LMFDB - Embedded newform 1200.4.f.d.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.d.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +8.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +8.00000i q^{7} -9.00000 q^{9} -36.0000 q^{11} -10.0000i q^{13} -18.0000i q^{17} -100.000 q^{19} +24.0000 q^{21} -72.0000i q^{23} +27.0000i q^{27} +234.000 q^{29} +16.0000 q^{31} +108.000i q^{33} +226.000i q^{37} -30.0000 q^{39} +90.0000 q^{41} -452.000i q^{43} +432.000i q^{47} +279.000 q^{49} -54.0000 q^{51} +414.000i q^{53} +300.000i q^{57} -684.000 q^{59} +422.000 q^{61} -72.0000i q^{63} +332.000i q^{67} -216.000 q^{69} +360.000 q^{71} +26.0000i q^{73} -288.000i q^{77} +512.000 q^{79} +81.0000 q^{81} +1188.00i q^{83} -702.000i q^{87} +630.000 q^{89} +80.0000 q^{91} -48.0000i q^{93} +1054.00i q^{97} +324.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 72 q^{11} - 200 q^{19} + 48 q^{21} + 468 q^{29} + 32 q^{31} - 60 q^{39} + 180 q^{41} + 558 q^{49} - 108 q^{51} - 1368 q^{59} + 844 q^{61} - 432 q^{69} + 720 q^{71} + 1024 q^{79} + 162 q^{81} + 1260 q^{89} + 160 q^{91} + 648 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 72 * q^11 - 200 * q^19 + 48 * q^21 + 468 * q^29 + 32 * q^31 - 60 * q^39 + 180 * q^41 + 558 * q^49 - 108 * q^51 - 1368 * q^59 + 844 * q^61 - 432 * q^69 + 720 * q^71 + 1024 * q^79 + 162 * q^81 + 1260 * q^89 + 160 * q^91 + 648 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	8.00000i	0.431959i	0.976398	+	0.215980i	$0.0692945\pi$
$7$	8.00000i	0.431959i	−0.976398	+	0.215980i	$0.930705\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	− 10.0000i	− 0.213346i	−0.994294	−	0.106673i	$-0.965980\pi$
$13$	− 10.0000i	− 0.213346i	0.994294	−	0.106673i	$-0.0340198\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18.0000i	− 0.256802i	−0.991722	−	0.128401i	$-0.959015\pi$
$17$	− 18.0000i	− 0.256802i	0.991722	−	0.128401i	$-0.0409845\pi$
$18$	0	0
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	24.0000	0.249392
$22$	0	0
$23$	− 72.0000i	− 0.652741i	−0.945242	−	0.326370i	$-0.894174\pi$
$23$	− 72.0000i	− 0.652741i	0.945242	−	0.326370i	$-0.105826\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	234.000	1.49837	0.749185	−	0.662361i	$-0.230446\pi$
$29$	234.000	1.49837	0.749185	+	0.662361i	$0.230446\pi$
$30$	0	0
$31$	16.0000	0.0926995	0.0463498	−	0.998925i	$-0.485241\pi$
$31$	16.0000	0.0926995	0.0463498	+	0.998925i	$0.485241\pi$
$32$	0	0
$33$	108.000i	0.569709i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	226.000i	1.00417i	0.864819	+	0.502083i	$0.167433\pi$
$37$	226.000i	1.00417i	−0.864819	+	0.502083i	$0.832567\pi$
$38$	0	0
$39$	−30.0000	−0.123176
$40$	0	0
$41$	90.0000	0.342820	0.171410	−	0.985200i	$-0.445168\pi$
$41$	90.0000	0.342820	0.171410	+	0.985200i	$0.445168\pi$
$42$	0	0
$43$	− 452.000i	− 1.60301i	−0.597989	−	0.801504i	$-0.704033\pi$
$43$	− 452.000i	− 1.60301i	0.597989	−	0.801504i	$-0.295967\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	432.000i	1.34072i	0.742038	+	0.670358i	$0.233860\pi$
$47$	432.000i	1.34072i	−0.742038	+	0.670358i	$0.766140\pi$
$48$	0	0
$49$	279.000	0.813411
$50$	0	0
$51$	−54.0000	−0.148265
$52$	0	0
$53$	414.000i	1.07297i	0.843911	+	0.536484i	$0.180248\pi$
$53$	414.000i	1.07297i	−0.843911	+	0.536484i	$0.819752\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	300.000i	0.697122i
$58$	0	0
$59$	−684.000	−1.50931	−0.754654	−	0.656123i	$-0.772195\pi$
$59$	−684.000	−1.50931	−0.754654	+	0.656123i	$0.772195\pi$
$60$	0	0
$61$	422.000	0.885763	0.442882	−	0.896580i	$-0.353956\pi$
$61$	422.000	0.885763	0.442882	+	0.896580i	$0.353956\pi$
$62$	0	0
$63$	− 72.0000i	− 0.143986i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	332.000i	0.605377i	0.953090	+	0.302688i	$0.0978842\pi$
$67$	332.000i	0.605377i	−0.953090	+	0.302688i	$0.902116\pi$
$68$	0	0
$69$	−216.000	−0.376860
$70$	0	0
$71$	360.000	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$72$	0	0
$73$	26.0000i	0.0416859i	0.999783	+	0.0208429i	$0.00663500\pi$
$73$	26.0000i	0.0416859i	−0.999783	+	0.0208429i	$0.993365\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 288.000i	− 0.426242i
$78$	0	0
$79$	512.000	0.729171	0.364585	−	0.931170i	$-0.381211\pi$
$79$	512.000	0.729171	0.364585	+	0.931170i	$0.381211\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1188.00i	1.57108i	0.618809	+	0.785542i	$0.287616\pi$
$83$	1188.00i	1.57108i	−0.618809	+	0.785542i	$0.712384\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 702.000i	− 0.865084i
$88$	0	0
$89$	630.000	0.750336	0.375168	−	0.926957i	$-0.377585\pi$
$89$	630.000	0.750336	0.375168	+	0.926957i	$0.377585\pi$
$90$	0	0
$91$	80.0000	0.0921569
$92$	0	0
$93$	− 48.0000i	− 0.0535201i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1054.00i	1.10327i	0.834085	+	0.551637i	$0.185996\pi$
$97$	1054.00i	1.10327i	−0.834085	+	0.551637i	$0.814004\pi$
$98$	0	0
$99$	324.000	0.328921
$100$	0	0
$101$	558.000	0.549733	0.274867	−	0.961482i	$-0.411366\pi$
$101$	558.000	0.549733	0.274867	+	0.961482i	$0.411366\pi$
$102$	0	0
$103$	− 8.00000i	− 0.00765304i	−0.999993	−	0.00382652i	$-0.998782\pi$
$103$	− 8.00000i	− 0.00765304i	0.999993	−	0.00382652i	$-0.00121802\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1764.00i	1.59376i	0.604138	+	0.796880i	$0.293518\pi$
$107$	1764.00i	1.59376i	−0.604138	+	0.796880i	$0.706482\pi$
$108$	0	0
$109$	−1622.00	−1.42532	−0.712658	−	0.701512i	$-0.752509\pi$
$109$	−1622.00	−1.42532	−0.712658	+	0.701512i	$0.752509\pi$
$110$	0	0
$111$	678.000	0.579756
$112$	0	0
$113$	− 1134.00i	− 0.944051i	−0.881585	−	0.472025i	$-0.843523\pi$
$113$	− 1134.00i	− 0.944051i	0.881585	−	0.472025i	$-0.156477\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	90.0000i	0.0711154i
$118$	0	0
$119$	144.000	0.110928
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	− 270.000i	− 0.197927i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 592.000i	− 0.413634i	−0.978380	−	0.206817i	$-0.933690\pi$
$127$	− 592.000i	− 0.413634i	0.978380	−	0.206817i	$-0.0663105\pi$
$128$	0	0
$129$	−1356.00	−0.925497
$130$	0	0
$131$	1908.00	1.27254	0.636270	−	0.771466i	$-0.280476\pi$
$131$	1908.00	1.27254	0.636270	+	0.771466i	$0.280476\pi$
$132$	0	0
$133$	− 800.000i	− 0.521570i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 954.000i	− 0.594932i	−0.954732	−	0.297466i	$-0.903858\pi$
$137$	− 954.000i	− 0.594932i	0.954732	−	0.297466i	$-0.0961415\pi$
$138$	0	0
$139$	2564.00	1.56457	0.782286	−	0.622919i	$-0.214053\pi$
$139$	2564.00	1.56457	0.782286	+	0.622919i	$0.214053\pi$
$140$	0	0
$141$	1296.00	0.774063
$142$	0	0
$143$	360.000i	0.210522i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 837.000i	− 0.469623i
$148$	0	0
$149$	738.000	0.405767	0.202884	−	0.979203i	$-0.434969\pi$
$149$	738.000	0.405767	0.202884	+	0.979203i	$0.434969\pi$
$150$	0	0
$151$	2440.00	1.31500	0.657498	−	0.753456i	$-0.271615\pi$
$151$	2440.00	1.31500	0.657498	+	0.753456i	$0.271615\pi$
$152$	0	0
$153$	162.000i	0.0856008i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2554.00i	1.29829i	0.760665	+	0.649145i	$0.224873\pi$
$157$	2554.00i	1.29829i	−0.760665	+	0.649145i	$0.775127\pi$
$158$	0	0
$159$	1242.00	0.619478
$160$	0	0
$161$	576.000	0.281958
$162$	0	0
$163$	820.000i	0.394033i	0.980400	+	0.197016i	$0.0631252\pi$
$163$	820.000i	0.394033i	−0.980400	+	0.197016i	$0.936875\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1944.00i	0.900786i	0.892830	+	0.450393i	$0.148716\pi$
$167$	1944.00i	0.900786i	−0.892830	+	0.450393i	$0.851284\pi$
$168$	0	0
$169$	2097.00	0.954483
$170$	0	0
$171$	900.000	0.402484
$172$	0	0
$173$	− 1242.00i	− 0.545824i	−0.962039	−	0.272912i	$-0.912013\pi$
$173$	− 1242.00i	− 0.545824i	0.962039	−	0.272912i	$-0.0879867\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2052.00i	0.871400i
$178$	0	0
$179$	1116.00	0.465999	0.232999	−	0.972477i	$-0.425146\pi$
$179$	1116.00	0.465999	0.232999	+	0.972477i	$0.425146\pi$
$180$	0	0
$181$	1070.00	0.439406	0.219703	−	0.975567i	$-0.429491\pi$
$181$	1070.00	0.439406	0.219703	+	0.975567i	$0.429491\pi$
$182$	0	0
$183$	− 1266.00i	− 0.511396i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	648.000i	0.253403i
$188$	0	0
$189$	−216.000	−0.0831306
$190$	0	0
$191$	576.000	0.218209	0.109104	−	0.994030i	$-0.465202\pi$
$191$	576.000	0.218209	0.109104	+	0.994030i	$0.465202\pi$
$192$	0	0
$193$	− 1342.00i	− 0.500514i	−0.968179	−	0.250257i	$-0.919485\pi$
$193$	− 1342.00i	− 0.500514i	0.968179	−	0.250257i	$-0.0805152\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1422.00i	− 0.514281i	−0.966374	−	0.257140i	$-0.917220\pi$
$197$	− 1422.00i	− 0.514281i	0.966374	−	0.257140i	$-0.0827803\pi$
$198$	0	0
$199$	872.000	0.310625	0.155313	−	0.987865i	$-0.450361\pi$
$199$	872.000	0.310625	0.155313	+	0.987865i	$0.450361\pi$
$200$	0	0
$201$	996.000	0.349515
$202$	0	0
$203$	1872.00i	0.647235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	648.000i	0.217580i
$208$	0	0
$209$	3600.00	1.19147
$210$	0	0
$211$	−1340.00	−0.437201	−0.218600	−	0.975814i	$-0.570149\pi$
$211$	−1340.00	−0.437201	−0.218600	+	0.975814i	$0.570149\pi$
$212$	0	0
$213$	− 1080.00i	− 0.347420i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	128.000i	0.0400424i
$218$	0	0
$219$	78.0000	0.0240674
$220$	0	0
$221$	−180.000	−0.0547878
$222$	0	0
$223$	− 4880.00i	− 1.46542i	−0.680540	−	0.732711i	$-0.738255\pi$
$223$	− 4880.00i	− 1.46542i	0.680540	−	0.732711i	$-0.261745\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2700.00i	0.789451i	0.918799	+	0.394725i	$0.129160\pi$
$227$	2700.00i	0.789451i	−0.918799	+	0.394725i	$0.870840\pi$
$228$	0	0
$229$	−254.000	−0.0732960	−0.0366480	−	0.999328i	$-0.511668\pi$
$229$	−254.000	−0.0732960	−0.0366480	+	0.999328i	$0.511668\pi$
$230$	0	0
$231$	−864.000	−0.246091
$232$	0	0
$233$	4410.00i	1.23995i	0.784621	+	0.619976i	$0.212858\pi$
$233$	4410.00i	1.23995i	−0.784621	+	0.619976i	$0.787142\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1536.00i	− 0.420987i
$238$	0	0
$239$	−3888.00	−1.05228	−0.526138	−	0.850399i	$-0.676360\pi$
$239$	−3888.00	−1.05228	−0.526138	+	0.850399i	$0.676360\pi$
$240$	0	0
$241$	5138.00	1.37331	0.686655	−	0.726984i	$-0.259078\pi$
$241$	5138.00	1.37331	0.686655	+	0.726984i	$0.259078\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1000.00i	0.257605i
$248$	0	0
$249$	3564.00	0.907066
$250$	0	0
$251$	−4788.00	−1.20405	−0.602024	−	0.798478i	$-0.705639\pi$
$251$	−4788.00	−1.20405	−0.602024	+	0.798478i	$0.705639\pi$
$252$	0	0
$253$	2592.00i	0.644101i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5886.00i	1.42863i	0.699823	+	0.714316i	$0.253262\pi$
$257$	5886.00i	1.42863i	−0.699823	+	0.714316i	$0.746738\pi$
$258$	0	0
$259$	−1808.00	−0.433759
$260$	0	0
$261$	−2106.00	−0.499456
$262$	0	0
$263$	− 2232.00i	− 0.523312i	−0.965161	−	0.261656i	$-0.915731\pi$
$263$	− 2232.00i	− 0.523312i	0.965161	−	0.261656i	$-0.0842686\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1890.00i	− 0.433206i
$268$	0	0
$269$	666.000	0.150954	0.0754772	−	0.997148i	$-0.475952\pi$
$269$	666.000	0.150954	0.0754772	+	0.997148i	$0.475952\pi$
$270$	0	0
$271$	5536.00	1.24092	0.620458	−	0.784240i	$-0.286947\pi$
$271$	5536.00	1.24092	0.620458	+	0.784240i	$0.286947\pi$
$272$	0	0
$273$	− 240.000i	− 0.0532068i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2126.00i	− 0.461151i	−0.973054	−	0.230576i	$-0.925939\pi$
$277$	− 2126.00i	− 0.461151i	0.973054	−	0.230576i	$-0.0740609\pi$
$278$	0	0
$279$	−144.000	−0.0308998
$280$	0	0
$281$	−2934.00	−0.622875	−0.311437	−	0.950267i	$-0.600810\pi$
$281$	−2934.00	−0.622875	−0.311437	+	0.950267i	$0.600810\pi$
$282$	0	0
$283$	− 2036.00i	− 0.427659i	−0.976871	−	0.213830i	$-0.931406\pi$
$283$	− 2036.00i	− 0.427659i	0.976871	−	0.213830i	$-0.0685938\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	720.000i	0.148085i
$288$	0	0
$289$	4589.00	0.934053
$290$	0	0
$291$	3162.00	0.636975
$292$	0	0
$293$	2286.00i	0.455800i	0.973684	+	0.227900i	$0.0731860\pi$
$293$	2286.00i	0.455800i	−0.973684	+	0.227900i	$0.926814\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 972.000i	− 0.189903i
$298$	0	0
$299$	−720.000	−0.139260
$300$	0	0
$301$	3616.00	0.692434
$302$	0	0
$303$	− 1674.00i	− 0.317389i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1244.00i	0.231267i	0.993292	+	0.115633i	$0.0368897\pi$
$307$	1244.00i	0.231267i	−0.993292	+	0.115633i	$0.963110\pi$
$308$	0	0
$309$	−24.0000	−0.00441849
$310$	0	0
$311$	−1224.00	−0.223173	−0.111586	−	0.993755i	$-0.535593\pi$
$311$	−1224.00	−0.223173	−0.111586	+	0.993755i	$0.535593\pi$
$312$	0	0
$313$	1898.00i	0.342752i	0.985206	+	0.171376i	$0.0548213\pi$
$313$	1898.00i	0.342752i	−0.985206	+	0.171376i	$0.945179\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9162.00i	1.62331i	0.584137	+	0.811655i	$0.301433\pi$
$317$	9162.00i	1.62331i	−0.584137	+	0.811655i	$0.698567\pi$
$318$	0	0
$319$	−8424.00	−1.47854
$320$	0	0
$321$	5292.00	0.920158
$322$	0	0
$323$	1800.00i	0.310076i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4866.00i	0.822906i
$328$	0	0
$329$	−3456.00	−0.579135
$330$	0	0
$331$	4348.00	0.722017	0.361009	−	0.932562i	$-0.382432\pi$
$331$	4348.00	0.722017	0.361009	+	0.932562i	$0.382432\pi$
$332$	0	0
$333$	− 2034.00i	− 0.334722i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7154.00i	− 1.15639i	−0.815899	−	0.578195i	$-0.803757\pi$
$337$	− 7154.00i	− 1.15639i	0.815899	−	0.578195i	$-0.196243\pi$
$338$	0	0
$339$	−3402.00	−0.545048
$340$	0	0
$341$	−576.000	−0.0914726
$342$	0	0
$343$	4976.00i	0.783320i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1836.00i	− 0.284039i	−0.989864	−	0.142020i	$-0.954640\pi$
$347$	− 1836.00i	− 0.284039i	0.989864	−	0.142020i	$-0.0453596\pi$
$348$	0	0
$349$	−5894.00	−0.904007	−0.452004	−	0.892016i	$-0.649291\pi$
$349$	−5894.00	−0.904007	−0.452004	+	0.892016i	$0.649291\pi$
$350$	0	0
$351$	270.000	0.0410585
$352$	0	0
$353$	11106.0i	1.67454i	0.546789	+	0.837270i	$0.315850\pi$
$353$	11106.0i	1.67454i	−0.546789	+	0.837270i	$0.684150\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 432.000i	− 0.0640444i
$358$	0	0
$359$	13176.0	1.93705	0.968527	−	0.248907i	$-0.0800713\pi$
$359$	13176.0	1.93705	0.968527	+	0.248907i	$0.0800713\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	105.000i	0.0151820i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6112.00i	− 0.869329i	−0.900592	−	0.434665i	$-0.856867\pi$
$367$	− 6112.00i	− 0.869329i	0.900592	−	0.434665i	$-0.143133\pi$
$368$	0	0
$369$	−810.000	−0.114273
$370$	0	0
$371$	−3312.00	−0.463478
$372$	0	0
$373$	− 13618.0i	− 1.89038i	−0.326515	−	0.945192i	$-0.605874\pi$
$373$	− 13618.0i	− 1.89038i	0.326515	−	0.945192i	$-0.394126\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2340.00i	− 0.319671i
$378$	0	0
$379$	692.000	0.0937880	0.0468940	−	0.998900i	$-0.485068\pi$
$379$	692.000	0.0937880	0.0468940	+	0.998900i	$0.485068\pi$
$380$	0	0
$381$	−1776.00	−0.238812
$382$	0	0
$383$	8064.00i	1.07585i	0.842992	+	0.537926i	$0.180792\pi$
$383$	8064.00i	1.07585i	−0.842992	+	0.537926i	$0.819208\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4068.00i	0.534336i
$388$	0	0
$389$	−12654.0	−1.64931	−0.824657	−	0.565633i	$-0.808632\pi$
$389$	−12654.0	−1.64931	−0.824657	+	0.565633i	$0.808632\pi$
$390$	0	0
$391$	−1296.00	−0.167625
$392$	0	0
$393$	− 5724.00i	− 0.734701i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	106.000i	0.0134005i	0.999978	+	0.00670024i	$0.00213277\pi$
$397$	106.000i	0.0134005i	−0.999978	+	0.00670024i	$0.997867\pi$
$398$	0	0
$399$	−2400.00	−0.301129
$400$	0	0
$401$	−4014.00	−0.499874	−0.249937	−	0.968262i	$-0.580410\pi$
$401$	−4014.00	−0.499874	−0.249937	+	0.968262i	$0.580410\pi$
$402$	0	0
$403$	− 160.000i	− 0.0197771i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8136.00i	− 0.990876i
$408$	0	0
$409$	−3914.00	−0.473190	−0.236595	−	0.971608i	$-0.576032\pi$
$409$	−3914.00	−0.473190	−0.236595	+	0.971608i	$0.576032\pi$
$410$	0	0
$411$	−2862.00	−0.343484
$412$	0	0
$413$	− 5472.00i	− 0.651960i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 7692.00i	− 0.903307i
$418$	0	0
$419$	4428.00	0.516282	0.258141	−	0.966107i	$-0.416890\pi$
$419$	4428.00	0.516282	0.258141	+	0.966107i	$0.416890\pi$
$420$	0	0
$421$	−15490.0	−1.79320	−0.896599	−	0.442843i	$-0.853970\pi$
$421$	−15490.0	−1.79320	−0.896599	+	0.442843i	$0.853970\pi$
$422$	0	0
$423$	− 3888.00i	− 0.446906i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3376.00i	0.382614i
$428$	0	0
$429$	1080.00	0.121545
$430$	0	0
$431$	−6768.00	−0.756388	−0.378194	−	0.925726i	$-0.623455\pi$
$431$	−6768.00	−0.756388	−0.378194	+	0.925726i	$0.623455\pi$
$432$	0	0
$433$	1298.00i	0.144060i	0.997402	+	0.0720299i	$0.0229477\pi$
$433$	1298.00i	0.144060i	−0.997402	+	0.0720299i	$0.977052\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7200.00i	0.788153i
$438$	0	0
$439$	−2248.00	−0.244399	−0.122200	−	0.992506i	$-0.538995\pi$
$439$	−2248.00	−0.244399	−0.122200	+	0.992506i	$0.538995\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	9612.00i	1.03088i	0.856926	+	0.515440i	$0.172372\pi$
$443$	9612.00i	1.03088i	−0.856926	+	0.515440i	$0.827628\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2214.00i	− 0.234270i
$448$	0	0
$449$	−162.000	−0.0170273	−0.00851364	−	0.999964i	$-0.502710\pi$
$449$	−162.000	−0.0170273	−0.00851364	+	0.999964i	$0.502710\pi$
$450$	0	0
$451$	−3240.00	−0.338283
$452$	0	0
$453$	− 7320.00i	− 0.759213i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1370.00i	− 0.140232i	−0.997539	−	0.0701159i	$-0.977663\pi$
$457$	− 1370.00i	− 0.140232i	0.997539	−	0.0701159i	$-0.0223369\pi$
$458$	0	0
$459$	486.000	0.0494217
$460$	0	0
$461$	−15354.0	−1.55121	−0.775604	−	0.631220i	$-0.782555\pi$
$461$	−15354.0	−1.55121	−0.775604	+	0.631220i	$0.782555\pi$
$462$	0	0
$463$	13024.0i	1.30729i	0.756800	+	0.653646i	$0.226762\pi$
$463$	13024.0i	1.30729i	−0.756800	+	0.653646i	$0.773238\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14436.0i	− 1.43045i	−0.698896	−	0.715223i	$-0.746325\pi$
$467$	− 14436.0i	− 1.43045i	0.698896	−	0.715223i	$-0.253675\pi$
$468$	0	0
$469$	−2656.00	−0.261498
$470$	0	0
$471$	7662.00	0.749568
$472$	0	0
$473$	16272.0i	1.58179i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3726.00i	− 0.357656i
$478$	0	0
$479$	12096.0	1.15382	0.576911	−	0.816807i	$-0.304258\pi$
$479$	12096.0	1.15382	0.576911	+	0.816807i	$0.304258\pi$
$480$	0	0
$481$	2260.00	0.214235
$482$	0	0
$483$	− 1728.00i	− 0.162788i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6056.00i	0.563498i	0.959488	+	0.281749i	$0.0909146\pi$
$487$	6056.00i	0.563498i	−0.959488	+	0.281749i	$0.909085\pi$
$488$	0	0
$489$	2460.00	0.227495
$490$	0	0
$491$	−7524.00	−0.691555	−0.345777	−	0.938317i	$-0.612385\pi$
$491$	−7524.00	−0.691555	−0.345777	+	0.938317i	$0.612385\pi$
$492$	0	0
$493$	− 4212.00i	− 0.384785i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2880.00i	0.259931i
$498$	0	0
$499$	5276.00	0.473319	0.236660	−	0.971593i	$-0.423947\pi$
$499$	5276.00	0.473319	0.236660	+	0.971593i	$0.423947\pi$
$500$	0	0
$501$	5832.00	0.520069
$502$	0	0
$503$	− 4968.00i	− 0.440382i	−0.975457	−	0.220191i	$-0.929332\pi$
$503$	− 4968.00i	− 0.440382i	0.975457	−	0.220191i	$-0.0706681\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6291.00i	− 0.551071i
$508$	0	0
$509$	−10998.0	−0.957717	−0.478858	−	0.877892i	$-0.658949\pi$
$509$	−10998.0	−0.957717	−0.478858	+	0.877892i	$0.658949\pi$
$510$	0	0
$511$	−208.000	−0.0180066
$512$	0	0
$513$	− 2700.00i	− 0.232374i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 15552.0i	− 1.32297i
$518$	0	0
$519$	−3726.00	−0.315131
$520$	0	0
$521$	−8838.00	−0.743186	−0.371593	−	0.928396i	$-0.621188\pi$
$521$	−8838.00	−0.743186	−0.371593	+	0.928396i	$0.621188\pi$
$522$	0	0
$523$	− 22436.0i	− 1.87583i	−0.346869	−	0.937914i	$-0.612755\pi$
$523$	− 22436.0i	− 1.87583i	0.346869	−	0.937914i	$-0.387245\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 288.000i	− 0.0238055i
$528$	0	0
$529$	6983.00	0.573929
$530$	0	0
$531$	6156.00	0.503103
$532$	0	0
$533$	− 900.000i	− 0.0731395i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3348.00i	− 0.269044i
$538$	0	0
$539$	−10044.0	−0.802645
$540$	0	0
$541$	−4762.00	−0.378437	−0.189218	−	0.981935i	$-0.560595\pi$
$541$	−4762.00	−0.378437	−0.189218	+	0.981935i	$0.560595\pi$
$542$	0	0
$543$	− 3210.00i	− 0.253691i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 6004.00i	− 0.469310i	−0.972079	−	0.234655i	$-0.924604\pi$
$547$	− 6004.00i	− 0.469310i	0.972079	−	0.234655i	$-0.0753960\pi$
$548$	0	0
$549$	−3798.00	−0.295254
$550$	0	0
$551$	−23400.0	−1.80921
$552$	0	0
$553$	4096.00i	0.314972i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5274.00i	0.401197i	0.979674	+	0.200598i	$0.0642886\pi$
$557$	5274.00i	0.401197i	−0.979674	+	0.200598i	$0.935711\pi$
$558$	0	0
$559$	−4520.00	−0.341996
$560$	0	0
$561$	1944.00	0.146303
$562$	0	0
$563$	12420.0i	0.929735i	0.885380	+	0.464867i	$0.153898\pi$
$563$	12420.0i	0.929735i	−0.885380	+	0.464867i	$0.846102\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	648.000i	0.0479955i
$568$	0	0
$569$	21366.0	1.57418	0.787091	−	0.616837i	$-0.211586\pi$
$569$	21366.0	1.57418	0.787091	+	0.616837i	$0.211586\pi$
$570$	0	0
$571$	−21140.0	−1.54935	−0.774677	−	0.632357i	$-0.782088\pi$
$571$	−21140.0	−1.54935	−0.774677	+	0.632357i	$0.782088\pi$
$572$	0	0
$573$	− 1728.00i	− 0.125983i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3266.00i	− 0.235642i	−0.993035	−	0.117821i	$-0.962409\pi$
$577$	− 3266.00i	− 0.235642i	0.993035	−	0.117821i	$-0.0375909\pi$
$578$	0	0
$579$	−4026.00	−0.288972
$580$	0	0
$581$	−9504.00	−0.678644
$582$	0	0
$583$	− 14904.0i	− 1.05877i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17028.0i	1.19731i	0.801007	+	0.598655i	$0.204298\pi$
$587$	17028.0i	1.19731i	−0.801007	+	0.598655i	$0.795702\pi$
$588$	0	0
$589$	−1600.00	−0.111930
$590$	0	0
$591$	−4266.00	−0.296920
$592$	0	0
$593$	9522.00i	0.659396i	0.944086	+	0.329698i	$0.106947\pi$
$593$	9522.00i	0.659396i	−0.944086	+	0.329698i	$0.893053\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2616.00i	− 0.179340i
$598$	0	0
$599$	−10296.0	−0.702309	−0.351155	−	0.936318i	$-0.614211\pi$
$599$	−10296.0	−0.702309	−0.351155	+	0.936318i	$0.614211\pi$
$600$	0	0
$601$	−3382.00	−0.229542	−0.114771	−	0.993392i	$-0.536613\pi$
$601$	−3382.00	−0.229542	−0.114771	+	0.993392i	$0.536613\pi$
$602$	0	0
$603$	− 2988.00i	− 0.201792i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 20656.0i	− 1.38122i	−0.723227	−	0.690611i	$-0.757342\pi$
$607$	− 20656.0i	− 1.38122i	0.723227	−	0.690611i	$-0.242658\pi$
$608$	0	0
$609$	5616.00	0.373681
$610$	0	0
$611$	4320.00	0.286037
$612$	0	0
$613$	− 22114.0i	− 1.45706i	−0.685015	−	0.728529i	$-0.740205\pi$
$613$	− 22114.0i	− 1.45706i	0.685015	−	0.728529i	$-0.259795\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19962.0i	− 1.30250i	−0.758865	−	0.651248i	$-0.774246\pi$
$617$	− 19962.0i	− 1.30250i	0.758865	−	0.651248i	$-0.225754\pi$
$618$	0	0
$619$	−604.000	−0.0392194	−0.0196097	−	0.999808i	$-0.506242\pi$
$619$	−604.000	−0.0392194	−0.0196097	+	0.999808i	$0.506242\pi$
$620$	0	0
$621$	1944.00	0.125620
$622$	0	0
$623$	5040.00i	0.324115i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 10800.0i	− 0.687895i
$628$	0	0
$629$	4068.00	0.257872
$630$	0	0
$631$	−152.000	−0.00958958	−0.00479479	−	0.999989i	$-0.501526\pi$
$631$	−152.000	−0.00958958	−0.00479479	+	0.999989i	$0.501526\pi$
$632$	0	0
$633$	4020.00i	0.252418i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2790.00i	− 0.173538i
$638$	0	0
$639$	−3240.00	−0.200583
$640$	0	0
$641$	4194.00	0.258429	0.129215	−	0.991617i	$-0.458754\pi$
$641$	4194.00	0.258429	0.129215	+	0.991617i	$0.458754\pi$
$642$	0	0
$643$	7252.00i	0.444776i	0.974958	+	0.222388i	$0.0713852\pi$
$643$	7252.00i	0.444776i	−0.974958	+	0.222388i	$0.928615\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 6696.00i	− 0.406873i	−0.979088	−	0.203437i	$-0.934789\pi$
$647$	− 6696.00i	− 0.406873i	0.979088	−	0.203437i	$-0.0652111\pi$
$648$	0	0
$649$	24624.0	1.48933
$650$	0	0
$651$	384.000	0.0231185
$652$	0	0
$653$	28422.0i	1.70328i	0.524131	+	0.851638i	$0.324390\pi$
$653$	28422.0i	1.70328i	−0.524131	+	0.851638i	$0.675610\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 234.000i	− 0.0138953i
$658$	0	0
$659$	−19908.0	−1.17679	−0.588396	−	0.808573i	$-0.700240\pi$
$659$	−19908.0	−1.17679	−0.588396	+	0.808573i	$0.700240\pi$
$660$	0	0
$661$	14318.0	0.842520	0.421260	−	0.906940i	$-0.361588\pi$
$661$	14318.0	0.842520	0.421260	+	0.906940i	$0.361588\pi$
$662$	0	0
$663$	540.000i	0.0316318i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16848.0i	− 0.978047i
$668$	0	0
$669$	−14640.0	−0.846061
$670$	0	0
$671$	−15192.0	−0.874040
$672$	0	0
$673$	30050.0i	1.72116i	0.509313	+	0.860581i	$0.329899\pi$
$673$	30050.0i	1.72116i	−0.509313	+	0.860581i	$0.670101\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22158.0i	− 1.25790i	−0.777444	−	0.628952i	$-0.783484\pi$
$677$	− 22158.0i	− 1.25790i	0.777444	−	0.628952i	$-0.216516\pi$
$678$	0	0
$679$	−8432.00	−0.476569
$680$	0	0
$681$	8100.00	0.455790
$682$	0	0
$683$	3132.00i	0.175465i	0.996144	+	0.0877325i	$0.0279621\pi$
$683$	3132.00i	0.175465i	−0.996144	+	0.0877325i	$0.972038\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	762.000i	0.0423175i
$688$	0	0
$689$	4140.00	0.228914
$690$	0	0
$691$	20932.0	1.15237	0.576187	−	0.817318i	$-0.304540\pi$
$691$	20932.0	1.15237	0.576187	+	0.817318i	$0.304540\pi$
$692$	0	0
$693$	2592.00i	0.142081i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1620.00i	− 0.0880371i
$698$	0	0
$699$	13230.0	0.715886
$700$	0	0
$701$	−21834.0	−1.17640	−0.588202	−	0.808714i	$-0.700164\pi$
$701$	−21834.0	−1.17640	−0.588202	+	0.808714i	$0.700164\pi$
$702$	0	0
$703$	− 22600.0i	− 1.21248i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4464.00i	0.237463i
$708$	0	0
$709$	−12446.0	−0.659266	−0.329633	−	0.944109i	$-0.606925\pi$
$709$	−12446.0	−0.659266	−0.329633	+	0.944109i	$0.606925\pi$
$710$	0	0
$711$	−4608.00	−0.243057
$712$	0	0
$713$	− 1152.00i	− 0.0605088i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11664.0i	0.607531i
$718$	0	0
$719$	−12528.0	−0.649813	−0.324907	−	0.945746i	$-0.605333\pi$
$719$	−12528.0	−0.649813	−0.324907	+	0.945746i	$0.605333\pi$
$720$	0	0
$721$	64.0000	0.00330580
$722$	0	0
$723$	− 15414.0i	− 0.792881i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11576.0i	0.590550i	0.955412	+	0.295275i	$0.0954113\pi$
$727$	11576.0i	0.590550i	−0.955412	+	0.295275i	$0.904589\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−8136.00	−0.411656
$732$	0	0
$733$	− 29338.0i	− 1.47834i	−0.673519	−	0.739170i	$-0.735218\pi$
$733$	− 29338.0i	− 1.47834i	0.673519	−	0.739170i	$-0.264782\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11952.0i	− 0.597364i
$738$	0	0
$739$	2540.00	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$739$	2540.00	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$740$	0	0
$741$	3000.00	0.148728
$742$	0	0
$743$	18792.0i	0.927876i	0.885868	+	0.463938i	$0.153564\pi$
$743$	18792.0i	0.927876i	−0.885868	+	0.463938i	$0.846436\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10692.0i	− 0.523695i
$748$	0	0
$749$	−14112.0	−0.688440
$750$	0	0
$751$	−4832.00	−0.234783	−0.117392	−	0.993086i	$-0.537453\pi$
$751$	−4832.00	−0.234783	−0.117392	+	0.993086i	$0.537453\pi$
$752$	0	0
$753$	14364.0i	0.695157i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20818.0i	0.999529i	0.866161	+	0.499764i	$0.166580\pi$
$757$	20818.0i	0.999529i	−0.866161	+	0.499764i	$0.833420\pi$
$758$	0	0
$759$	7776.00	0.371872
$760$	0	0
$761$	12042.0	0.573617	0.286808	−	0.957988i	$-0.407406\pi$
$761$	12042.0	0.573617	0.286808	+	0.957988i	$0.407406\pi$
$762$	0	0
$763$	− 12976.0i	− 0.615679i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6840.00i	0.322005i
$768$	0	0
$769$	−13058.0	−0.612332	−0.306166	−	0.951978i	$-0.599046\pi$
$769$	−13058.0	−0.612332	−0.306166	+	0.951978i	$0.599046\pi$
$770$	0	0
$771$	17658.0	0.824821
$772$	0	0
$773$	− 11826.0i	− 0.550261i	−0.961407	−	0.275130i	$-0.911279\pi$
$773$	− 11826.0i	− 0.550261i	0.961407	−	0.275130i	$-0.0887210\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5424.00i	0.250431i
$778$	0	0
$779$	−9000.00	−0.413939
$780$	0	0
$781$	−12960.0	−0.593784
$782$	0	0
$783$	6318.00i	0.288361i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11996.0i	0.543343i	0.962390	+	0.271672i	$0.0875765\pi$
$787$	11996.0i	0.543343i	−0.962390	+	0.271672i	$0.912424\pi$
$788$	0	0
$789$	−6696.00	−0.302134
$790$	0	0
$791$	9072.00	0.407792
$792$	0	0
$793$	− 4220.00i	− 0.188974i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6966.00i	− 0.309596i	−0.987946	−	0.154798i	$-0.950527\pi$
$797$	− 6966.00i	− 0.309596i	0.987946	−	0.154798i	$-0.0494727\pi$
$798$	0	0
$799$	7776.00	0.344299
$800$	0	0
$801$	−5670.00	−0.250112
$802$	0	0
$803$	− 936.000i	− 0.0411342i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1998.00i	− 0.0871536i
$808$	0	0
$809$	40806.0	1.77338	0.886689	−	0.462367i	$-0.153000\pi$
$809$	40806.0	1.77338	0.886689	+	0.462367i	$0.153000\pi$
$810$	0	0
$811$	17980.0	0.778500	0.389250	−	0.921132i	$-0.372734\pi$
$811$	17980.0	0.778500	0.389250	+	0.921132i	$0.372734\pi$
$812$	0	0
$813$	− 16608.0i	− 0.716443i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	45200.0i	1.93555i
$818$	0	0
$819$	−720.000	−0.0307190
$820$	0	0
$821$	−12834.0	−0.545566	−0.272783	−	0.962076i	$-0.587944\pi$
$821$	−12834.0	−0.545566	−0.272783	+	0.962076i	$0.587944\pi$
$822$	0	0
$823$	37864.0i	1.60371i	0.597516	+	0.801857i	$0.296154\pi$
$823$	37864.0i	1.60371i	−0.597516	+	0.801857i	$0.703846\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42516.0i	1.78770i	0.448368	+	0.893849i	$0.352005\pi$
$827$	42516.0i	1.78770i	−0.448368	+	0.893849i	$0.647995\pi$
$828$	0	0
$829$	−45638.0	−1.91203	−0.956015	−	0.293317i	$-0.905241\pi$
$829$	−45638.0	−1.91203	−0.956015	+	0.293317i	$0.905241\pi$
$830$	0	0
$831$	−6378.00	−0.266246
$832$	0	0
$833$	− 5022.00i	− 0.208886i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	432.000i	0.0178400i
$838$	0	0
$839$	17496.0	0.719939	0.359970	−	0.932964i	$-0.382787\pi$
$839$	17496.0	0.719939	0.359970	+	0.932964i	$0.382787\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	0	0
$843$	8802.00i	0.359617i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 280.000i	− 0.0113588i
$848$	0	0
$849$	−6108.00	−0.246909
$850$	0	0
$851$	16272.0	0.655461
$852$	0	0
$853$	32174.0i	1.29146i	0.763565	+	0.645731i	$0.223447\pi$
$853$	32174.0i	1.29146i	−0.763565	+	0.645731i	$0.776553\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38934.0i	1.55188i	0.630807	+	0.775939i	$0.282724\pi$
$857$	38934.0i	1.55188i	−0.630807	+	0.775939i	$0.717276\pi$
$858$	0	0
$859$	29780.0	1.18286	0.591432	−	0.806355i	$-0.298563\pi$
$859$	29780.0	1.18286	0.591432	+	0.806355i	$0.298563\pi$
$860$	0	0
$861$	2160.00	0.0854966
$862$	0	0
$863$	48096.0i	1.89711i	0.316611	+	0.948556i	$0.397455\pi$
$863$	48096.0i	1.89711i	−0.316611	+	0.948556i	$0.602545\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13767.0i	− 0.539275i
$868$	0	0
$869$	−18432.0	−0.719520
$870$	0	0
$871$	3320.00	0.129155
$872$	0	0
$873$	− 9486.00i	− 0.367758i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 21302.0i	− 0.820202i	−0.912040	−	0.410101i	$-0.865493\pi$
$877$	− 21302.0i	− 0.820202i	0.912040	−	0.410101i	$-0.134507\pi$
$878$	0	0
$879$	6858.00	0.263157
$880$	0	0
$881$	−7470.00	−0.285665	−0.142832	−	0.989747i	$-0.545621\pi$
$881$	−7470.00	−0.285665	−0.142832	+	0.989747i	$0.545621\pi$
$882$	0	0
$883$	− 764.000i	− 0.0291174i	−0.999894	−	0.0145587i	$-0.995366\pi$
$883$	− 764.000i	− 0.0291174i	0.999894	−	0.0145587i	$-0.00463434\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32328.0i	1.22375i	0.790954	+	0.611876i	$0.209585\pi$
$887$	32328.0i	1.22375i	−0.790954	+	0.611876i	$0.790415\pi$
$888$	0	0
$889$	4736.00	0.178673
$890$	0	0
$891$	−2916.00	−0.109640
$892$	0	0
$893$	− 43200.0i	− 1.61885i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2160.00i	0.0804017i
$898$	0	0
$899$	3744.00	0.138898
$900$	0	0
$901$	7452.00	0.275541
$902$	0	0
$903$	− 10848.0i	− 0.399777i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 36316.0i	− 1.32950i	−0.747068	−	0.664748i	$-0.768539\pi$
$907$	− 36316.0i	− 1.32950i	0.747068	−	0.664748i	$-0.231461\pi$
$908$	0	0
$909$	−5022.00	−0.183244
$910$	0	0
$911$	13392.0	0.487044	0.243522	−	0.969895i	$-0.421697\pi$
$911$	13392.0	0.487044	0.243522	+	0.969895i	$0.421697\pi$
$912$	0	0
$913$	− 42768.0i	− 1.55029i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15264.0i	0.549686i
$918$	0	0
$919$	38072.0	1.36657	0.683286	−	0.730151i	$-0.260550\pi$
$919$	38072.0	1.36657	0.683286	+	0.730151i	$0.260550\pi$
$920$	0	0
$921$	3732.00	0.133522
$922$	0	0
$923$	− 3600.00i	− 0.128381i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	72.0000i	0.00255101i
$928$	0	0
$929$	12798.0	0.451979	0.225990	−	0.974130i	$-0.427438\pi$
$929$	12798.0	0.451979	0.225990	+	0.974130i	$0.427438\pi$
$930$	0	0
$931$	−27900.0	−0.982154
$932$	0	0
$933$	3672.00i	0.128849i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 34874.0i	− 1.21588i	−0.793981	−	0.607942i	$-0.791995\pi$
$937$	− 34874.0i	− 1.21588i	0.793981	−	0.607942i	$-0.208005\pi$
$938$	0	0
$939$	5694.00	0.197888
$940$	0	0
$941$	17190.0	0.595513	0.297757	−	0.954642i	$-0.403762\pi$
$941$	17190.0	0.595513	0.297757	+	0.954642i	$0.403762\pi$
$942$	0	0
$943$	− 6480.00i	− 0.223773i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40284.0i	1.38232i	0.722703	+	0.691158i	$0.242899\pi$
$947$	40284.0i	1.38232i	−0.722703	+	0.691158i	$0.757101\pi$
$948$	0	0
$949$	260.000	0.00889353
$950$	0	0
$951$	27486.0	0.937218
$952$	0	0
$953$	15498.0i	0.526789i	0.964688	+	0.263394i	$0.0848420\pi$
$953$	15498.0i	0.526789i	−0.964688	+	0.263394i	$0.915158\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	25272.0i	0.853634i
$958$	0	0
$959$	7632.00	0.256987
$960$	0	0
$961$	−29535.0	−0.991407
$962$	0	0
$963$	− 15876.0i	− 0.531253i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37160.0i	1.23577i	0.786270	+	0.617883i	$0.212009\pi$
$967$	37160.0i	1.23577i	−0.786270	+	0.617883i	$0.787991\pi$
$968$	0	0
$969$	5400.00	0.179023
$970$	0	0
$971$	−18468.0	−0.610367	−0.305183	−	0.952294i	$-0.598718\pi$
$971$	−18468.0	−0.610367	−0.305183	+	0.952294i	$0.598718\pi$
$972$	0	0
$973$	20512.0i	0.675832i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10386.0i	− 0.340100i	−0.985435	−	0.170050i	$-0.945607\pi$
$977$	− 10386.0i	− 0.340100i	0.985435	−	0.170050i	$-0.0543929\pi$
$978$	0	0
$979$	−22680.0	−0.740404
$980$	0	0
$981$	14598.0	0.475105
$982$	0	0
$983$	− 44136.0i	− 1.43206i	−0.698067	−	0.716032i	$-0.745956\pi$
$983$	− 44136.0i	− 1.43206i	0.698067	−	0.716032i	$-0.254044\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10368.0i	0.334364i
$988$	0	0
$989$	−32544.0	−1.04635
$990$	0	0
$991$	28432.0	0.911375	0.455687	−	0.890140i	$-0.349394\pi$
$991$	28432.0	0.911375	0.455687	+	0.890140i	$0.349394\pi$
$992$	0	0
$993$	− 13044.0i	− 0.416857i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39778.0i	1.26357i	0.775143	+	0.631786i	$0.217678\pi$
$997$	39778.0i	1.26357i	−0.775143	+	0.631786i	$0.782322\pi$
$998$	0	0
$999$	−6102.00	−0.193252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.d.49.1		2
4.3	odd	2		300.4.d.e.49.2		2
5.2	odd	4		48.4.a.a.1.1		1
5.3	odd	4		1200.4.a.be.1.1		1
5.4	even	2	inner	1200.4.f.d.49.2		2
12.11	even	2		900.4.d.c.649.1		2
15.2	even	4		144.4.a.g.1.1		1
20.3	even	4		300.4.a.b.1.1		1
20.7	even	4		12.4.a.a.1.1	✓	1
20.19	odd	2		300.4.d.e.49.1		2
35.27	even	4		2352.4.a.bk.1.1		1
40.27	even	4		192.4.a.f.1.1		1
40.37	odd	4		192.4.a.l.1.1		1
60.23	odd	4		900.4.a.g.1.1		1
60.47	odd	4		36.4.a.a.1.1		1
60.59	even	2		900.4.d.c.649.2		2
80.27	even	4		768.4.d.g.385.1		2
80.37	odd	4		768.4.d.j.385.2		2
80.67	even	4		768.4.d.g.385.2		2
80.77	odd	4		768.4.d.j.385.1		2
120.77	even	4		576.4.a.a.1.1		1
120.107	odd	4		576.4.a.b.1.1		1
140.27	odd	4		588.4.a.c.1.1		1
140.47	odd	12		588.4.i.e.361.1		2
140.67	even	12		588.4.i.d.373.1		2
140.87	odd	12		588.4.i.e.373.1		2
140.107	even	12		588.4.i.d.361.1		2
180.7	even	12		324.4.e.h.109.1		2
180.47	odd	12		324.4.e.a.109.1		2
180.67	even	12		324.4.e.h.217.1		2
180.167	odd	12		324.4.e.a.217.1		2
220.87	odd	4		1452.4.a.d.1.1		1
260.47	odd	4		2028.4.b.c.337.1		2
260.187	odd	4		2028.4.b.c.337.2		2
260.207	even	4		2028.4.a.c.1.1		1
420.47	even	12		1764.4.k.o.361.1		2
420.107	odd	12		1764.4.k.b.361.1		2
420.167	even	4		1764.4.a.b.1.1		1
420.227	even	12		1764.4.k.o.1549.1		2
420.347	odd	12		1764.4.k.b.1549.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.a.a.1.1	✓	1	20.7	even	4
36.4.a.a.1.1		1	60.47	odd	4
48.4.a.a.1.1		1	5.2	odd	4
144.4.a.g.1.1		1	15.2	even	4
192.4.a.f.1.1		1	40.27	even	4
192.4.a.l.1.1		1	40.37	odd	4
300.4.a.b.1.1		1	20.3	even	4
300.4.d.e.49.1		2	20.19	odd	2
300.4.d.e.49.2		2	4.3	odd	2
324.4.e.a.109.1		2	180.47	odd	12
324.4.e.a.217.1		2	180.167	odd	12
324.4.e.h.109.1		2	180.7	even	12
324.4.e.h.217.1		2	180.67	even	12
576.4.a.a.1.1		1	120.77	even	4
576.4.a.b.1.1		1	120.107	odd	4
588.4.a.c.1.1		1	140.27	odd	4
588.4.i.d.361.1		2	140.107	even	12
588.4.i.d.373.1		2	140.67	even	12
588.4.i.e.361.1		2	140.47	odd	12
588.4.i.e.373.1		2	140.87	odd	12
768.4.d.g.385.1		2	80.27	even	4
768.4.d.g.385.2		2	80.67	even	4
768.4.d.j.385.1		2	80.77	odd	4
768.4.d.j.385.2		2	80.37	odd	4
900.4.a.g.1.1		1	60.23	odd	4
900.4.d.c.649.1		2	12.11	even	2
900.4.d.c.649.2		2	60.59	even	2
1200.4.a.be.1.1		1	5.3	odd	4
1200.4.f.d.49.1		2	1.1	even	1	trivial
1200.4.f.d.49.2		2	5.4	even	2	inner
1452.4.a.d.1.1		1	220.87	odd	4
1764.4.a.b.1.1		1	420.167	even	4
1764.4.k.b.361.1		2	420.107	odd	12
1764.4.k.b.1549.1		2	420.347	odd	12
1764.4.k.o.361.1		2	420.47	even	12
1764.4.k.o.1549.1		2	420.227	even	12
2028.4.a.c.1.1		1	260.207	even	4
2028.4.b.c.337.1		2	260.47	odd	4
2028.4.b.c.337.2		2	260.187	odd	4
2352.4.a.bk.1.1		1	35.27	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +8.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +8.00000i q^{7} -9.00000 q^{9} -36.0000 q^{11} -10.0000i q^{13} -18.0000i q^{17} -100.000 q^{19} +24.0000 q^{21} -72.0000i q^{23} +27.0000i q^{27} +234.000 q^{29} +16.0000 q^{31} +108.000i q^{33} +226.000i q^{37} -30.0000 q^{39} +90.0000 q^{41} -452.000i q^{43} +432.000i q^{47} +279.000 q^{49} -54.0000 q^{51} +414.000i q^{53} +300.000i q^{57} -684.000 q^{59} +422.000 q^{61} -72.0000i q^{63} +332.000i q^{67} -216.000 q^{69} +360.000 q^{71} +26.0000i q^{73} -288.000i q^{77} +512.000 q^{79} +81.0000 q^{81} +1188.00i q^{83} -702.000i q^{87} +630.000 q^{89} +80.0000 q^{91} -48.0000i q^{93} +1054.00i q^{97} +324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 72 q^{11} - 200 q^{19} + 48 q^{21} + 468 q^{29} + 32 q^{31} - 60 q^{39} + 180 q^{41} + 558 q^{49} - 108 q^{51} - 1368 q^{59} + 844 q^{61} - 432 q^{69} + 720 q^{71} + 1024 q^{79} + 162 q^{81} + 1260 q^{89} + 160 q^{91} + 648 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 72 * q^11 - 200 * q^19 + 48 * q^21 + 468 * q^29 + 32 * q^31 - 60 * q^39 + 180 * q^41 + 558 * q^49 - 108 * q^51 - 1368 * q^59 + 844 * q^61 - 432 * q^69 + 720 * q^71 + 1024 * q^79 + 162 * q^81 + 1260 * q^89 + 160 * q^91 + 648 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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