35.9309i	1.48303i	−0.670939	+	0.741513i	$0.734109\pi$
$588$	0	0
$589$	55.4773	2.28590
$590$	0	0
$591$	−16.1771	−0.665436
$592$	0	0
$593$	25.6155i	1.05190i	0.850514	+	0.525952i	$0.176291\pi$
$593$	25.6155i	1.05190i	−0.850514	+	0.525952i	$0.823709\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 25.6155i	− 1.04837i
$598$	0	0
$599$	−32.9848	−1.34772	−0.673862	−	0.738857i	$-0.735366\pi$
$599$	−32.9848	−1.34772	−0.673862	+	0.738857i	$0.735366\pi$
$600$	0	0
$601$	11.6307	0.474425	0.237213	−	0.971458i	$-0.423766\pi$
$601$	11.6307	0.474425	0.237213	+	0.971458i	$0.423766\pi$
$602$	0	0
$603$	15.8078i	0.643742i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 24.4924i	− 0.994117i	−0.867717	−	0.497058i	$-0.834413\pi$
$607$	− 24.4924i	− 0.994117i	0.867717	−	0.497058i	$-0.165587\pi$
$608$	0	0
$609$	−8.49242	−0.344130
$610$	0	0
$611$	−4.31534	−0.174580
$612$	0	0
$613$	− 35.6155i	− 1.43850i	−0.694753	−	0.719249i	$-0.744486\pi$
$613$	− 35.6155i	− 1.43850i	0.694753	−	0.719249i	$-0.255514\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.56155i	− 0.143383i	−0.997427	−	0.0716914i	$-0.977160\pi$
$617$	− 3.56155i	− 0.143383i	0.997427	−	0.0716914i	$-0.0228397\pi$
$618$	0	0
$619$	20.4924	0.823660	0.411830	−	0.911261i	$-0.364890\pi$
$619$	20.4924	0.823660	0.411830	+	0.911261i	$0.364890\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	− 3.50758i	− 0.140528i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	30.7386i	1.22758i
$628$	0	0
$629$	−0.684658	−0.0272991
$630$	0	0
$631$	13.7538	0.547530	0.273765	−	0.961797i	$-0.411731\pi$
$631$	13.7538	0.547530	0.273765	+	0.961797i	$0.411731\pi$
$632$	0	0
$633$	− 32.4924i	− 1.29146i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.56155i	− 0.101492i
$638$	0	0
$639$	−6.68466	−0.264441
$640$	0	0
$641$	15.1231	0.597327	0.298663	−	0.954359i	$-0.403459\pi$
$641$	15.1231	0.597327	0.298663	+	0.954359i	$0.403459\pi$
$642$	0	0
$643$	41.4233i	1.63358i	0.576939	+	0.816788i	$0.304247\pi$
$643$	41.4233i	1.63358i	−0.576939	+	0.816788i	$0.695753\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.6847i	0.695256i	0.937633	+	0.347628i	$0.113013\pi$
$647$	17.6847i	0.695256i	−0.937633	+	0.347628i	$0.886987\pi$
$648$	0	0
$649$	−17.3693	−0.681805
$650$	0	0
$651$	36.9848	1.44955
$652$	0	0
$653$	38.6695i	1.51325i	0.653846	+	0.756627i	$0.273154\pi$
$653$	38.6695i	1.51325i	−0.653846	+	0.756627i	$0.726846\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 30.4924i	− 1.18962i
$658$	0	0
$659$	9.36932	0.364977	0.182488	−	0.983208i	$-0.441585\pi$
$659$	9.36932	0.364977	0.182488	+	0.983208i	$0.441585\pi$
$660$	0	0
$661$	26.4924	1.03044	0.515218	−	0.857059i	$-0.327711\pi$
$661$	26.4924	1.03044	0.515218	+	0.857059i	$0.327711\pi$
$662$	0	0
$663$	2.24621i	0.0872356i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.12311i	− 0.0822070i
$668$	0	0
$669$	−60.4924	−2.33877
$670$	0	0
$671$	2.24621	0.0867140
$672$	0	0
$673$	35.5464i	1.37021i	0.728443	+	0.685106i	$0.240244\pi$
$673$	35.5464i	1.37021i	−0.728443	+	0.685106i	$0.759756\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.19224i	0.237987i	0.992895	+	0.118993i	$0.0379668\pi$
$677$	6.19224i	0.237987i	−0.992895	+	0.118993i	$0.962033\pi$
$678$	0	0
$679$	−7.61553	−0.292257
$680$	0	0
$681$	24.9848	0.957421
$682$	0	0
$683$	− 4.94602i	− 0.189254i	−0.995513	−	0.0946272i	$-0.969834\pi$
$683$	− 4.94602i	− 0.189254i	0.995513	−	0.0946272i	$-0.0301659\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	58.2462i	2.22223i
$688$	0	0
$689$	0.246211	0.00937990
$690$	0	0
$691$	−16.4924	−0.627401	−0.313701	−	0.949522i	$-0.601569\pi$
$691$	−16.4924	−0.627401	−0.313701	+	0.949522i	$0.601569\pi$
$692$	0	0
$693$	11.1231i	0.422532i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.43845i	0.243874i
$698$	0	0
$699$	−29.9309	−1.13209
$700$	0	0
$701$	−18.2462	−0.689150	−0.344575	−	0.938759i	$-0.611977\pi$
$701$	−18.2462	−0.689150	−0.344575	+	0.938759i	$0.611977\pi$
$702$	0	0
$703$	2.63068i	0.0992181i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.30019i	0.312161i
$708$	0	0
$709$	−46.2462	−1.73681	−0.868406	−	0.495853i	$-0.834855\pi$
$709$	−46.2462	−1.73681	−0.868406	+	0.495853i	$0.834855\pi$
$710$	0	0
$711$	46.7386	1.75284
$712$	0	0
$713$	9.24621i	0.346273i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.3002i	1.84115i
$718$	0	0
$719$	18.0540	0.673300	0.336650	−	0.941630i	$-0.390706\pi$
$719$	18.0540	0.673300	0.336650	+	0.941630i	$0.390706\pi$
$720$	0	0
$721$	24.9848	0.930484
$722$	0	0
$723$	− 15.3693i	− 0.571591i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 41.8078i	− 1.55056i	−0.631615	−	0.775282i	$-0.717608\pi$
$727$	− 41.8078i	− 1.55056i	0.631615	−	0.775282i	$-0.282392\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 12.6847i	− 0.468519i	−0.972174	−	0.234259i	$-0.924734\pi$
$733$	− 12.6847i	− 0.468519i	0.972174	−	0.234259i	$-0.0752665\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.87689i	− 0.326985i
$738$	0	0
$739$	40.6155	1.49407	0.747033	−	0.664787i	$-0.231478\pi$
$739$	40.6155	1.49407	0.747033	+	0.664787i	$0.231478\pi$
$740$	0	0
$741$	8.63068	0.317056
$742$	0	0
$743$	− 36.4924i	− 1.33878i	−0.742912	−	0.669389i	$-0.766556\pi$
$743$	− 36.4924i	− 1.33878i	0.742912	−	0.669389i	$-0.233444\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	53.1771i	1.94565i
$748$	0	0
$749$	11.8078	0.431446
$750$	0	0
$751$	−4.87689	−0.177960	−0.0889802	−	0.996033i	$-0.528361\pi$
$751$	−4.87689	−0.177960	−0.0889802	+	0.996033i	$0.528361\pi$
$752$	0	0
$753$	− 44.4924i	− 1.62139i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.43845i	0.306701i	0.988172	+	0.153350i	$0.0490063\pi$
$757$	8.43845i	0.306701i	−0.988172	+	0.153350i	$0.950994\pi$
$758$	0	0
$759$	−5.12311	−0.185957
$760$	0	0
$761$	21.9848	0.796950	0.398475	−	0.917179i	$-0.369540\pi$
$761$	21.9848	0.796950	0.398475	+	0.917179i	$0.369540\pi$
$762$	0	0
$763$	− 14.6307i	− 0.529666i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.87689i	0.176094i
$768$	0	0
$769$	35.3693	1.27545	0.637725	−	0.770264i	$-0.279876\pi$
$769$	35.3693	1.27545	0.637725	+	0.770264i	$0.279876\pi$
$770$	0	0
$771$	28.6695	1.03251
$772$	0	0
$773$	32.8769i	1.18250i	0.806488	+	0.591250i	$0.201365\pi$
$773$	32.8769i	1.18250i	−0.806488	+	0.591250i	$0.798635\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.75379i	0.0629168i
$778$	0	0
$779$	24.7386	0.886354
$780$	0	0
$781$	3.75379	0.134321
$782$	0	0
$783$	3.05398i	0.109140i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 27.3153i	− 0.973687i	−0.873489	−	0.486843i	$-0.838148\pi$
$787$	− 27.3153i	− 0.973687i	0.873489	−	0.486843i	$-0.161852\pi$
$788$	0	0
$789$	−74.1080	−2.63831
$790$	0	0
$791$	12.1922	0.433506
$792$	0	0
$793$	− 0.630683i	− 0.0223962i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.42329i	0.333790i	0.985975	+	0.166895i	$0.0533741\pi$
$797$	9.42329i	0.333790i	−0.985975	+	0.166895i	$0.946626\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	17.1231i	0.604261i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.3002i	0.608995i
$808$	0	0
$809$	8.82292	0.310197	0.155099	−	0.987899i	$-0.450430\pi$
$809$	8.82292	0.310197	0.155099	+	0.987899i	$0.450430\pi$
$810$	0	0
$811$	30.7538	1.07991	0.539956	−	0.841693i	$-0.318441\pi$
$811$	30.7538	1.07991	0.539956	+	0.841693i	$0.318441\pi$
$812$	0	0
$813$	17.7538i	0.622653i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	3.12311	0.109130
$820$	0	0
$821$	−1.50758	−0.0526148	−0.0263074	−	0.999654i	$-0.508375\pi$
$821$	−1.50758	−0.0526148	−0.0263074	+	0.999654i	$0.508375\pi$
$822$	0	0
$823$	0.946025i	0.0329763i	0.999864	+	0.0164882i	$0.00524859\pi$
$823$	0.946025i	0.0329763i	−0.999864	+	0.0164882i	$0.994751\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.31534i	0.254379i	0.991878	+	0.127190i	$0.0405957\pi$
$827$	7.31534i	0.254379i	−0.991878	+	0.127190i	$0.959404\pi$
$828$	0	0
$829$	−40.5464	−1.40823	−0.704117	−	0.710084i	$-0.748657\pi$
$829$	−40.5464	−1.40823	−0.704117	+	0.710084i	$0.748657\pi$
$830$	0	0
$831$	14.5616	0.505135
$832$	0	0
$833$	− 7.12311i	− 0.246801i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 13.3002i	− 0.459722i
$838$	0	0
$839$	51.1231	1.76497	0.882483	−	0.470345i	$-0.155870\pi$
$839$	51.1231	1.76497	0.882483	+	0.470345i	$0.155870\pi$
$840$	0	0
$841$	−24.4924	−0.844566
$842$	0	0
$843$	38.7386i	1.33423i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.9309i	0.375589i
$848$	0	0
$849$	14.8769	0.510574
$850$	0	0
$851$	−0.438447	−0.0150298
$852$	0	0
$853$	3.75379i	0.128527i	0.997933	+	0.0642636i	$0.0204699\pi$
$853$	3.75379i	0.128527i	−0.997933	+	0.0642636i	$0.979530\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 45.6847i	− 1.56056i	−0.625431	−	0.780279i	$-0.715077\pi$
$857$	− 45.6847i	− 1.56056i	0.625431	−	0.780279i	$-0.284923\pi$
$858$	0	0
$859$	47.4924	1.62042	0.810210	−	0.586139i	$-0.199353\pi$
$859$	47.4924	1.62042	0.810210	+	0.586139i	$0.199353\pi$
$860$	0	0
$861$	16.4924	0.562060
$862$	0	0
$863$	3.43845i	0.117046i	0.998286	+	0.0585231i	$0.0186391\pi$
$863$	3.43845i	0.117046i	−0.998286	+	0.0585231i	$0.981361\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 37.3002i	− 1.26678i
$868$	0	0
$869$	−26.2462	−0.890342
$870$	0	0
$871$	−2.49242	−0.0844525
$872$	0	0
$873$	17.3693i	0.587862i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.9848i	1.58657i	0.608853	+	0.793283i	$0.291630\pi$
$877$	46.9848i	1.58657i	−0.608853	+	0.793283i	$0.708370\pi$
$878$	0	0
$879$	−9.75379	−0.328987
$880$	0	0
$881$	−57.4773	−1.93646	−0.968229	−	0.250065i	$-0.919548\pi$
$881$	−57.4773	−1.93646	−0.968229	+	0.250065i	$0.919548\pi$
$882$	0	0
$883$	38.7386i	1.30366i	0.758366	+	0.651829i	$0.225998\pi$
$883$	38.7386i	1.30366i	−0.758366	+	0.651829i	$0.774002\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 43.7926i	− 1.47041i	−0.677844	−	0.735206i	$-0.737085\pi$
$887$	− 43.7926i	− 1.47041i	0.677844	−	0.735206i	$-0.262915\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	−14.0000	−0.469018
$892$	0	0
$893$	46.1080i	1.54294i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.43845i	0.0480284i
$898$	0	0
$899$	−19.6307	−0.654720
$900$	0	0
$901$	0.684658	0.0228093
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.31534i	0.309311i	0.987968	+	0.154655i	$0.0494267\pi$
$907$	9.31534i	0.309311i	−0.987968	+	0.154655i	$0.950573\pi$
$908$	0	0
$909$	18.9309	0.627897
$910$	0	0
$911$	5.12311	0.169736	0.0848680	−	0.996392i	$-0.472953\pi$
$911$	5.12311	0.169736	0.0848680	+	0.996392i	$0.472953\pi$
$912$	0	0
$913$	− 29.8617i	− 0.988279i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 32.4924i	− 1.07299i
$918$	0	0
$919$	2.73863	0.0903392	0.0451696	−	0.998979i	$-0.485617\pi$
$919$	2.73863	0.0903392	0.0451696	+	0.998979i	$0.485617\pi$
$920$	0	0
$921$	−32.9848	−1.08689
$922$	0	0
$923$	− 1.05398i	− 0.0346920i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 56.9848i	− 1.87163i
$928$	0	0
$929$	−58.7235	−1.92665	−0.963327	−	0.268329i	$-0.913529\pi$
$929$	−58.7235	−1.92665	−0.963327	+	0.268329i	$0.913529\pi$
$930$	0	0
$931$	−27.3693	−0.896993
$932$	0	0
$933$	9.43845i	0.309001i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.246211i	0.00804337i	0.999992	+	0.00402169i	$0.00128015\pi$
$937$	0.246211i	0.00804337i	−0.999992	+	0.00402169i	$0.998720\pi$
$938$	0	0
$939$	−50.7386	−1.65579
$940$	0	0
$941$	−54.2462	−1.76838	−0.884188	−	0.467131i	$-0.845288\pi$
$941$	−54.2462	−1.76838	−0.884188	+	0.467131i	$0.845288\pi$
$942$	0	0
$943$	4.12311i	0.134267i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.3153i	1.44006i	0.693945	+	0.720028i	$0.255871\pi$
$947$	44.3153i	1.44006i	−0.693945	+	0.720028i	$0.744129\pi$
$948$	0	0
$949$	4.80776	0.156067
$950$	0	0
$951$	17.6155	0.571223
$952$	0	0
$953$	5.50758i	0.178408i	0.996013	+	0.0892040i	$0.0284323\pi$
$953$	5.50758i	0.178408i	−0.996013	+	0.0892040i	$0.971568\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 10.8769i	− 0.351600i
$958$	0	0
$959$	23.6155	0.762585
$960$	0	0
$961$	54.4924	1.75782
$962$	0	0
$963$	− 26.9309i	− 0.867835i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 44.3153i	− 1.42509i	−0.701629	−	0.712543i	$-0.747543\pi$
$967$	− 44.3153i	− 1.42509i	0.701629	−	0.712543i	$-0.252457\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	−47.3693	−1.52015	−0.760077	−	0.649833i	$-0.774839\pi$
$971$	−47.3693	−1.52015	−0.760077	+	0.649833i	$0.774839\pi$
$972$	0	0
$973$	8.19224i	0.262631i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 26.7926i	− 0.857172i	−0.903501	−	0.428586i	$-0.859012\pi$
$977$	− 26.7926i	− 0.857172i	0.903501	−	0.428586i	$-0.140988\pi$
$978$	0	0
$979$	4.49242	0.143578
$980$	0	0
$981$	−33.3693	−1.06540
$982$	0	0
$983$	− 0.0539753i	− 0.00172155i	−1.00000		0.000860773i	$-0.999726\pi$
$983$	− 0.0539753i	− 0.00172155i	1.00000		0.000860773i	$-0.000273992\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.7386i	0.978421i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−0.300187	−0.00953574	−0.00476787	−	0.999989i	$-0.501518\pi$
$991$	−0.300187	−0.00953574	−0.00476787	+	0.999989i	$0.501518\pi$
$992$	0	0
$993$	32.3153i	1.02550i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.3845i	0.455561i	0.973713	+	0.227780i	$0.0731468\pi$
$997$	14.3845i	0.455561i	−0.973713	+	0.227780i	$0.926853\pi$
$998$	0	0
$999$	0.630683	0.0199539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.c.h.1749.1		4
5.2	odd	4		2300.2.a.i.1.1		2
5.3	odd	4		460.2.a.e.1.2	✓	2
5.4	even	2	inner	2300.2.c.h.1749.4		4
15.8	even	4		4140.2.a.m.1.1		2
20.3	even	4		1840.2.a.m.1.1		2
20.7	even	4		9200.2.a.bv.1.2		2
40.3	even	4		7360.2.a.bo.1.2		2
40.13	odd	4		7360.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.a.e.1.2	✓	2	5.3	odd	4
1840.2.a.m.1.1		2	20.3	even	4
2300.2.a.i.1.1		2	5.2	odd	4
2300.2.c.h.1749.1		4	1.1	even	1	trivial
2300.2.c.h.1749.4		4	5.4	even	2	inner
4140.2.a.m.1.1		2	15.8	even	4
7360.2.a.bi.1.1		2	40.13	odd	4
7360.2.a.bo.1.2		2	40.3	even	4
9200.2.a.bv.1.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56155i q^{3} -1.56155i q^{7} -3.56155 q^{9} +O(q^{10})\)	\(q-2.56155i q^{3} -1.56155i q^{7} -3.56155 q^{9} +2.00000 q^{11} -0.561553i q^{13} -1.56155i q^{17} -6.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} +1.43845i q^{27} +2.12311 q^{29} -9.24621 q^{31} -5.12311i q^{33} -0.438447i q^{37} -1.43845 q^{39} -4.12311 q^{41} -7.68466i q^{47} +4.56155 q^{49} -4.00000 q^{51} +0.438447i q^{53} +15.3693i q^{57} -8.68466 q^{59} +1.12311 q^{61} +5.56155i q^{63} -4.43845i q^{67} -2.56155 q^{69} +1.87689 q^{71} +8.56155i q^{73} -3.12311i q^{77} -13.1231 q^{79} -7.00000 q^{81} -14.9309i q^{83} -5.43845i q^{87} +2.24621 q^{89} -0.876894 q^{91} +23.6847i q^{93} -4.87689i q^{97} -7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} + 8 q^{11} - 24 q^{19} - 16 q^{21} - 8 q^{29} - 4 q^{31} - 14 q^{39} + 10 q^{49} - 16 q^{51} - 10 q^{59} - 12 q^{61} - 2 q^{69} + 24 q^{71} - 36 q^{79} - 28 q^{81} - 24 q^{89} - 20 q^{91} - 12 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 + 8 * q^11 - 24 * q^19 - 16 * q^21 - 8 * q^29 - 4 * q^31 - 14 * q^39 + 10 * q^49 - 16 * q^51 - 10 * q^59 - 12 * q^61 - 2 * q^69 + 24 * q^71 - 36 * q^79 - 28 * q^81 - 24 * q^89 - 20 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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