LMFDB - Embedded newform 1200.4.f.b.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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
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.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} -52.0000 q^{11} +22.0000i q^{13} +14.0000i q^{17} -20.0000 q^{19} -72.0000 q^{21} +168.000i q^{23} +27.0000i q^{27} -230.000 q^{29} +288.000 q^{31} +156.000i q^{33} +34.0000i q^{37} +66.0000 q^{39} +122.000 q^{41} +188.000i q^{43} +256.000i q^{47} -233.000 q^{49} +42.0000 q^{51} -338.000i q^{53} +60.0000i q^{57} +100.000 q^{59} +742.000 q^{61} +216.000i q^{63} -84.0000i q^{67} +504.000 q^{69} +328.000 q^{71} -38.0000i q^{73} +1248.00i q^{77} -240.000 q^{79} +81.0000 q^{81} -1212.00i q^{83} +690.000i q^{87} -330.000 q^{89} +528.000 q^{91} -864.000i q^{93} -866.000i q^{97} +468.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 104 * q^11 - 40 * q^19 - 144 * q^21 - 460 * q^29 + 576 * q^31 + 132 * q^39 + 244 * q^41 - 466 * q^49 + 84 * q^51 + 200 * q^59 + 1484 * q^61 + 1008 * q^69 + 656 * q^71 - 480 * q^79 + 162 * q^81 - 660 * q^89 + 1056 * q^91 + 936 * 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
$7$	− 24.0000i	− 1.29588i	−0.761692	−	0.647939i	$-0.775631\pi$
$7$	− 24.0000i	− 1.29588i	0.761692	−	0.647939i	$-0.224369\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	0	0
$13$	22.0000i	0.469362i	0.972072	+	0.234681i	$0.0754045\pi$
$13$	22.0000i	0.469362i	−0.972072	+	0.234681i	$0.924595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.0000i	0.199735i	0.995001	+	0.0998676i	$0.0318419\pi$
$17$	14.0000i	0.199735i	−0.995001	+	0.0998676i	$0.968158\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	−72.0000	−0.748176
$22$	0	0
$23$	168.000i	1.52306i	0.648129	+	0.761531i	$0.275552\pi$
$23$	168.000i	1.52306i	−0.648129	+	0.761531i	$0.724448\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−230.000	−1.47276	−0.736378	−	0.676570i	$-0.763465\pi$
$29$	−230.000	−1.47276	−0.736378	+	0.676570i	$0.763465\pi$
$30$	0	0
$31$	288.000	1.66859	0.834296	−	0.551317i	$-0.185875\pi$
$31$	288.000	1.66859	0.834296	+	0.551317i	$0.185875\pi$
$32$	0	0
$33$	156.000i	0.822913i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	34.0000i	0.151069i	0.997143	+	0.0755347i	$0.0240664\pi$
$37$	34.0000i	0.151069i	−0.997143	+	0.0755347i	$0.975934\pi$
$38$	0	0
$39$	66.0000	0.270986
$40$	0	0
$41$	122.000	0.464712	0.232356	−	0.972631i	$-0.425357\pi$
$41$	122.000	0.464712	0.232356	+	0.972631i	$0.425357\pi$
$42$	0	0
$43$	188.000i	0.666738i	0.942796	+	0.333369i	$0.108185\pi$
$43$	188.000i	0.666738i	−0.942796	+	0.333369i	$0.891815\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	256.000i	0.794499i	0.917711	+	0.397249i	$0.130035\pi$
$47$	256.000i	0.794499i	−0.917711	+	0.397249i	$0.869965\pi$
$48$	0	0
$49$	−233.000	−0.679300
$50$	0	0
$51$	42.0000	0.115317
$52$	0	0
$53$	− 338.000i	− 0.875998i	−0.898976	−	0.437999i	$-0.855687\pi$
$53$	− 338.000i	− 0.875998i	0.898976	−	0.437999i	$-0.144313\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	60.0000i	0.139424i
$58$	0	0
$59$	100.000	0.220659	0.110330	−	0.993895i	$-0.464809\pi$
$59$	100.000	0.220659	0.110330	+	0.993895i	$0.464809\pi$
$60$	0	0
$61$	742.000	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$61$	742.000	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$62$	0	0
$63$	216.000i	0.431959i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 84.0000i	− 0.153168i	−0.997063	−	0.0765838i	$-0.975599\pi$
$67$	− 84.0000i	− 0.153168i	0.997063	−	0.0765838i	$-0.0244013\pi$
$68$	0	0
$69$	504.000	0.879340
$70$	0	0
$71$	328.000	0.548260	0.274130	−	0.961693i	$-0.411610\pi$
$71$	328.000	0.548260	0.274130	+	0.961693i	$0.411610\pi$
$72$	0	0
$73$	− 38.0000i	− 0.0609255i	−0.999536	−	0.0304628i	$-0.990302\pi$
$73$	− 38.0000i	− 0.0609255i	0.999536	−	0.0304628i	$-0.00969810\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1248.00i	1.84705i
$78$	0	0
$79$	−240.000	−0.341799	−0.170899	−	0.985288i	$-0.554667\pi$
$79$	−240.000	−0.341799	−0.170899	+	0.985288i	$0.554667\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1212.00i	− 1.60282i	−0.598114	−	0.801411i	$-0.704083\pi$
$83$	− 1212.00i	− 1.60282i	0.598114	−	0.801411i	$-0.295917\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	690.000i	0.850296i
$88$	0	0
$89$	−330.000	−0.393033	−0.196516	−	0.980501i	$-0.562963\pi$
$89$	−330.000	−0.393033	−0.196516	+	0.980501i	$0.562963\pi$
$90$	0	0
$91$	528.000	0.608236
$92$	0	0
$93$	− 864.000i	− 0.963362i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 866.000i	− 0.906484i	−0.891387	−	0.453242i	$-0.850267\pi$
$97$	− 866.000i	− 0.906484i	0.891387	−	0.453242i	$-0.149733\pi$
$98$	0	0
$99$	468.000	0.475109
$100$	0	0
$101$	−1218.00	−1.19996	−0.599978	−	0.800017i	$-0.704824\pi$
$101$	−1218.00	−1.19996	−0.599978	+	0.800017i	$0.704824\pi$
$102$	0	0
$103$	88.0000i	0.0841835i	0.999114	+	0.0420917i	$0.0134022\pi$
$103$	88.0000i	0.0841835i	−0.999114	+	0.0420917i	$0.986598\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	36.0000i	0.0325257i	0.999868	+	0.0162629i	$0.00517686\pi$
$107$	36.0000i	0.0325257i	−0.999868	+	0.0162629i	$0.994823\pi$
$108$	0	0
$109$	970.000	0.852378	0.426189	−	0.904634i	$-0.359856\pi$
$109$	970.000	0.852378	0.426189	+	0.904634i	$0.359856\pi$
$110$	0	0
$111$	102.000	0.0872199
$112$	0	0
$113$	1042.00i	0.867461i	0.901043	+	0.433731i	$0.142803\pi$
$113$	1042.00i	0.867461i	−0.901043	+	0.433731i	$0.857197\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 198.000i	− 0.156454i
$118$	0	0
$119$	336.000	0.258833
$120$	0	0
$121$	1373.00	1.03156
$122$	0	0
$123$	− 366.000i	− 0.268302i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1936.00i	1.35269i	0.736583	+	0.676347i	$0.236438\pi$
$127$	1936.00i	1.35269i	−0.736583	+	0.676347i	$0.763562\pi$
$128$	0	0
$129$	564.000	0.384941
$130$	0	0
$131$	−732.000	−0.488207	−0.244104	−	0.969749i	$-0.578494\pi$
$131$	−732.000	−0.488207	−0.244104	+	0.969749i	$0.578494\pi$
$132$	0	0
$133$	480.000i	0.312942i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2214.00i	1.38069i	0.723479	+	0.690346i	$0.242542\pi$
$137$	2214.00i	1.38069i	−0.723479	+	0.690346i	$0.757458\pi$
$138$	0	0
$139$	20.0000	0.0122042	0.00610208	−	0.999981i	$-0.498058\pi$
$139$	20.0000	0.0122042	0.00610208	+	0.999981i	$0.498058\pi$
$140$	0	0
$141$	768.000	0.458704
$142$	0	0
$143$	− 1144.00i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	699.000i	0.392194i
$148$	0	0
$149$	1330.00	0.731261	0.365630	−	0.930760i	$-0.380853\pi$
$149$	1330.00	0.731261	0.365630	+	0.930760i	$0.380853\pi$
$150$	0	0
$151$	1208.00	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	1208.00	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	− 126.000i	− 0.0665784i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3514.00i	1.78629i	0.449768	+	0.893146i	$0.351507\pi$
$157$	3514.00i	1.78629i	−0.449768	+	0.893146i	$0.648493\pi$
$158$	0	0
$159$	−1014.00	−0.505757
$160$	0	0
$161$	4032.00	1.97370
$162$	0	0
$163$	2068.00i	0.993732i	0.867827	+	0.496866i	$0.165516\pi$
$163$	2068.00i	0.993732i	−0.867827	+	0.496866i	$0.834484\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 24.0000i	− 0.0111208i	−0.999985	−	0.00556041i	$-0.998230\pi$
$167$	− 24.0000i	− 0.0111208i	0.999985	−	0.00556041i	$-0.00176994\pi$
$168$	0	0
$169$	1713.00	0.779700
$170$	0	0
$171$	180.000	0.0804967
$172$	0	0
$173$	− 618.000i	− 0.271593i	−0.990737	−	0.135797i	$-0.956641\pi$
$173$	− 618.000i	− 0.271593i	0.990737	−	0.135797i	$-0.0433594\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 300.000i	− 0.127398i
$178$	0	0
$179$	3340.00	1.39466	0.697328	−	0.716752i	$-0.254372\pi$
$179$	3340.00	1.39466	0.697328	+	0.716752i	$0.254372\pi$
$180$	0	0
$181$	−178.000	−0.0730974	−0.0365487	−	0.999332i	$-0.511636\pi$
$181$	−178.000	−0.0730974	−0.0365487	+	0.999332i	$0.511636\pi$
$182$	0	0
$183$	− 2226.00i	− 0.899184i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 728.000i	− 0.284688i
$188$	0	0
$189$	648.000	0.249392
$190$	0	0
$191$	1888.00	0.715240	0.357620	−	0.933867i	$-0.383588\pi$
$191$	1888.00	0.715240	0.357620	+	0.933867i	$0.383588\pi$
$192$	0	0
$193$	1922.00i	0.716832i	0.933562	+	0.358416i	$0.116683\pi$
$193$	1922.00i	0.716832i	−0.933562	+	0.358416i	$0.883317\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2526.00i	− 0.913554i	−0.889581	−	0.456777i	$-0.849004\pi$
$197$	− 2526.00i	− 0.913554i	0.889581	−	0.456777i	$-0.150996\pi$
$198$	0	0
$199$	−1160.00	−0.413217	−0.206609	−	0.978424i	$-0.566243\pi$
$199$	−1160.00	−0.413217	−0.206609	+	0.978424i	$0.566243\pi$
$200$	0	0
$201$	−252.000	−0.0884314
$202$	0	0
$203$	5520.00i	1.90851i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1512.00i	− 0.507687i
$208$	0	0
$209$	1040.00	0.344202
$210$	0	0
$211$	4468.00	1.45777	0.728886	−	0.684635i	$-0.240039\pi$
$211$	4468.00	1.45777	0.728886	+	0.684635i	$0.240039\pi$
$212$	0	0
$213$	− 984.000i	− 0.316538i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 6912.00i	− 2.16229i
$218$	0	0
$219$	−114.000	−0.0351754
$220$	0	0
$221$	−308.000	−0.0937481
$222$	0	0
$223$	− 6032.00i	− 1.81136i	−0.423965	−	0.905678i	$-0.639362\pi$
$223$	− 6032.00i	− 1.81136i	0.423965	−	0.905678i	$-0.360638\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2636.00i	0.770738i	0.922763	+	0.385369i	$0.125926\pi$
$227$	2636.00i	0.770738i	−0.922763	+	0.385369i	$0.874074\pi$
$228$	0	0
$229$	−4830.00	−1.39378	−0.696889	−	0.717179i	$-0.745433\pi$
$229$	−4830.00	−1.39378	−0.696889	+	0.717179i	$0.745433\pi$
$230$	0	0
$231$	3744.00	1.06639
$232$	0	0
$233$	2682.00i	0.754093i	0.926194	+	0.377046i	$0.123060\pi$
$233$	2682.00i	0.754093i	−0.926194	+	0.377046i	$0.876940\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	720.000i	0.197338i
$238$	0	0
$239$	2320.00	0.627901	0.313950	−	0.949439i	$-0.398347\pi$
$239$	2320.00	0.627901	0.313950	+	0.949439i	$0.398347\pi$
$240$	0	0
$241$	2002.00	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$241$	2002.00	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 440.000i	− 0.113346i
$248$	0	0
$249$	−3636.00	−0.925390
$250$	0	0
$251$	−132.000	−0.0331943	−0.0165971	−	0.999862i	$-0.505283\pi$
$251$	−132.000	−0.0331943	−0.0165971	+	0.999862i	$0.505283\pi$
$252$	0	0
$253$	− 8736.00i	− 2.17086i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7614.00i	1.84805i	0.382335	+	0.924024i	$0.375120\pi$
$257$	7614.00i	1.84805i	−0.382335	+	0.924024i	$0.624880\pi$
$258$	0	0
$259$	816.000	0.195767
$260$	0	0
$261$	2070.00	0.490919
$262$	0	0
$263$	4888.00i	1.14603i	0.819543	+	0.573017i	$0.194227\pi$
$263$	4888.00i	1.14603i	−0.819543	+	0.573017i	$0.805773\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	990.000i	0.226918i
$268$	0	0
$269$	−1270.00	−0.287856	−0.143928	−	0.989588i	$-0.545973\pi$
$269$	−1270.00	−0.287856	−0.143928	+	0.989588i	$0.545973\pi$
$270$	0	0
$271$	−1072.00	−0.240293	−0.120146	−	0.992756i	$-0.538336\pi$
$271$	−1072.00	−0.240293	−0.120146	+	0.992756i	$0.538336\pi$
$272$	0	0
$273$	− 1584.00i	− 0.351165i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5394.00i	1.17001i	0.811028	+	0.585007i	$0.198908\pi$
$277$	5394.00i	1.17001i	−0.811028	+	0.585007i	$0.801092\pi$
$278$	0	0
$279$	−2592.00	−0.556197
$280$	0	0
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	0	0
$283$	− 2772.00i	− 0.582255i	−0.956684	−	0.291128i	$-0.905970\pi$
$283$	− 2772.00i	− 0.582255i	0.956684	−	0.291128i	$-0.0940305\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2928.00i	− 0.602210i
$288$	0	0
$289$	4717.00	0.960106
$290$	0	0
$291$	−2598.00	−0.523359
$292$	0	0
$293$	4542.00i	0.905619i	0.891607	+	0.452810i	$0.149578\pi$
$293$	4542.00i	0.905619i	−0.891607	+	0.452810i	$0.850422\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1404.00i	− 0.274304i
$298$	0	0
$299$	−3696.00	−0.714867
$300$	0	0
$301$	4512.00	0.864011
$302$	0	0
$303$	3654.00i	0.692795i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5116.00i	0.951093i	0.879691	+	0.475546i	$0.157750\pi$
$307$	5116.00i	0.951093i	−0.879691	+	0.475546i	$0.842250\pi$
$308$	0	0
$309$	264.000	0.0486034
$310$	0	0
$311$	2808.00	0.511984	0.255992	−	0.966679i	$-0.417598\pi$
$311$	2808.00	0.511984	0.255992	+	0.966679i	$0.417598\pi$
$312$	0	0
$313$	− 7318.00i	− 1.32153i	−0.750594	−	0.660763i	$-0.770233\pi$
$313$	− 7318.00i	− 1.32153i	0.750594	−	0.660763i	$-0.229767\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2246.00i	− 0.397943i	−0.980005	−	0.198971i	$-0.936240\pi$
$317$	− 2246.00i	− 0.397943i	0.980005	−	0.198971i	$-0.0637601\pi$
$318$	0	0
$319$	11960.0	2.09916
$320$	0	0
$321$	108.000	0.0187787
$322$	0	0
$323$	− 280.000i	− 0.0482341i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2910.00i	− 0.492120i
$328$	0	0
$329$	6144.00	1.02957
$330$	0	0
$331$	−1332.00	−0.221188	−0.110594	−	0.993866i	$-0.535275\pi$
$331$	−1332.00	−0.221188	−0.110594	+	0.993866i	$0.535275\pi$
$332$	0	0
$333$	− 306.000i	− 0.0503564i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11534.0i	1.86438i	0.361966	+	0.932191i	$0.382106\pi$
$337$	11534.0i	1.86438i	−0.361966	+	0.932191i	$0.617894\pi$
$338$	0	0
$339$	3126.00	0.500829
$340$	0	0
$341$	−14976.0	−2.37829
$342$	0	0
$343$	− 2640.00i	− 0.415588i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11956.0i	1.84966i	0.380382	+	0.924830i	$0.375793\pi$
$347$	11956.0i	1.84966i	−0.380382	+	0.924830i	$0.624207\pi$
$348$	0	0
$349$	−4870.00	−0.746949	−0.373474	−	0.927640i	$-0.621834\pi$
$349$	−4870.00	−0.746949	−0.373474	+	0.927640i	$0.621834\pi$
$350$	0	0
$351$	−594.000	−0.0903287
$352$	0	0
$353$	10722.0i	1.61664i	0.588742	+	0.808321i	$0.299623\pi$
$353$	10722.0i	1.61664i	−0.588742	+	0.808321i	$0.700377\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1008.00i	− 0.149437i
$358$	0	0
$359$	120.000	0.0176417	0.00882083	−	0.999961i	$-0.497192\pi$
$359$	120.000	0.0176417	0.00882083	+	0.999961i	$0.497192\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	− 4119.00i	− 0.595569i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3936.00i	0.559830i	0.960025	+	0.279915i	$0.0903063\pi$
$367$	3936.00i	0.559830i	−0.960025	+	0.279915i	$0.909694\pi$
$368$	0	0
$369$	−1098.00	−0.154904
$370$	0	0
$371$	−8112.00	−1.13519
$372$	0	0
$373$	3022.00i	0.419499i	0.977755	+	0.209750i	$0.0672649\pi$
$373$	3022.00i	0.419499i	−0.977755	+	0.209750i	$0.932735\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5060.00i	− 0.691255i
$378$	0	0
$379$	−13340.0	−1.80799	−0.903997	−	0.427539i	$-0.859381\pi$
$379$	−13340.0	−1.80799	−0.903997	+	0.427539i	$0.859381\pi$
$380$	0	0
$381$	5808.00	0.780979
$382$	0	0
$383$	1008.00i	0.134481i	0.997737	+	0.0672407i	$0.0214195\pi$
$383$	1008.00i	0.134481i	−0.997737	+	0.0672407i	$0.978580\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1692.00i	− 0.222246i
$388$	0	0
$389$	−9630.00	−1.25517	−0.627584	−	0.778549i	$-0.715956\pi$
$389$	−9630.00	−1.25517	−0.627584	+	0.778549i	$0.715956\pi$
$390$	0	0
$391$	−2352.00	−0.304209
$392$	0	0
$393$	2196.00i	0.281867i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7126.00i	− 0.900866i	−0.892810	−	0.450433i	$-0.851270\pi$
$397$	− 7126.00i	− 0.900866i	0.892810	−	0.450433i	$-0.148730\pi$
$398$	0	0
$399$	1440.00	0.180677
$400$	0	0
$401$	−8718.00	−1.08568	−0.542838	−	0.839837i	$-0.682650\pi$
$401$	−8718.00	−1.08568	−0.542838	+	0.839837i	$0.682650\pi$
$402$	0	0
$403$	6336.00i	0.783173i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1768.00i	− 0.215323i
$408$	0	0
$409$	10870.0	1.31415	0.657074	−	0.753826i	$-0.271794\pi$
$409$	10870.0	1.31415	0.657074	+	0.753826i	$0.271794\pi$
$410$	0	0
$411$	6642.00	0.797143
$412$	0	0
$413$	− 2400.00i	− 0.285947i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 60.0000i	− 0.00704607i
$418$	0	0
$419$	−9700.00	−1.13097	−0.565484	−	0.824759i	$-0.691311\pi$
$419$	−9700.00	−1.13097	−0.565484	+	0.824759i	$0.691311\pi$
$420$	0	0
$421$	862.000	0.0997893	0.0498947	−	0.998754i	$-0.484111\pi$
$421$	862.000	0.0997893	0.0498947	+	0.998754i	$0.484111\pi$
$422$	0	0
$423$	− 2304.00i	− 0.264833i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 17808.0i	− 2.01824i
$428$	0	0
$429$	−3432.00	−0.386244
$430$	0	0
$431$	−15792.0	−1.76490	−0.882452	−	0.470402i	$-0.844109\pi$
$431$	−15792.0	−1.76490	−0.882452	+	0.470402i	$0.844109\pi$
$432$	0	0
$433$	11602.0i	1.28766i	0.765169	+	0.643830i	$0.222655\pi$
$433$	11602.0i	1.28766i	−0.765169	+	0.643830i	$0.777345\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3360.00i	− 0.367805i
$438$	0	0
$439$	−440.000	−0.0478361	−0.0239181	−	0.999714i	$-0.507614\pi$
$439$	−440.000	−0.0478361	−0.0239181	+	0.999714i	$0.507614\pi$
$440$	0	0
$441$	2097.00	0.226433
$442$	0	0
$443$	10188.0i	1.09266i	0.837571	+	0.546328i	$0.183975\pi$
$443$	10188.0i	1.09266i	−0.837571	+	0.546328i	$0.816025\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3990.00i	− 0.422194i
$448$	0	0
$449$	13310.0	1.39897	0.699485	−	0.714647i	$-0.253413\pi$
$449$	13310.0	1.39897	0.699485	+	0.714647i	$0.253413\pi$
$450$	0	0
$451$	−6344.00	−0.662367
$452$	0	0
$453$	− 3624.00i	− 0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3226.00i	− 0.330210i	−0.986276	−	0.165105i	$-0.947204\pi$
$457$	− 3226.00i	− 0.330210i	0.986276	−	0.165105i	$-0.0527963\pi$
$458$	0	0
$459$	−378.000	−0.0384391
$460$	0	0
$461$	6582.00	0.664977	0.332488	−	0.943107i	$-0.392112\pi$
$461$	6582.00	0.664977	0.332488	+	0.943107i	$0.392112\pi$
$462$	0	0
$463$	− 15072.0i	− 1.51286i	−0.654073	−	0.756431i	$-0.726941\pi$
$463$	− 15072.0i	− 1.51286i	0.654073	−	0.756431i	$-0.273059\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	476.000i	0.0471663i	0.999722	+	0.0235831i	$0.00750744\pi$
$467$	476.000i	0.0471663i	−0.999722	+	0.0235831i	$0.992493\pi$
$468$	0	0
$469$	−2016.00	−0.198487
$470$	0	0
$471$	10542.0	1.03132
$472$	0	0
$473$	− 9776.00i	− 0.950319i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3042.00i	0.291999i
$478$	0	0
$479$	−19680.0	−1.87725	−0.938624	−	0.344941i	$-0.887899\pi$
$479$	−19680.0	−1.87725	−0.938624	+	0.344941i	$0.887899\pi$
$480$	0	0
$481$	−748.000	−0.0709062
$482$	0	0
$483$	− 12096.0i	− 1.13952i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5944.00i	− 0.553077i	−0.961003	−	0.276538i	$-0.910813\pi$
$487$	− 5944.00i	− 0.553077i	0.961003	−	0.276538i	$-0.0891873\pi$
$488$	0	0
$489$	6204.00	0.573731
$490$	0	0
$491$	−10772.0	−0.990089	−0.495044	−	0.868868i	$-0.664848\pi$
$491$	−10772.0	−0.990089	−0.495044	+	0.868868i	$0.664848\pi$
$492$	0	0
$493$	− 3220.00i	− 0.294161i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7872.00i	− 0.710478i
$498$	0	0
$499$	8140.00	0.730253	0.365127	−	0.930958i	$-0.381026\pi$
$499$	8140.00	0.730253	0.365127	+	0.930958i	$0.381026\pi$
$500$	0	0
$501$	−72.0000	−0.00642060
$502$	0	0
$503$	13768.0i	1.22045i	0.792229	+	0.610223i	$0.208920\pi$
$503$	13768.0i	1.22045i	−0.792229	+	0.610223i	$0.791080\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5139.00i	− 0.450160i
$508$	0	0
$509$	−22150.0	−1.92884	−0.964422	−	0.264368i	$-0.914837\pi$
$509$	−22150.0	−1.92884	−0.964422	+	0.264368i	$0.914837\pi$
$510$	0	0
$511$	−912.000	−0.0789521
$512$	0	0
$513$	− 540.000i	− 0.0464748i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13312.0i	− 1.13242i
$518$	0	0
$519$	−1854.00	−0.156805
$520$	0	0
$521$	1562.00	0.131348	0.0656741	−	0.997841i	$-0.479080\pi$
$521$	1562.00	0.131348	0.0656741	+	0.997841i	$0.479080\pi$
$522$	0	0
$523$	668.000i	0.0558501i	0.999610	+	0.0279250i	$0.00888997\pi$
$523$	668.000i	0.0558501i	−0.999610	+	0.0279250i	$0.991110\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4032.00i	0.333276i
$528$	0	0
$529$	−16057.0	−1.31972
$530$	0	0
$531$	−900.000	−0.0735531
$532$	0	0
$533$	2684.00i	0.218118i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 10020.0i	− 0.805205i
$538$	0	0
$539$	12116.0	0.968225
$540$	0	0
$541$	−6138.00	−0.487788	−0.243894	−	0.969802i	$-0.578425\pi$
$541$	−6138.00	−0.487788	−0.243894	+	0.969802i	$0.578425\pi$
$542$	0	0
$543$	534.000i	0.0422028i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10484.0i	− 0.819494i	−0.912199	−	0.409747i	$-0.865617\pi$
$547$	− 10484.0i	− 0.819494i	0.912199	−	0.409747i	$-0.134383\pi$
$548$	0	0
$549$	−6678.00	−0.519144
$550$	0	0
$551$	4600.00	0.355656
$552$	0	0
$553$	5760.00i	0.442930i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3606.00i	− 0.274311i	−0.990550	−	0.137155i	$-0.956204\pi$
$557$	− 3606.00i	− 0.274311i	0.990550	−	0.137155i	$-0.0437960\pi$
$558$	0	0
$559$	−4136.00	−0.312941
$560$	0	0
$561$	−2184.00	−0.164365
$562$	0	0
$563$	− 12252.0i	− 0.917159i	−0.888654	−	0.458579i	$-0.848359\pi$
$563$	− 12252.0i	− 0.917159i	0.888654	−	0.458579i	$-0.151641\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1944.00i	− 0.143986i
$568$	0	0
$569$	14550.0	1.07200	0.536000	−	0.844218i	$-0.319935\pi$
$569$	14550.0	1.07200	0.536000	+	0.844218i	$0.319935\pi$
$570$	0	0
$571$	25468.0	1.86655	0.933277	−	0.359157i	$-0.116936\pi$
$571$	25468.0	1.86655	0.933277	+	0.359157i	$0.116936\pi$
$572$	0	0
$573$	− 5664.00i	− 0.412944i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 12866.0i	− 0.928282i	−0.885761	−	0.464141i	$-0.846363\pi$
$577$	− 12866.0i	− 0.928282i	0.885761	−	0.464141i	$-0.153637\pi$
$578$	0	0
$579$	5766.00	0.413863
$580$	0	0
$581$	−29088.0	−2.07706
$582$	0	0
$583$	17576.0i	1.24858i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14844.0i	− 1.04374i	−0.853024	−	0.521872i	$-0.825234\pi$
$587$	− 14844.0i	− 1.04374i	0.853024	−	0.521872i	$-0.174766\pi$
$588$	0	0
$589$	−5760.00	−0.402948
$590$	0	0
$591$	−7578.00	−0.527440
$592$	0	0
$593$	20402.0i	1.41283i	0.707797	+	0.706416i	$0.249689\pi$
$593$	20402.0i	1.41283i	−0.707797	+	0.706416i	$0.750311\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3480.00i	0.238571i
$598$	0	0
$599$	10760.0	0.733959	0.366980	−	0.930229i	$-0.380392\pi$
$599$	10760.0	0.733959	0.366980	+	0.930229i	$0.380392\pi$
$600$	0	0
$601$	14282.0	0.969343	0.484671	−	0.874696i	$-0.338939\pi$
$601$	14282.0	0.969343	0.484671	+	0.874696i	$0.338939\pi$
$602$	0	0
$603$	756.000i	0.0510559i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11056.0i	0.739290i	0.929173	+	0.369645i	$0.120521\pi$
$607$	11056.0i	0.739290i	−0.929173	+	0.369645i	$0.879479\pi$
$608$	0	0
$609$	16560.0	1.10188
$610$	0	0
$611$	−5632.00	−0.372907
$612$	0	0
$613$	− 16418.0i	− 1.08176i	−0.841101	−	0.540878i	$-0.818092\pi$
$613$	− 16418.0i	− 1.08176i	0.841101	−	0.540878i	$-0.181908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10374.0i	0.676891i	0.940986	+	0.338445i	$0.109901\pi$
$617$	10374.0i	0.676891i	−0.940986	+	0.338445i	$0.890099\pi$
$618$	0	0
$619$	−5260.00	−0.341546	−0.170773	−	0.985310i	$-0.554627\pi$
$619$	−5260.00	−0.341546	−0.170773	+	0.985310i	$0.554627\pi$
$620$	0	0
$621$	−4536.00	−0.293113
$622$	0	0
$623$	7920.00i	0.509323i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3120.00i	− 0.198725i
$628$	0	0
$629$	−476.000	−0.0301739
$630$	0	0
$631$	−21352.0	−1.34708	−0.673542	−	0.739149i	$-0.735228\pi$
$631$	−21352.0	−1.34708	−0.673542	+	0.739149i	$0.735228\pi$
$632$	0	0
$633$	− 13404.0i	− 0.841645i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5126.00i	− 0.318838i
$638$	0	0
$639$	−2952.00	−0.182753
$640$	0	0
$641$	−29118.0	−1.79422	−0.897108	−	0.441812i	$-0.854336\pi$
$641$	−29118.0	−1.79422	−0.897108	+	0.441812i	$0.854336\pi$
$642$	0	0
$643$	− 5772.00i	− 0.354005i	−0.984210	−	0.177003i	$-0.943360\pi$
$643$	− 5772.00i	− 0.354005i	0.984210	−	0.177003i	$-0.0566401\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 14264.0i	− 0.866732i	−0.901218	−	0.433366i	$-0.857326\pi$
$647$	− 14264.0i	− 0.866732i	0.901218	−	0.433366i	$-0.142674\pi$
$648$	0	0
$649$	−5200.00	−0.314511
$650$	0	0
$651$	−20736.0	−1.24840
$652$	0	0
$653$	6902.00i	0.413623i	0.978381	+	0.206812i	$0.0663088\pi$
$653$	6902.00i	0.413623i	−0.978381	+	0.206812i	$0.933691\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	342.000i	0.0203085i
$658$	0	0
$659$	20140.0	1.19051	0.595253	−	0.803539i	$-0.297052\pi$
$659$	20140.0	1.19051	0.595253	+	0.803539i	$0.297052\pi$
$660$	0	0
$661$	−3218.00	−0.189358	−0.0946790	−	0.995508i	$-0.530182\pi$
$661$	−3218.00	−0.189358	−0.0946790	+	0.995508i	$0.530182\pi$
$662$	0	0
$663$	924.000i	0.0541255i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 38640.0i	− 2.24310i
$668$	0	0
$669$	−18096.0	−1.04579
$670$	0	0
$671$	−38584.0	−2.21985
$672$	0	0
$673$	− 7518.00i	− 0.430606i	−0.976547	−	0.215303i	$-0.930926\pi$
$673$	− 7518.00i	− 0.430606i	0.976547	−	0.215303i	$-0.0690739\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18114.0i	1.02833i	0.857692	+	0.514164i	$0.171898\pi$
$677$	18114.0i	1.02833i	−0.857692	+	0.514164i	$0.828102\pi$
$678$	0	0
$679$	−20784.0	−1.17469
$680$	0	0
$681$	7908.00	0.444986
$682$	0	0
$683$	23868.0i	1.33716i	0.743638	+	0.668582i	$0.233099\pi$
$683$	23868.0i	1.33716i	−0.743638	+	0.668582i	$0.766901\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14490.0i	0.804699i
$688$	0	0
$689$	7436.00	0.411160
$690$	0	0
$691$	−172.000	−0.00946916	−0.00473458	−	0.999989i	$-0.501507\pi$
$691$	−172.000	−0.00946916	−0.00473458	+	0.999989i	$0.501507\pi$
$692$	0	0
$693$	− 11232.0i	− 0.615683i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1708.00i	0.0928194i
$698$	0	0
$699$	8046.00	0.435376
$700$	0	0
$701$	−22138.0	−1.19278	−0.596391	−	0.802694i	$-0.703399\pi$
$701$	−22138.0	−1.19278	−0.596391	+	0.802694i	$0.703399\pi$
$702$	0	0
$703$	− 680.000i	− 0.0364818i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29232.0i	1.55500i
$708$	0	0
$709$	−3070.00	−0.162618	−0.0813091	−	0.996689i	$-0.525910\pi$
$709$	−3070.00	−0.162618	−0.0813091	+	0.996689i	$0.525910\pi$
$710$	0	0
$711$	2160.00	0.113933
$712$	0	0
$713$	48384.0i	2.54137i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6960.00i	− 0.362519i
$718$	0	0
$719$	15600.0	0.809154	0.404577	−	0.914504i	$-0.367419\pi$
$719$	15600.0	0.809154	0.404577	+	0.914504i	$0.367419\pi$
$720$	0	0
$721$	2112.00	0.109092
$722$	0	0
$723$	− 6006.00i	− 0.308943i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20696.0i	1.05581i	0.849304	+	0.527904i	$0.177022\pi$
$727$	20696.0i	1.05581i	−0.849304	+	0.527904i	$0.822978\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−2632.00	−0.133171
$732$	0	0
$733$	− 30778.0i	− 1.55090i	−0.631408	−	0.775451i	$-0.717522\pi$
$733$	− 30778.0i	− 1.55090i	0.631408	−	0.775451i	$-0.282478\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4368.00i	0.218314i
$738$	0	0
$739$	11740.0	0.584388	0.292194	−	0.956359i	$-0.405615\pi$
$739$	11740.0	0.584388	0.292194	+	0.956359i	$0.405615\pi$
$740$	0	0
$741$	−1320.00	−0.0654405
$742$	0	0
$743$	− 2632.00i	− 0.129958i	−0.997887	−	0.0649789i	$-0.979302\pi$
$743$	− 2632.00i	− 0.129958i	0.997887	−	0.0649789i	$-0.0206980\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10908.0i	0.534274i
$748$	0	0
$749$	864.000	0.0421494
$750$	0	0
$751$	20528.0	0.997440	0.498720	−	0.866763i	$-0.333804\pi$
$751$	20528.0	0.997440	0.498720	+	0.866763i	$0.333804\pi$
$752$	0	0
$753$	396.000i	0.0191647i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 21646.0i	− 1.03928i	−0.854384	−	0.519642i	$-0.826066\pi$
$757$	− 21646.0i	− 1.03928i	0.854384	−	0.519642i	$-0.173934\pi$
$758$	0	0
$759$	−26208.0	−1.25335
$760$	0	0
$761$	18282.0	0.870857	0.435428	−	0.900223i	$-0.356597\pi$
$761$	18282.0	0.870857	0.435428	+	0.900223i	$0.356597\pi$
$762$	0	0
$763$	− 23280.0i	− 1.10458i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2200.00i	0.103569i
$768$	0	0
$769$	24190.0	1.13435	0.567174	−	0.823598i	$-0.308037\pi$
$769$	24190.0	1.13435	0.567174	+	0.823598i	$0.308037\pi$
$770$	0	0
$771$	22842.0	1.06697
$772$	0	0
$773$	− 25698.0i	− 1.19572i	−0.801600	−	0.597861i	$-0.796018\pi$
$773$	− 25698.0i	− 1.19572i	0.801600	−	0.597861i	$-0.203982\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2448.00i	− 0.113026i
$778$	0	0
$779$	−2440.00	−0.112223
$780$	0	0
$781$	−17056.0	−0.781449
$782$	0	0
$783$	− 6210.00i	− 0.283432i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33436.0i	1.51444i	0.653160	+	0.757220i	$0.273443\pi$
$787$	33436.0i	1.51444i	−0.653160	+	0.757220i	$0.726557\pi$
$788$	0	0
$789$	14664.0	0.661663
$790$	0	0
$791$	25008.0	1.12412
$792$	0	0
$793$	16324.0i	0.730999i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37594.0i	1.67083i	0.549623	+	0.835413i	$0.314771\pi$
$797$	37594.0i	1.67083i	−0.549623	+	0.835413i	$0.685229\pi$
$798$	0	0
$799$	−3584.00	−0.158689
$800$	0	0
$801$	2970.00	0.131011
$802$	0	0
$803$	1976.00i	0.0868388i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3810.00i	0.166194i
$808$	0	0
$809$	−4730.00	−0.205560	−0.102780	−	0.994704i	$-0.532774\pi$
$809$	−4730.00	−0.205560	−0.102780	+	0.994704i	$0.532774\pi$
$810$	0	0
$811$	8748.00	0.378772	0.189386	−	0.981903i	$-0.439350\pi$
$811$	8748.00	0.378772	0.189386	+	0.981903i	$0.439350\pi$
$812$	0	0
$813$	3216.00i	0.138733i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3760.00i	− 0.161011i
$818$	0	0
$819$	−4752.00	−0.202745
$820$	0	0
$821$	44142.0	1.87645	0.938226	−	0.346024i	$-0.112468\pi$
$821$	44142.0	1.87645	0.938226	+	0.346024i	$0.112468\pi$
$822$	0	0
$823$	− 3992.00i	− 0.169079i	−0.996420	−	0.0845397i	$-0.973058\pi$
$823$	− 3992.00i	− 0.169079i	0.996420	−	0.0845397i	$-0.0269420\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14444.0i	− 0.607336i	−0.952778	−	0.303668i	$-0.901789\pi$
$827$	− 14444.0i	− 0.607336i	0.952778	−	0.303668i	$-0.0982114\pi$
$828$	0	0
$829$	−42150.0	−1.76590	−0.882949	−	0.469468i	$-0.844446\pi$
$829$	−42150.0	−1.76590	−0.882949	+	0.469468i	$0.844446\pi$
$830$	0	0
$831$	16182.0	0.675508
$832$	0	0
$833$	− 3262.00i	− 0.135680i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7776.00i	0.321121i
$838$	0	0
$839$	13400.0	0.551394	0.275697	−	0.961245i	$-0.411091\pi$
$839$	13400.0	0.551394	0.275697	+	0.961245i	$0.411091\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	0	0
$843$	− 7326.00i	− 0.299313i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 32952.0i	− 1.33677i
$848$	0	0
$849$	−8316.00	−0.336165
$850$	0	0
$851$	−5712.00	−0.230088
$852$	0	0
$853$	− 8658.00i	− 0.347531i	−0.984787	−	0.173766i	$-0.944406\pi$
$853$	− 8658.00i	− 0.347531i	0.984787	−	0.173766i	$-0.0555935\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 42826.0i	− 1.70701i	−0.521084	−	0.853505i	$-0.674472\pi$
$857$	− 42826.0i	− 1.70701i	0.521084	−	0.853505i	$-0.325528\pi$
$858$	0	0
$859$	−35900.0	−1.42595	−0.712976	−	0.701189i	$-0.752653\pi$
$859$	−35900.0	−1.42595	−0.712976	+	0.701189i	$0.752653\pi$
$860$	0	0
$861$	−8784.00	−0.347686
$862$	0	0
$863$	3088.00i	0.121804i	0.998144	+	0.0609019i	$0.0193977\pi$
$863$	3088.00i	0.121804i	−0.998144	+	0.0609019i	$0.980602\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14151.0i	− 0.554317i
$868$	0	0
$869$	12480.0	0.487175
$870$	0	0
$871$	1848.00	0.0718910
$872$	0	0
$873$	7794.00i	0.302161i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35274.0i	1.35817i	0.734058	+	0.679087i	$0.237624\pi$
$877$	35274.0i	1.35817i	−0.734058	+	0.679087i	$0.762376\pi$
$878$	0	0
$879$	13626.0	0.522860
$880$	0	0
$881$	25042.0	0.957646	0.478823	−	0.877911i	$-0.341064\pi$
$881$	25042.0	0.957646	0.478823	+	0.877911i	$0.341064\pi$
$882$	0	0
$883$	− 12572.0i	− 0.479141i	−0.970879	−	0.239570i	$-0.922993\pi$
$883$	− 12572.0i	− 0.479141i	0.970879	−	0.239570i	$-0.0770066\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 21864.0i	− 0.827645i	−0.910358	−	0.413823i	$-0.864193\pi$
$887$	− 21864.0i	− 0.827645i	0.910358	−	0.413823i	$-0.135807\pi$
$888$	0	0
$889$	46464.0	1.75293
$890$	0	0
$891$	−4212.00	−0.158370
$892$	0	0
$893$	− 5120.00i	− 0.191864i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11088.0i	0.412729i
$898$	0	0
$899$	−66240.0	−2.45743
$900$	0	0
$901$	4732.00	0.174968
$902$	0	0
$903$	− 13536.0i	− 0.498837i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	31236.0i	1.14352i	0.820420	+	0.571761i	$0.193740\pi$
$907$	31236.0i	1.14352i	−0.820420	+	0.571761i	$0.806260\pi$
$908$	0	0
$909$	10962.0	0.399985
$910$	0	0
$911$	−8272.00	−0.300838	−0.150419	−	0.988622i	$-0.548062\pi$
$911$	−8272.00	−0.300838	−0.150419	+	0.988622i	$0.548062\pi$
$912$	0	0
$913$	63024.0i	2.28455i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17568.0i	0.632657i
$918$	0	0
$919$	20200.0	0.725067	0.362533	−	0.931971i	$-0.381912\pi$
$919$	20200.0	0.725067	0.362533	+	0.931971i	$0.381912\pi$
$920$	0	0
$921$	15348.0	0.549114
$922$	0	0
$923$	7216.00i	0.257332i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 792.000i	− 0.0280612i
$928$	0	0
$929$	−31010.0	−1.09516	−0.547581	−	0.836753i	$-0.684451\pi$
$929$	−31010.0	−1.09516	−0.547581	+	0.836753i	$0.684451\pi$
$930$	0	0
$931$	4660.00	0.164044
$932$	0	0
$933$	− 8424.00i	− 0.295594i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39174.0i	1.36580i	0.730510	+	0.682902i	$0.239283\pi$
$937$	39174.0i	1.36580i	−0.730510	+	0.682902i	$0.760717\pi$
$938$	0	0
$939$	−21954.0	−0.762984
$940$	0	0
$941$	−4138.00	−0.143353	−0.0716764	−	0.997428i	$-0.522835\pi$
$941$	−4138.00	−0.143353	−0.0716764	+	0.997428i	$0.522835\pi$
$942$	0	0
$943$	20496.0i	0.707785i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23676.0i	0.812425i	0.913779	+	0.406213i	$0.133151\pi$
$947$	23676.0i	0.812425i	−0.913779	+	0.406213i	$0.866849\pi$
$948$	0	0
$949$	836.000	0.0285961
$950$	0	0
$951$	−6738.00	−0.229752
$952$	0	0
$953$	18922.0i	0.643173i	0.946880	+	0.321586i	$0.104216\pi$
$953$	18922.0i	0.643173i	−0.946880	+	0.321586i	$0.895784\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 35880.0i	− 1.21195i
$958$	0	0
$959$	53136.0	1.78921
$960$	0	0
$961$	53153.0	1.78420
$962$	0	0
$963$	− 324.000i	− 0.0108419i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39656.0i	1.31877i	0.751805	+	0.659385i	$0.229183\pi$
$967$	39656.0i	1.31877i	−0.751805	+	0.659385i	$0.770817\pi$
$968$	0	0
$969$	−840.000	−0.0278480
$970$	0	0
$971$	33228.0	1.09818	0.549092	−	0.835762i	$-0.314974\pi$
$971$	33228.0	1.09818	0.549092	+	0.835762i	$0.314974\pi$
$972$	0	0
$973$	− 480.000i	− 0.0158151i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	974.000i	0.0318946i	0.999873	+	0.0159473i	$0.00507640\pi$
$977$	974.000i	0.0318946i	−0.999873	+	0.0159473i	$0.994924\pi$
$978$	0	0
$979$	17160.0	0.560200
$980$	0	0
$981$	−8730.00	−0.284126
$982$	0	0
$983$	13608.0i	0.441534i	0.975327	+	0.220767i	$0.0708560\pi$
$983$	13608.0i	0.441534i	−0.975327	+	0.220767i	$0.929144\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 18432.0i	− 0.594425i
$988$	0	0
$989$	−31584.0	−1.01548
$990$	0	0
$991$	−13472.0	−0.431839	−0.215919	−	0.976411i	$-0.569275\pi$
$991$	−13472.0	−0.431839	−0.215919	+	0.976411i	$0.569275\pi$
$992$	0	0
$993$	3996.00i	0.127703i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3234.00i	0.102730i	0.998680	+	0.0513650i	$0.0163572\pi$
$997$	3234.00i	0.102730i	−0.998680	+	0.0513650i	$0.983643\pi$
$998$	0	0
$999$	−918.000	−0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.b.49.1		2
4.3	odd	2		75.4.b.b.49.1		2
5.2	odd	4		240.4.a.e.1.1		1
5.3	odd	4		1200.4.a.t.1.1		1
5.4	even	2	inner	1200.4.f.b.49.2		2
12.11	even	2		225.4.b.e.199.2		2
15.2	even	4		720.4.a.n.1.1		1
20.3	even	4		75.4.a.b.1.1		1
20.7	even	4		15.4.a.a.1.1	✓	1
20.19	odd	2		75.4.b.b.49.2		2
40.27	even	4		960.4.a.b.1.1		1
40.37	odd	4		960.4.a.ba.1.1		1
60.23	odd	4		225.4.a.f.1.1		1
60.47	odd	4		45.4.a.c.1.1		1
60.59	even	2		225.4.b.e.199.1		2
140.27	odd	4		735.4.a.e.1.1		1
180.7	even	12		405.4.e.g.271.1		2
180.47	odd	12		405.4.e.i.271.1		2
180.67	even	12		405.4.e.g.136.1		2
180.167	odd	12		405.4.e.i.136.1		2
220.87	odd	4		1815.4.a.e.1.1		1
420.167	even	4		2205.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	20.7	even	4
45.4.a.c.1.1		1	60.47	odd	4
75.4.a.b.1.1		1	20.3	even	4
75.4.b.b.49.1		2	4.3	odd	2
75.4.b.b.49.2		2	20.19	odd	2
225.4.a.f.1.1		1	60.23	odd	4
225.4.b.e.199.1		2	60.59	even	2
225.4.b.e.199.2		2	12.11	even	2
240.4.a.e.1.1		1	5.2	odd	4
405.4.e.g.136.1		2	180.67	even	12
405.4.e.g.271.1		2	180.7	even	12
405.4.e.i.136.1		2	180.167	odd	12
405.4.e.i.271.1		2	180.47	odd	12
720.4.a.n.1.1		1	15.2	even	4
735.4.a.e.1.1		1	140.27	odd	4
960.4.a.b.1.1		1	40.27	even	4
960.4.a.ba.1.1		1	40.37	odd	4
1200.4.a.t.1.1		1	5.3	odd	4
1200.4.f.b.49.1		2	1.1	even	1	trivial
1200.4.f.b.49.2		2	5.4	even	2	inner
1815.4.a.e.1.1		1	220.87	odd	4
2205.4.a.l.1.1		1	420.167	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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} -52.0000 q^{11} +22.0000i q^{13} +14.0000i q^{17} -20.0000 q^{19} -72.0000 q^{21} +168.000i q^{23} +27.0000i q^{27} -230.000 q^{29} +288.000 q^{31} +156.000i q^{33} +34.0000i q^{37} +66.0000 q^{39} +122.000 q^{41} +188.000i q^{43} +256.000i q^{47} -233.000 q^{49} +42.0000 q^{51} -338.000i q^{53} +60.0000i q^{57} +100.000 q^{59} +742.000 q^{61} +216.000i q^{63} -84.0000i q^{67} +504.000 q^{69} +328.000 q^{71} -38.0000i q^{73} +1248.00i q^{77} -240.000 q^{79} +81.0000 q^{81} -1212.00i q^{83} +690.000i q^{87} -330.000 q^{89} +528.000 q^{91} -864.000i q^{93} -866.000i q^{97} +468.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 104 * q^11 - 40 * q^19 - 144 * q^21 - 460 * q^29 + 576 * q^31 + 132 * q^39 + 244 * q^41 - 466 * q^49 + 84 * q^51 + 200 * q^59 + 1484 * q^61 + 1008 * q^69 + 656 * q^71 - 480 * q^79 + 162 * q^81 - 660 * q^89 + 1056 * q^91 + 936 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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