LMFDB - Embedded newform 960.4.h.d.191.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(191,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.h (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:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.15
Character	$\chi$	$=$	960.191
Dual form			960.4.h.d.191.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.95519 - 4.81427i) q^{3} +5.00000i q^{5} +20.3642i q^{7} +(-19.3544 - 18.8257i) q^{9} +O(q^{10})$	\(q+(1.95519 - 4.81427i) q^{3} +5.00000i q^{5} +20.3642i q^{7} +(-19.3544 - 18.8257i) q^{9} +15.2982 q^{11} -27.7054 q^{13} +(24.0714 + 9.77596i) q^{15} +92.5031i q^{17} -127.926i q^{19} +(98.0390 + 39.8160i) q^{21} +51.1779 q^{23} -25.0000 q^{25} +(-128.474 + 56.3698i) q^{27} -99.2953i q^{29} +25.8075i q^{31} +(29.9110 - 73.6499i) q^{33} -101.821 q^{35} -356.629 q^{37} +(-54.1693 + 133.381i) q^{39} -292.895i q^{41} +521.476i q^{43} +(94.1283 - 96.7722i) q^{45} -573.546 q^{47} -71.7025 q^{49} +(445.335 + 180.861i) q^{51} -305.765i q^{53} +76.4912i q^{55} +(-615.871 - 250.120i) q^{57} -295.688 q^{59} -326.449 q^{61} +(383.370 - 394.139i) q^{63} -138.527i q^{65} -299.605i q^{67} +(100.063 - 246.384i) q^{69} -653.840 q^{71} +504.190 q^{73} +(-48.8798 + 120.357i) q^{75} +311.537i q^{77} -110.758i q^{79} +(20.1889 + 728.720i) q^{81} -1479.78 q^{83} -462.515 q^{85} +(-478.035 - 194.141i) q^{87} -772.934i q^{89} -564.199i q^{91} +(124.244 + 50.4585i) q^{93} +639.631 q^{95} -26.1154 q^{97} +(-296.089 - 288.000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 20 q^{9}+O(q^{10}) $ 24 * q + 20 * q^9	$ 24 q + 20 q^{9} - 72 q^{13} + 68 q^{21} - 600 q^{25} + 848 q^{33} - 504 q^{37} + 220 q^{45} - 2256 q^{49} + 1416 q^{57} - 1992 q^{61} + 1548 q^{69} - 2304 q^{73} + 3840 q^{81} - 240 q^{85} + 4384 q^{93} - 2448 q^{97}+O(q^{100}) $ 24 * q + 20 * q^9 - 72 * q^13 + 68 * q^21 - 600 * q^25 + 848 * q^33 - 504 * q^37 + 220 * q^45 - 2256 * q^49 + 1416 * q^57 - 1992 * q^61 + 1548 * q^69 - 2304 * q^73 + 3840 * q^81 - 240 * q^85 + 4384 * q^93 - 2448 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	1.95519	−	4.81427i	0.376277	−	0.926507i
$3$	1.95519	−	4.81427i	0.376277	−	0.926507i
$4$		0			0
$5$			5.00000i			0.447214i
$5$			5.00000i			0.447214i
$6$		0			0
$7$			20.3642i			1.09957i	0.835308	+	0.549783i	$0.185290\pi$
$7$			20.3642i			1.09957i	−0.835308	+	0.549783i	$0.814710\pi$
$8$		0			0
$9$	−19.3544	−	18.8257i	−0.716831	−	0.697247i
$10$		0			0
$11$	15.2982			0.419327			0.209663	−	0.977774i	$-0.432763\pi$
$11$	15.2982			0.419327			0.209663	+	0.977774i	$0.432763\pi$
$12$		0			0
$13$	−27.7054			−0.591084			−0.295542	−	0.955330i	$-0.595500\pi$
$13$	−27.7054			−0.591084			−0.295542	+	0.955330i	$0.595500\pi$
$14$		0			0
$15$	24.0714	+	9.77596i	0.414347	+	0.168276i
$16$		0			0
$17$			92.5031i			1.31972i	0.751387	+	0.659861i	$0.229385\pi$
$17$			92.5031i			1.31972i	−0.751387	+	0.659861i	$0.770615\pi$
$18$		0			0
$19$		−	127.926i		−	1.54465i	−0.635230	−	0.772323i	$-0.719095\pi$
$19$		−	127.926i		−	1.54465i	0.635230	−	0.772323i	$-0.280905\pi$
$20$		0			0
$21$	98.0390	+	39.8160i	1.01876	+	0.413741i
$22$		0			0
$23$	51.1779			0.463971			0.231985	−	0.972719i	$-0.425478\pi$
$23$	51.1779			0.463971			0.231985	+	0.972719i	$0.425478\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−128.474	+	56.3698i	−0.915731	+	0.401791i
$28$		0			0
$29$		−	99.2953i		−	0.635816i	−0.948121	−	0.317908i	$-0.897020\pi$
$29$		−	99.2953i		−	0.635816i	0.948121	−	0.317908i	$-0.102980\pi$
$30$		0			0
$31$			25.8075i			0.149521i	0.997202	+	0.0747606i	$0.0238193\pi$
$31$			25.8075i			0.149521i	−0.997202	+	0.0747606i	$0.976181\pi$
$32$		0			0
$33$	29.9110	−	73.6499i	0.157783	−	0.388509i
$34$		0			0
$35$	−101.821			−0.491741
$36$		0			0
$37$	−356.629			−1.58458			−0.792290	−	0.610144i	$-0.791112\pi$
$37$	−356.629			−1.58458			−0.792290	+	0.610144i	$0.791112\pi$
$38$		0			0
$39$	−54.1693	+	133.381i	−0.222411	+	0.547643i
$40$		0			0
$41$		−	292.895i		−	1.11567i	−0.829952	−	0.557835i	$-0.811632\pi$
$41$		−	292.895i		−	1.11567i	0.829952	−	0.557835i	$-0.188368\pi$
$42$		0			0
$43$			521.476i			1.84940i	0.380694	+	0.924701i	$0.375685\pi$
$43$			521.476i			1.84940i	−0.380694	+	0.924701i	$0.624315\pi$
$44$		0			0
$45$	94.1283	−	96.7722i	0.311818	−	0.320577i
$46$		0			0
$47$	−573.546			−1.78001			−0.890003	−	0.455955i	$-0.849298\pi$
$47$	−573.546			−1.78001			−0.890003	+	0.455955i	$0.849298\pi$
$48$		0			0
$49$	−71.7025			−0.209045
$50$		0			0
$51$	445.335	+	180.861i	1.22273	+	0.496581i
$52$		0			0
$53$		−	305.765i		−	0.792454i	−0.918153	−	0.396227i	$-0.870319\pi$
$53$		−	305.765i		−	0.792454i	0.918153	−	0.396227i	$-0.129681\pi$
$54$		0			0
$55$			76.4912i			0.187529i
$56$		0			0
$57$	−615.871	−	250.120i	−1.43113	−	0.581215i
$58$		0			0
$59$	−295.688			−0.652462			−0.326231	−	0.945290i	$-0.605779\pi$
$59$	−295.688			−0.652462			−0.326231	+	0.945290i	$0.605779\pi$
$60$		0			0
$61$	−326.449			−0.685204			−0.342602	−	0.939481i	$-0.611308\pi$
$61$	−326.449			−0.685204			−0.342602	+	0.939481i	$0.611308\pi$
$62$		0			0
$63$	383.370	−	394.139i	0.766669	−	0.788203i
$64$		0			0
$65$		−	138.527i		−	0.264341i
$66$		0			0
$67$		−	299.605i		−	0.546307i	−0.961971	−	0.273153i	$-0.911933\pi$
$67$		−	299.605i		−	0.546307i	0.961971	−	0.273153i	$-0.0880666\pi$
$68$		0			0
$69$	100.063	−	246.384i	0.174581	−	0.429872i
$70$		0			0
$71$	−653.840			−1.09291			−0.546454	−	0.837489i	$-0.684023\pi$
$71$	−653.840			−1.09291			−0.546454	+	0.837489i	$0.684023\pi$
$72$		0			0
$73$	504.190			0.808369			0.404184	−	0.914677i	$-0.367555\pi$
$73$	504.190			0.808369			0.404184	+	0.914677i	$0.367555\pi$
$74$		0			0
$75$	−48.8798	+	120.357i	−0.0752554	+	0.185301i
$76$		0			0
$77$			311.537i			0.461077i
$78$		0			0
$79$		−	110.758i		−	0.157738i	−0.996885	−	0.0788688i	$-0.974869\pi$
$79$		−	110.758i		−	0.157738i	0.996885	−	0.0788688i	$-0.0251308\pi$
$80$		0			0
$81$	20.1889	+	728.720i	0.0276939	+	0.999616i
$82$		0			0
$83$	−1479.78			−1.95695			−0.978474	−	0.206371i	$-0.933835\pi$
$83$	−1479.78			−1.95695			−0.978474	+	0.206371i	$0.933835\pi$
$84$		0			0
$85$	−462.515			−0.590198
$86$		0			0
$87$	−478.035	−	194.141i	−0.589089	−	0.239243i
$88$		0			0
$89$		−	772.934i		−	0.920572i	−0.887771	−	0.460286i	$-0.847747\pi$
$89$		−	772.934i		−	0.920572i	0.887771	−	0.460286i	$-0.152253\pi$
$90$		0			0
$91$		−	564.199i		−	0.649935i
$92$		0			0
$93$	124.244	+	50.4585i	0.138532	+	0.0562614i
$94$		0			0
$95$	639.631			0.690787
$96$		0			0
$97$	−26.1154			−0.0273362			−0.0136681	−	0.999907i	$-0.504351\pi$
$97$	−26.1154			−0.0273362			−0.0136681	+	0.999907i	$0.504351\pi$
$98$		0			0
$99$	−296.089	−	288.000i	−0.300586	−	0.292374i
$100$		0			0
$101$		−	196.046i		−	0.193141i	−0.995326	−	0.0965706i	$-0.969213\pi$
$101$		−	196.046i		−	0.193141i	0.995326	−	0.0965706i	$-0.0307874\pi$
$102$		0			0
$103$			685.393i			0.655668i	0.944735	+	0.327834i	$0.106319\pi$
$103$			685.393i			0.655668i	−0.944735	+	0.327834i	$0.893681\pi$
$104$		0			0
$105$	−199.080	+	490.195i	−0.185031	+	0.455601i
$106$		0			0
$107$	−503.806			−0.455185			−0.227593	−	0.973756i	$-0.573085\pi$
$107$	−503.806			−0.455185			−0.227593	+	0.973756i	$0.573085\pi$
$108$		0			0
$109$	928.609			0.816006			0.408003	−	0.912981i	$-0.366225\pi$
$109$	928.609			0.816006			0.408003	+	0.912981i	$0.366225\pi$
$110$		0			0
$111$	−697.279	+	1716.91i	−0.596241	+	1.46813i
$112$		0			0
$113$			451.244i			0.375659i	0.982202	+	0.187830i	$0.0601453\pi$
$113$			451.244i			0.375659i	−0.982202	+	0.187830i	$0.939855\pi$
$114$		0			0
$115$			255.889i			0.207494i
$116$		0			0
$117$	536.222	+	521.572i	0.423707	+	0.412131i
$118$		0			0
$119$	−1883.75			−1.45112
$120$		0			0
$121$	−1096.96			−0.824165
$122$		0			0
$123$	−1410.08	−	572.666i	−1.03368	−	0.419801i
$124$		0			0
$125$		−	125.000i		−	0.0894427i
$126$		0			0
$127$			1069.78i			0.747463i	0.927537	+	0.373732i	$0.121922\pi$
$127$			1069.78i			0.747463i	−0.927537	+	0.373732i	$0.878078\pi$
$128$		0			0
$129$	2510.53	+	1019.59i	1.71348	+	0.695888i
$130$		0			0
$131$	728.141			0.485633			0.242817	−	0.970072i	$-0.421929\pi$
$131$	728.141			0.485633			0.242817	+	0.970072i	$0.421929\pi$
$132$		0			0
$133$	2605.12			1.69844
$134$		0			0
$135$	−281.849	−	642.368i	−0.179687	−	0.409527i
$136$		0			0
$137$			1335.43i			0.832801i	0.909181	+	0.416400i	$0.136709\pi$
$137$			1335.43i			0.832801i	−0.909181	+	0.416400i	$0.863291\pi$
$138$		0			0
$139$			2006.84i			1.22459i	0.790629	+	0.612296i	$0.209754\pi$
$139$			2006.84i			1.22459i	−0.790629	+	0.612296i	$0.790246\pi$
$140$		0			0
$141$	−1121.39	+	2761.21i	−0.669775	+	1.64919i
$142$		0			0
$143$	−423.843			−0.247857
$144$		0			0
$145$	496.477			0.284346
$146$		0			0
$147$	−140.192	+	345.195i	−0.0786588	+	0.193682i
$148$		0			0
$149$			2021.25i			1.11132i	0.831409	+	0.555661i	$0.187535\pi$
$149$			2021.25i			1.11132i	−0.831409	+	0.555661i	$0.812465\pi$
$150$		0			0
$151$		−	264.606i		−	0.142605i	−0.997455	−	0.0713024i	$-0.977284\pi$
$151$		−	264.606i		−	0.142605i	0.997455	−	0.0713024i	$-0.0227155\pi$
$152$		0			0
$153$	1741.43	−	1790.35i	0.920172	−	0.946019i
$154$		0			0
$155$	−129.037			−0.0668679
$156$		0			0
$157$	−2339.73			−1.18937			−0.594684	−	0.803960i	$-0.702723\pi$
$157$	−2339.73			−1.18937			−0.594684	+	0.803960i	$0.702723\pi$
$158$		0			0
$159$	−1472.04	−	597.830i	−0.734215	−	0.298182i
$160$		0			0
$161$			1042.20i			0.510166i
$162$		0			0
$163$			1541.95i			0.740948i	0.928843	+	0.370474i	$0.120805\pi$
$163$			1541.95i			0.740948i	−0.928843	+	0.370474i	$0.879195\pi$
$164$		0			0
$165$	368.250	+	149.555i	0.173747	+	0.0705627i
$166$		0			0
$167$	−151.105			−0.0700173			−0.0350086	−	0.999387i	$-0.511146\pi$
$167$	−151.105			−0.0700173			−0.0350086	+	0.999387i	$0.511146\pi$
$168$		0			0
$169$	−1429.41			−0.650620
$170$		0			0
$171$	−2408.29	+	2475.94i	−1.07700	+	1.10725i
$172$		0			0
$173$		−	4436.11i		−	1.94954i	−0.223206	−	0.974771i	$-0.571652\pi$
$173$		−	4436.11i		−	1.94954i	0.223206	−	0.974771i	$-0.428348\pi$
$174$		0			0
$175$		−	509.106i		−	0.219913i
$176$		0			0
$177$	−578.127	+	1423.52i	−0.245507	+	0.604511i
$178$		0			0
$179$	−2609.23			−1.08951			−0.544757	−	0.838594i	$-0.683378\pi$
$179$	−2609.23			−1.08951			−0.544757	+	0.838594i	$0.683378\pi$
$180$		0			0
$181$	1528.41			0.627657			0.313828	−	0.949480i	$-0.398388\pi$
$181$	1528.41			0.627657			0.313828	+	0.949480i	$0.398388\pi$
$182$		0			0
$183$	−638.270	+	1571.61i	−0.257827	+	0.634847i
$184$		0			0
$185$		−	1783.15i		−	0.708646i
$186$		0			0
$187$			1415.13i			0.553395i
$188$		0			0
$189$	−1147.93	−	2616.27i	−0.441796	−	1.00691i
$190$		0			0
$191$	−832.060			−0.315213			−0.157607	−	0.987502i	$-0.550378\pi$
$191$	−832.060			−0.315213			−0.157607	+	0.987502i	$0.550378\pi$
$192$		0			0
$193$	659.353			0.245913			0.122957	−	0.992412i	$-0.460762\pi$
$193$	659.353			0.245913			0.122957	+	0.992412i	$0.460762\pi$
$194$		0			0
$195$	−666.906	−	270.847i	−0.244913	−	0.0994653i
$196$		0			0
$197$			963.336i			0.348400i	0.984710	+	0.174200i	$0.0557340\pi$
$197$			963.336i			0.348400i	−0.984710	+	0.174200i	$0.944266\pi$
$198$		0			0
$199$		−	870.488i		−	0.310087i	−0.987908	−	0.155043i	$-0.950448\pi$
$199$		−	870.488i		−	0.310087i	0.987908	−	0.155043i	$-0.0495517\pi$
$200$		0			0
$201$	−1442.38	−	585.785i	−0.506157	−	0.205563i
$202$		0			0
$203$	2022.07			0.699122
$204$		0			0
$205$	1464.47			0.498943
$206$		0			0
$207$	−990.519	−	963.457i	−0.332589	−	0.323502i
$208$		0			0
$209$		−	1957.05i		−	0.647711i
$210$		0			0
$211$		−	102.524i		−	0.0334503i	−0.999860	−	0.0167252i	$-0.994676\pi$
$211$		−	102.524i		−	0.0334503i	0.999860	−	0.0167252i	$-0.00532404\pi$
$212$		0			0
$213$	−1278.38	+	3147.76i	−0.411236	+	1.01259i
$214$		0			0
$215$	−2607.38			−0.827078
$216$		0			0
$217$	−525.549			−0.164408
$218$		0			0
$219$	985.788	−	2427.31i	0.304171	−	0.748960i
$220$		0			0
$221$		−	2562.83i		−	0.780067i
$222$		0			0
$223$			1477.75i			0.443756i	0.975074	+	0.221878i	$0.0712187\pi$
$223$			1477.75i			0.443756i	−0.975074	+	0.221878i	$0.928781\pi$
$224$		0			0
$225$	483.861	+	470.642i	0.143366	+	0.139449i
$226$		0			0
$227$	817.892			0.239143			0.119571	−	0.992826i	$-0.461848\pi$
$227$	817.892			0.239143			0.119571	+	0.992826i	$0.461848\pi$
$228$		0			0
$229$	−1154.61			−0.333182			−0.166591	−	0.986026i	$-0.553276\pi$
$229$	−1154.61			−0.333182			−0.166591	+	0.986026i	$0.553276\pi$
$230$		0			0
$231$	1499.82	+	609.115i	0.427191	+	0.173493i
$232$		0			0
$233$		−	3871.61i		−	1.08857i	−0.838899	−	0.544287i	$-0.816800\pi$
$233$		−	3871.61i		−	1.08857i	0.838899	−	0.544287i	$-0.183200\pi$
$234$		0			0
$235$		−	2867.73i		−	0.796043i
$236$		0			0
$237$	−533.220	−	216.554i	−0.146145	−	0.0593530i
$238$		0			0
$239$	4418.94			1.19597			0.597987	−	0.801506i	$-0.295968\pi$
$239$	4418.94			1.19597			0.597987	+	0.801506i	$0.295968\pi$
$240$		0			0
$241$	1310.51			0.350279			0.175139	−	0.984544i	$-0.443962\pi$
$241$	1310.51			0.350279			0.175139	+	0.984544i	$0.443962\pi$
$242$		0			0
$243$	3547.73	+	1327.59i	0.936572	+	0.350474i
$244$		0			0
$245$		−	358.512i		−	0.0934878i
$246$		0			0
$247$			3544.24i			0.913015i
$248$		0			0
$249$	−2893.25	+	7124.05i	−0.736354	+	1.81313i
$250$		0			0
$251$	−5925.09			−1.48999			−0.744997	−	0.667068i	$-0.767549\pi$
$251$	−5925.09			−1.48999			−0.744997	+	0.667068i	$0.767549\pi$
$252$		0			0
$253$	782.932			0.194555
$254$		0			0
$255$	−904.307	+	2226.67i	−0.222078	+	0.546823i
$256$		0			0
$257$		−	4401.57i		−	1.06834i	−0.845378	−	0.534168i	$-0.820625\pi$
$257$		−	4401.57i		−	1.06834i	0.845378	−	0.534168i	$-0.179375\pi$
$258$		0			0
$259$		−	7262.49i		−	1.74235i
$260$		0			0
$261$	−1869.30	+	1921.81i	−0.443321	+	0.455773i
$262$		0			0
$263$	−4384.53			−1.02799			−0.513996	−	0.857793i	$-0.671835\pi$
$263$	−4384.53			−1.02799			−0.513996	+	0.857793i	$0.671835\pi$
$264$		0			0
$265$	1528.83			0.354396
$266$		0			0
$267$	−3721.12	−	1511.24i	−0.852916	−	0.346390i
$268$		0			0
$269$			3387.40i			0.767783i	0.923378	+	0.383892i	$0.125416\pi$
$269$			3387.40i			0.767783i	−0.923378	+	0.383892i	$0.874584\pi$
$270$		0			0
$271$		−	4088.66i		−	0.916489i	−0.888826	−	0.458244i	$-0.848478\pi$
$271$		−	4088.66i		−	0.916489i	0.888826	−	0.458244i	$-0.151522\pi$
$272$		0			0
$273$	−2716.21	−	1103.12i	−0.602170	−	0.244556i
$274$		0			0
$275$	−382.456			−0.0838653
$276$		0			0
$277$	3485.20			0.755977			0.377988	−	0.925810i	$-0.376616\pi$
$277$	3485.20			0.755977			0.377988	+	0.925810i	$0.376616\pi$
$278$		0			0
$279$	485.842	−	499.489i	0.104253	−	0.107181i
$280$		0			0
$281$			4399.79i			0.934056i	0.884243	+	0.467028i	$0.154675\pi$
$281$			4399.79i			0.934056i	−0.884243	+	0.467028i	$0.845325\pi$
$282$		0			0
$283$			505.577i			0.106196i	0.998589	+	0.0530980i	$0.0169096\pi$
$283$			505.577i			0.106196i	−0.998589	+	0.0530980i	$0.983090\pi$
$284$		0			0
$285$	1250.60	−	3079.36i	0.259927	−	0.640019i
$286$		0			0
$287$	5964.58			1.22675
$288$		0			0
$289$	−3643.82			−0.741668
$290$		0			0
$291$	−51.0606	+	125.727i	−0.0102860	+	0.0253272i
$292$		0			0
$293$		−	6365.91i		−	1.26928i	−0.772806	−	0.634642i	$-0.781147\pi$
$293$		−	6365.91i		−	1.26928i	0.772806	−	0.634642i	$-0.218853\pi$
$294$		0			0
$295$		−	1478.44i		−	0.291790i
$296$		0			0
$297$	−1965.42	+	862.358i	−0.383991	+	0.168482i
$298$		0			0
$299$	−1417.90			−0.274245
$300$		0			0
$301$	−10619.5			−2.03354
$302$		0			0
$303$	−943.817	−	383.307i	−0.178947	−	0.0726746i
$304$		0			0
$305$		−	1632.24i		−	0.306433i
$306$		0			0
$307$			7389.54i			1.37376i	0.726772	+	0.686879i	$0.241020\pi$
$307$			7389.54i			1.37376i	−0.726772	+	0.686879i	$0.758980\pi$
$308$		0			0
$309$	3299.67	+	1340.08i	0.607481	+	0.246713i
$310$		0			0
$311$	1630.49			0.297288			0.148644	−	0.988891i	$-0.452509\pi$
$311$	1630.49			0.297288			0.148644	+	0.988891i	$0.452509\pi$
$312$		0			0
$313$	5922.15			1.06946			0.534728	−	0.845024i	$-0.320414\pi$
$313$	5922.15			1.06946			0.534728	+	0.845024i	$0.320414\pi$
$314$		0			0
$315$	1970.69	+	1916.85i	0.352495	+	0.342865i
$316$		0			0
$317$		−	7808.36i		−	1.38347i	−0.722149	−	0.691737i	$-0.756846\pi$
$317$		−	7808.36i		−	1.38347i	0.722149	−	0.691737i	$-0.243154\pi$
$318$		0			0
$319$		−	1519.04i		−	0.266615i
$320$		0			0
$321$	−985.039	+	2425.46i	−0.171276	+	0.421732i
$322$		0			0
$323$	11833.6			2.03850
$324$		0			0
$325$	692.634			0.118217
$326$		0			0
$327$	1815.61	−	4470.58i	0.307044	−	0.756035i
$328$		0			0
$329$		−	11679.8i		−	1.95723i
$330$		0			0
$331$			1569.05i			0.260552i	0.991478	+	0.130276i	$0.0415863\pi$
$331$			1569.05i			0.260552i	−0.991478	+	0.130276i	$0.958414\pi$
$332$		0			0
$333$	6902.36	+	6713.78i	1.13588	+	1.10484i
$334$		0			0
$335$	1498.02			0.244316
$336$		0			0
$337$	6488.72			1.04885			0.524426	−	0.851456i	$-0.324280\pi$
$337$	6488.72			1.04885			0.524426	+	0.851456i	$0.324280\pi$
$338$		0			0
$339$	2172.41	+	882.269i	0.348051	+	0.141352i
$340$		0			0
$341$			394.809i			0.0626982i
$342$		0			0
$343$			5524.77i			0.869707i
$344$		0			0
$345$	1231.92	+	500.313i	0.192245	+	0.0780752i
$346$		0			0
$347$	−595.599			−0.0921425			−0.0460712	−	0.998938i	$-0.514670\pi$
$347$	−595.599			−0.0921425			−0.0460712	+	0.998938i	$0.514670\pi$
$348$		0			0
$349$	−9358.18			−1.43534			−0.717668	−	0.696386i	$-0.754790\pi$
$349$	−9358.18			−1.43534			−0.717668	+	0.696386i	$0.754790\pi$
$350$		0			0
$351$	3559.41	−	1561.75i	0.541274	−	0.237492i
$352$		0			0
$353$			573.683i			0.0864988i	0.999064	+	0.0432494i	$0.0137710\pi$
$353$			573.683i			0.0864988i	−0.999064	+	0.0432494i	$0.986229\pi$
$354$		0			0
$355$		−	3269.20i		−	0.488763i
$356$		0			0
$357$	−3683.10	+	9068.91i	−0.546024	+	1.34448i
$358$		0			0
$359$	−8718.72			−1.28177			−0.640886	−	0.767636i	$-0.721433\pi$
$359$	−8718.72			−1.28177			−0.640886	+	0.767636i	$0.721433\pi$
$360$		0			0
$361$	−9506.11			−1.38593
$362$		0			0
$363$	−2144.78	+	5281.08i	−0.310114	+	0.763595i
$364$		0			0
$365$			2520.95i			0.361514i
$366$		0			0
$367$		−	3926.75i		−	0.558515i	−0.960216	−	0.279257i	$-0.909912\pi$
$367$		−	3926.75i		−	0.558515i	0.960216	−	0.279257i	$-0.0900882\pi$
$368$		0			0
$369$	−5513.94	+	5668.81i	−0.777897	+	0.799747i
$370$		0			0
$371$	6226.68			0.871356
$372$		0			0
$373$	−5278.89			−0.732789			−0.366395	−	0.930460i	$-0.619408\pi$
$373$	−5278.89			−0.732789			−0.366395	+	0.930460i	$0.619408\pi$
$374$		0			0
$375$	−601.784	−	244.399i	−0.0828693	−	0.0336552i
$376$		0			0
$377$			2751.01i			0.375821i
$378$		0			0
$379$			1153.98i			0.156401i	0.996938	+	0.0782003i	$0.0249174\pi$
$379$			1153.98i			0.156401i	−0.996938	+	0.0782003i	$0.975083\pi$
$380$		0			0
$381$	5150.23	+	2091.63i	0.692530	+	0.281253i
$382$		0			0
$383$	6431.26			0.858021			0.429010	−	0.903300i	$-0.358862\pi$
$383$	6431.26			0.858021			0.429010	+	0.903300i	$0.358862\pi$
$384$		0			0
$385$	−1557.69			−0.206200
$386$		0			0
$387$	9817.13	−	10092.9i	1.28949	−	1.32571i
$388$		0			0
$389$		−	9956.93i		−	1.29778i	−0.760882	−	0.648890i	$-0.775233\pi$
$389$		−	9956.93i		−	1.29778i	0.760882	−	0.648890i	$-0.224767\pi$
$390$		0			0
$391$			4734.11i			0.612313i
$392$		0			0
$393$	1423.66	−	3505.47i	0.182733	−	0.449943i
$394$		0			0
$395$	553.791			0.0705424
$396$		0			0
$397$	11143.6			1.40877			0.704386	−	0.709817i	$-0.251223\pi$
$397$	11143.6			1.40877			0.704386	+	0.709817i	$0.251223\pi$
$398$		0			0
$399$	5093.51	−	12541.8i	0.639084	−	1.57362i
$400$		0			0
$401$		−	4045.81i		−	0.503836i	−0.967749	−	0.251918i	$-0.918939\pi$
$401$		−	4045.81i		−	0.503836i	0.967749	−	0.251918i	$-0.0810613\pi$
$402$		0			0
$403$		−	715.005i		−	0.0883795i
$404$		0			0
$405$	−3643.60	+	100.944i	−0.447042	+	0.0123851i
$406$		0			0
$407$	−5455.80			−0.664457
$408$		0			0
$409$	12909.8			1.56076			0.780379	−	0.625307i	$-0.215026\pi$
$409$	12909.8			1.56076			0.780379	+	0.625307i	$0.215026\pi$
$410$		0			0
$411$	6429.14	+	2611.03i	0.771596	+	0.313364i
$412$		0			0
$413$		−	6021.46i		−	0.717425i
$414$		0			0
$415$		−	7398.89i		−	0.875174i
$416$		0			0
$417$	9661.49	+	3923.76i	1.13459	+	0.460786i
$418$		0			0
$419$	425.462			0.0496067			0.0248033	−	0.999692i	$-0.492104\pi$
$419$	425.462			0.0496067			0.0248033	+	0.999692i	$0.492104\pi$
$420$		0			0
$421$	6347.87			0.734860			0.367430	−	0.930051i	$-0.380238\pi$
$421$	6347.87			0.734860			0.367430	+	0.930051i	$0.380238\pi$
$422$		0			0
$423$	11100.7	+	10797.4i	1.27596	+	1.24110i
$424$		0			0
$425$		−	2312.58i		−	0.263945i
$426$		0			0
$427$		−	6647.88i		−	0.753427i
$428$		0			0
$429$	−828.696	+	2040.50i	−0.0932629	+	0.229641i
$430$		0			0
$431$	14572.3			1.62859			0.814295	−	0.580452i	$-0.197124\pi$
$431$	14572.3			1.62859			0.814295	+	0.580452i	$0.197124\pi$
$432$		0			0
$433$	−8416.25			−0.934086			−0.467043	−	0.884235i	$-0.654681\pi$
$433$	−8416.25			−0.934086			−0.467043	+	0.884235i	$0.654681\pi$
$434$		0			0
$435$	970.707	−	2390.17i	0.106993	−	0.263448i
$436$		0			0
$437$		−	6546.99i		−	0.716670i
$438$		0			0
$439$		−	10822.6i		−	1.17661i	−0.808638	−	0.588306i	$-0.799795\pi$
$439$		−	10822.6i		−	1.17661i	0.808638	−	0.588306i	$-0.200205\pi$
$440$		0			0
$441$	1387.76	+	1349.85i	0.149850	+	0.145756i
$442$		0			0
$443$	−11962.9			−1.28301			−0.641507	−	0.767117i	$-0.721691\pi$
$443$	−11962.9			−1.28301			−0.641507	+	0.767117i	$0.721691\pi$
$444$		0			0
$445$	3864.67			0.411692
$446$		0			0
$447$	9730.84	+	3951.93i	1.02965	+	0.418165i
$448$		0			0
$449$			3582.84i			0.376581i	0.982113	+	0.188291i	$0.0602947\pi$
$449$			3582.84i			0.376581i	−0.982113	+	0.188291i	$0.939705\pi$
$450$		0			0
$451$		−	4480.77i		−	0.467830i
$452$		0			0
$453$	−1273.89	−	517.356i	−0.132124	−	0.0536589i
$454$		0			0
$455$	2820.99			0.290660
$456$		0			0
$457$	−9998.12			−1.02340			−0.511699	−	0.859165i	$-0.670984\pi$
$457$	−9998.12			−1.02340			−0.511699	+	0.859165i	$0.670984\pi$
$458$		0			0
$459$	−5214.38	−	11884.2i	−0.530253	−	1.20851i
$460$		0			0
$461$			9035.94i			0.912897i	0.889750	+	0.456449i	$0.150879\pi$
$461$			9035.94i			0.912897i	−0.889750	+	0.456449i	$0.849121\pi$
$462$		0			0
$463$			2608.87i			0.261867i	0.991391	+	0.130933i	$0.0417974\pi$
$463$			2608.87i			0.261867i	−0.991391	+	0.130933i	$0.958203\pi$
$464$		0			0
$465$	−252.293	+	621.221i	−0.0251608	+	0.0619536i
$466$		0			0
$467$	9712.48			0.962398			0.481199	−	0.876611i	$-0.340201\pi$
$467$	9712.48			0.962398			0.481199	+	0.876611i	$0.340201\pi$
$468$		0			0
$469$	6101.22			0.600700
$470$		0			0
$471$	−4574.62	+	11264.1i	−0.447531	+	1.10196i
$472$		0			0
$473$			7977.66i			0.775504i
$474$		0			0
$475$			3198.15i			0.308929i
$476$		0			0
$477$	−5756.23	+	5917.91i	−0.552536	+	0.568056i
$478$		0			0
$479$	3430.18			0.327200			0.163600	−	0.986527i	$-0.447689\pi$
$479$	3430.18			0.327200			0.163600	+	0.986527i	$0.447689\pi$
$480$		0			0
$481$	9880.55			0.936620
$482$		0			0
$483$	5017.43	+	2037.70i	0.472673	+	0.191964i
$484$		0			0
$485$		−	130.577i		−	0.0122251i
$486$		0			0
$487$			7275.39i			0.676960i	0.940974	+	0.338480i	$0.109913\pi$
$487$			7275.39i			0.676960i	−0.940974	+	0.338480i	$0.890087\pi$
$488$		0			0
$489$	7423.35	+	3014.80i	0.686494	+	0.278802i
$490$		0			0
$491$	17601.4			1.61780			0.808902	−	0.587944i	$-0.200062\pi$
$491$	17601.4			1.61780			0.808902	+	0.587944i	$0.200062\pi$
$492$		0			0
$493$	9185.12			0.839102
$494$		0			0
$495$	1440.00	−	1480.44i	0.130754	−	0.134426i
$496$		0			0
$497$		−	13314.9i		−	1.20172i
$498$		0			0
$499$		−	17324.8i		−	1.55424i	−0.629355	−	0.777118i	$-0.716681\pi$
$499$		−	17324.8i		−	1.55424i	0.629355	−	0.777118i	$-0.283319\pi$
$500$		0			0
$501$	−295.440	+	727.462i	−0.0263459	+	0.0648715i
$502$		0			0
$503$	17802.1			1.57805			0.789023	−	0.614364i	$-0.210587\pi$
$503$	17802.1			1.57805			0.789023	+	0.614364i	$0.210587\pi$
$504$		0			0
$505$	980.228			0.0863754
$506$		0			0
$507$	−2794.78	+	6881.58i	−0.244813	+	0.602804i
$508$		0			0
$509$			14138.0i			1.23115i	0.788078	+	0.615576i	$0.211076\pi$
$509$			14138.0i			1.23115i	−0.788078	+	0.615576i	$0.788924\pi$
$510$		0			0
$511$			10267.4i			0.888855i
$512$		0			0
$513$	7211.17	+	16435.1i	0.620625	+	1.41448i
$514$		0			0
$515$	−3426.97			−0.293224
$516$		0			0
$517$	−8774.25			−0.746404
$518$		0			0
$519$	−21356.6	−	8673.44i	−1.80627	−	0.733568i
$520$		0			0
$521$		−	3032.05i		−	0.254965i	−0.991841	−	0.127482i	$-0.959310\pi$
$521$		−	3032.05i		−	0.254965i	0.991841	−	0.127482i	$-0.0406896\pi$
$522$		0			0
$523$			12845.0i			1.07394i	0.843601	+	0.536971i	$0.180432\pi$
$523$			12845.0i			1.07394i	−0.843601	+	0.536971i	$0.819568\pi$
$524$		0			0
$525$	−2450.98	−	995.401i	−0.203751	−	0.0827483i
$526$		0			0
$527$	−2387.27			−0.197326
$528$		0			0
$529$	−9547.82			−0.784731
$530$		0			0
$531$	5722.87	+	5566.52i	0.467705	+	0.454927i
$532$		0			0
$533$			8114.76i			0.659454i
$534$		0			0
$535$		−	2519.03i		−	0.203565i
$536$		0			0
$537$	−5101.55	+	12561.5i	−0.409959	+	1.00944i
$538$		0			0
$539$	−1096.92			−0.0876582
$540$		0			0
$541$	11333.2			0.900654			0.450327	−	0.892864i	$-0.351307\pi$
$541$	11333.2			0.900654			0.450327	+	0.892864i	$0.351307\pi$
$542$		0			0
$543$	2988.34	−	7358.18i	0.236173	−	0.581528i
$544$		0			0
$545$			4643.05i			0.364929i
$546$		0			0
$547$			7995.82i			0.625003i	0.949917	+	0.312501i	$0.101167\pi$
$547$			7995.82i			0.625003i	−0.949917	+	0.312501i	$0.898833\pi$
$548$		0			0
$549$	6318.23	+	6145.61i	0.491176	+	0.477757i
$550$		0			0
$551$	−12702.5			−0.982111
$552$		0			0
$553$	2255.51			0.173443
$554$		0			0
$555$	−8584.55	−	3486.40i	−0.656566	−	0.266647i
$556$		0			0
$557$			6218.87i			0.473074i	0.971623	+	0.236537i	$0.0760124\pi$
$557$			6218.87i			0.473074i	−0.971623	+	0.236537i	$0.923988\pi$
$558$		0			0
$559$		−	14447.7i		−	1.09315i
$560$		0			0
$561$	6812.84	+	2766.86i	0.512724	+	0.208230i
$562$		0			0
$563$	7704.32			0.576729			0.288364	−	0.957521i	$-0.406889\pi$
$563$	7704.32			0.576729			0.288364	+	0.957521i	$0.406889\pi$
$564$		0			0
$565$	−2256.22			−0.168000
$566$		0			0
$567$	−14839.8	+	411.131i	−1.09914	+	0.0304513i
$568$		0			0
$569$		−	4897.62i		−	0.360841i	−0.983590	−	0.180421i	$-0.942254\pi$
$569$		−	4897.62i		−	0.360841i	0.983590	−	0.180421i	$-0.0577459\pi$
$570$		0			0
$571$			7425.55i			0.544220i	0.962266	+	0.272110i	$0.0877215\pi$
$571$			7425.55i			0.544220i	−0.962266	+	0.272110i	$0.912279\pi$
$572$		0			0
$573$	−1626.84	+	4005.76i	−0.118608	+	0.292047i
$574$		0			0
$575$	−1279.45			−0.0927941
$576$		0			0
$577$	−13212.1			−0.953254			−0.476627	−	0.879106i	$-0.658141\pi$
$577$	−13212.1			−0.953254			−0.476627	+	0.879106i	$0.658141\pi$
$578$		0			0
$579$	1289.16	−	3174.30i	0.0925315	−	0.227840i
$580$		0			0
$581$		−	30134.5i		−	2.15179i
$582$		0			0
$583$		−	4677.67i		−	0.332297i
$584$		0			0
$585$	−2607.86	+	2681.11i	−0.184311	+	0.189488i
$586$		0			0
$587$	−7041.44			−0.495113			−0.247557	−	0.968873i	$-0.579628\pi$
$587$	−7041.44			−0.495113			−0.247557	+	0.968873i	$0.579628\pi$
$588$		0			0
$589$	3301.45			0.230957
$590$		0			0
$591$	4637.76	+	1883.51i	0.322795	+	0.131095i
$592$		0			0
$593$		−	18675.4i		−	1.29326i	−0.762802	−	0.646632i	$-0.776177\pi$
$593$		−	18675.4i		−	1.29326i	0.762802	−	0.646632i	$-0.223823\pi$
$594$		0			0
$595$		−	9418.77i		−	0.648962i
$596$		0			0
$597$	−4190.77	−	1701.97i	−0.287298	−	0.116679i
$598$		0			0
$599$	8073.52			0.550709			0.275355	−	0.961343i	$-0.411205\pi$
$599$	8073.52			0.550709			0.275355	+	0.961343i	$0.411205\pi$
$600$		0			0
$601$	8290.31			0.562677			0.281338	−	0.959609i	$-0.409222\pi$
$601$	8290.31			0.562677			0.281338	+	0.959609i	$0.409222\pi$
$602$		0			0
$603$	−5640.26	+	5798.68i	−0.380910	+	0.391610i
$604$		0			0
$605$		−	5484.82i		−	0.368578i
$606$		0			0
$607$		−	15779.1i		−	1.05511i	−0.849521	−	0.527556i	$-0.823109\pi$
$607$		−	15779.1i		−	1.05511i	0.849521	−	0.527556i	$-0.176891\pi$
$608$		0			0
$609$	3953.54	−	9734.82i	0.263064	−	0.647742i
$610$		0			0
$611$	15890.3			1.05213
$612$		0			0
$613$	−27380.0			−1.80403			−0.902014	−	0.431708i	$-0.857911\pi$
$613$	−27380.0			−1.80403			−0.902014	+	0.431708i	$0.857911\pi$
$614$		0			0
$615$	2863.33	−	7050.38i	0.187741	−	0.462274i
$616$		0			0
$617$			212.657i			0.0138756i	0.999976	+	0.00693782i	$0.00220839\pi$
$617$			212.657i			0.0138756i	−0.999976	+	0.00693782i	$0.997792\pi$
$618$		0			0
$619$		−	19326.6i		−	1.25493i	−0.778645	−	0.627465i	$-0.784092\pi$
$619$		−	19326.6i		−	1.25493i	0.778645	−	0.627465i	$-0.215908\pi$
$620$		0			0
$621$	−6575.00	+	2884.88i	−0.424872	+	0.186419i
$622$		0			0
$623$	15740.2			1.01223
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−9421.75	−	3826.40i	−0.600109	−	0.243719i
$628$		0			0
$629$		−	32989.3i		−	2.09121i
$630$		0			0
$631$		−	13797.7i		−	0.870486i	−0.900313	−	0.435243i	$-0.856662\pi$
$631$		−	13797.7i		−	0.870486i	0.900313	−	0.435243i	$-0.143338\pi$
$632$		0			0
$633$	−493.577	−	200.454i	−0.0309920	−	0.0125866i
$634$		0			0
$635$	−5348.91			−0.334276
$636$		0			0
$637$	1986.54			0.123563
$638$		0			0
$639$	12654.7	+	12309.0i	0.783431	+	0.762027i
$640$		0			0
$641$		−	25984.0i		−	1.60110i	−0.599264	−	0.800552i	$-0.704540\pi$
$641$		−	25984.0i		−	1.60110i	0.599264	−	0.800552i	$-0.295460\pi$
$642$		0			0
$643$			29221.5i			1.79220i	0.443853	+	0.896099i	$0.353611\pi$
$643$			29221.5i			1.79220i	−0.443853	+	0.896099i	$0.646389\pi$
$644$		0			0
$645$	−5097.93	+	12552.6i	−0.311210	+	0.766294i
$646$		0			0
$647$	3210.39			0.195075			0.0975375	−	0.995232i	$-0.468903\pi$
$647$	3210.39			0.195075			0.0975375	+	0.995232i	$0.468903\pi$
$648$		0			0
$649$	−4523.50			−0.273595
$650$		0			0
$651$	−1027.55	+	2530.14i	−0.0618631	+	0.152326i
$652$		0			0
$653$		−	1248.39i		−	0.0748136i	−0.999300	−	0.0374068i	$-0.988090\pi$
$653$		−	1248.39i		−	0.0748136i	0.999300	−	0.0374068i	$-0.0119097\pi$
$654$		0			0
$655$			3640.70i			0.217182i
$656$		0			0
$657$	−9758.31	−	9491.70i	−0.579464	−	0.563633i
$658$		0			0
$659$	9138.61			0.540197			0.270098	−	0.962833i	$-0.412944\pi$
$659$	9138.61			0.540197			0.270098	+	0.962833i	$0.412944\pi$
$660$		0			0
$661$	−4756.08			−0.279864			−0.139932	−	0.990161i	$-0.544688\pi$
$661$	−4756.08			−0.279864			−0.139932	+	0.990161i	$0.544688\pi$
$662$		0			0
$663$	−12338.2	−	5010.83i	−0.722737	−	0.293521i
$664$		0			0
$665$			13025.6i			0.759566i
$666$		0			0
$667$		−	5081.72i		−	0.295000i
$668$		0			0
$669$	7114.30	+	2889.29i	0.411143	+	0.166975i
$670$		0			0
$671$	−4994.09			−0.287324
$672$		0			0
$673$	−23520.9			−1.34720			−0.673598	−	0.739098i	$-0.735252\pi$
$673$	−23520.9			−1.34720			−0.673598	+	0.739098i	$0.735252\pi$
$674$		0			0
$675$	3211.84	−	1409.24i	0.183146	−	0.0803583i
$676$		0			0
$677$			3924.98i			0.222820i	0.993775	+	0.111410i	$0.0355367\pi$
$677$			3924.98i			0.222820i	−0.993775	+	0.111410i	$0.964463\pi$
$678$		0			0
$679$		−	531.820i		−	0.0300580i
$680$		0			0
$681$	1599.14	−	3937.56i	0.0899840	−	0.221568i
$682$		0			0
$683$	2097.25			0.117495			0.0587473	−	0.998273i	$-0.481289\pi$
$683$	2097.25			0.117495			0.0587473	+	0.998273i	$0.481289\pi$
$684$		0			0
$685$	−6677.16			−0.372440
$686$		0			0
$687$	−2257.48	+	5558.60i	−0.125369	+	0.308695i
$688$		0			0
$689$			8471.34i			0.468407i
$690$		0			0
$691$		−	595.762i		−	0.0327987i	−0.999866	−	0.0163993i	$-0.994780\pi$
$691$		−	595.762i		−	0.0327987i	0.999866	−	0.0163993i	$-0.00522030\pi$
$692$		0			0
$693$	5864.89	−	6029.63i	0.321485	−	0.330515i
$694$		0			0
$695$	−10034.2			−0.547654
$696$		0			0
$697$	27093.7			1.47238
$698$		0			0
$699$	−18639.0	−	7569.75i	−1.00857	−	0.409606i
$700$		0			0
$701$		−	9077.51i		−	0.489091i	−0.969638	−	0.244545i	$-0.921361\pi$
$701$		−	9077.51i		−	0.489091i	0.969638	−	0.244545i	$-0.0786387\pi$
$702$		0			0
$703$			45622.2i			2.44762i
$704$		0			0
$705$	−13806.0	−	5606.96i	−0.737539	−	0.299533i
$706$		0			0
$707$	3992.32			0.212371
$708$		0			0
$709$	8539.10			0.452317			0.226158	−	0.974091i	$-0.427383\pi$
$709$	8539.10			0.452317			0.226158	+	0.974091i	$0.427383\pi$
$710$		0			0
$711$	−2085.10	+	2143.66i	−0.109982	+	0.113071i
$712$		0			0
$713$			1320.77i			0.0693734i
$714$		0			0
$715$		−	2119.22i		−	0.110845i
$716$		0			0
$717$	8639.89	−	21274.0i	0.450017	−	1.10808i
$718$		0			0
$719$	−17913.1			−0.929130			−0.464565	−	0.885539i	$-0.653789\pi$
$719$	−17913.1			−0.929130			−0.464565	+	0.885539i	$0.653789\pi$
$720$		0			0
$721$	−13957.5			−0.720950
$722$		0			0
$723$	2562.29	−	6309.13i	0.131802	−	0.324536i
$724$		0			0
$725$			2482.38i			0.127163i
$726$		0			0
$727$			32276.8i			1.64660i	0.567605	+	0.823301i	$0.307870\pi$
$727$			32276.8i			1.64660i	−0.567605	+	0.823301i	$0.692130\pi$
$728$		0			0
$729$	13327.9	−	14484.0i	0.677127	−	0.735866i
$730$		0			0
$731$	−48238.1			−2.44070
$732$		0			0
$733$	−12015.9			−0.605481			−0.302740	−	0.953073i	$-0.597901\pi$
$733$	−12015.9			−0.605481			−0.302740	+	0.953073i	$0.597901\pi$
$734$		0			0
$735$	−1725.98	−	700.961i	−0.0866171	−	0.0351773i
$736$		0			0
$737$		−	4583.42i		−	0.229081i
$738$		0			0
$739$		−	18920.3i		−	0.941804i	−0.882185	−	0.470902i	$-0.843928\pi$
$739$		−	18920.3i		−	0.941804i	0.882185	−	0.470902i	$-0.156072\pi$
$740$		0			0
$741$	17062.9	+	6929.68i	0.845915	+	0.343547i
$742$		0			0
$743$	−32431.7			−1.60135			−0.800676	−	0.599097i	$-0.795526\pi$
$743$	−32431.7			−1.60135			−0.800676	+	0.599097i	$0.795526\pi$
$744$		0			0
$745$	−10106.2			−0.496999
$746$		0			0
$747$	28640.3	+	27857.8i	1.40280	+	1.36448i
$748$		0			0
$749$		−	10259.6i		−	0.500506i
$750$		0			0
$751$			23544.8i			1.14402i	0.820245	+	0.572012i	$0.193837\pi$
$751$			23544.8i			1.14402i	−0.820245	+	0.572012i	$0.806163\pi$
$752$		0			0
$753$	−11584.7	+	28525.0i	−0.560650	+	1.38049i
$754$		0			0
$755$	1323.03			0.0637748
$756$		0			0
$757$	−11963.2			−0.574388			−0.287194	−	0.957872i	$-0.592722\pi$
$757$	−11963.2			−0.574388			−0.287194	+	0.957872i	$0.592722\pi$
$758$		0			0
$759$	1530.78	−	3769.25i	0.0732067	−	0.180257i
$760$		0			0
$761$		−	2168.96i		−	0.103318i	−0.998665	−	0.0516588i	$-0.983549\pi$
$761$		−	2168.96i		−	0.103318i	0.998665	−	0.0516588i	$-0.0164508\pi$
$762$		0			0
$763$			18910.4i			0.897252i
$764$		0			0
$765$	8951.73	+	8707.16i	0.423072	+	0.411514i
$766$		0			0
$767$	8192.14			0.385660
$768$		0			0
$769$	6990.21			0.327794			0.163897	−	0.986477i	$-0.447594\pi$
$769$	6990.21			0.327794			0.163897	+	0.986477i	$0.447594\pi$
$770$		0			0
$771$	−21190.4	−	8605.92i	−0.989821	−	0.401990i
$772$		0			0
$773$			15944.4i			0.741887i	0.928655	+	0.370944i	$0.120966\pi$
$773$			15944.4i			0.741887i	−0.928655	+	0.370944i	$0.879034\pi$
$774$		0			0
$775$		−	645.186i		−	0.0299042i
$776$		0			0
$777$	−34963.6	−	14199.6i	−1.61430	−	0.655607i
$778$		0			0
$779$	−37468.9			−1.72332
$780$		0			0
$781$	−10002.6			−0.458286
$782$		0			0
$783$	5597.25	+	12756.8i	0.255466	+	0.582237i
$784$		0			0
$785$		−	11698.6i		−	0.531901i
$786$		0			0
$787$		−	13904.9i		−	0.629803i	−0.949124	−	0.314901i	$-0.898029\pi$
$787$		−	13904.9i		−	0.629803i	0.949124	−	0.314901i	$-0.101971\pi$
$788$		0			0
$789$	−8572.61	+	21108.3i	−0.386810	+	0.952442i
$790$		0			0
$791$	−9189.25			−0.413062
$792$		0			0
$793$	9044.38			0.405013
$794$		0			0
$795$	2989.15	−	7360.18i	0.133351	−	0.328351i
$796$		0			0
$797$		−	6202.60i		−	0.275668i	−0.990455	−	0.137834i	$-0.955986\pi$
$797$		−	6202.60i		−	0.275668i	0.990455	−	0.137834i	$-0.0440140\pi$
$798$		0			0
$799$		−	53054.8i		−	2.34911i
$800$		0			0
$801$	−14551.0	+	14959.7i	−0.641866	+	0.659894i
$802$		0			0
$803$	7713.22			0.338971
$804$		0			0
$805$	−5210.99			−0.228153
$806$		0			0
$807$	16307.9	+	6623.03i	0.711357	+	0.288899i
$808$		0			0
$809$			16847.6i			0.732175i	0.930580	+	0.366088i	$0.119303\pi$
$809$			16847.6i			0.732175i	−0.930580	+	0.366088i	$0.880697\pi$
$810$		0			0
$811$			33782.0i			1.46270i	0.682004	+	0.731348i	$0.261109\pi$
$811$			33782.0i			1.46270i	−0.682004	+	0.731348i	$0.738891\pi$
$812$		0			0
$813$	−19683.9	−	7994.12i	−0.849133	−	0.344854i
$814$		0			0
$815$	−7709.73			−0.331362
$816$		0			0
$817$	66710.4			2.85667
$818$		0			0
$819$	−10621.4	+	10919.8i	−0.453165	+	0.465894i
$820$		0			0
$821$			26588.7i			1.13027i	0.824998	+	0.565135i	$0.191176\pi$
$821$			26588.7i			1.13027i	−0.824998	+	0.565135i	$0.808824\pi$
$822$		0			0
$823$			33619.7i			1.42395i	0.702205	+	0.711975i	$0.252199\pi$
$823$			33619.7i			1.42395i	−0.702205	+	0.711975i	$0.747801\pi$
$824$		0			0
$825$	−747.775	+	1841.25i	−0.0315566	+	0.0777018i
$826$		0			0
$827$	−27338.2			−1.14951			−0.574753	−	0.818327i	$-0.694902\pi$
$827$	−27338.2			−1.14951			−0.574753	+	0.818327i	$0.694902\pi$
$828$		0			0
$829$	−18514.6			−0.775680			−0.387840	−	0.921727i	$-0.626779\pi$
$829$	−18514.6			−0.775680			−0.387840	+	0.921727i	$0.626779\pi$
$830$		0			0
$831$	6814.24	−	16778.7i	0.284457	−	0.700418i
$832$		0			0
$833$		−	6632.70i		−	0.275882i
$834$		0			0
$835$		−	755.527i		−	0.0313127i
$836$		0			0
$837$	−1454.76	−	3315.57i	−0.0600763	−	0.136921i
$838$		0			0
$839$	−42265.2			−1.73916			−0.869580	−	0.493792i	$-0.835610\pi$
$839$	−42265.2			−1.73916			−0.869580	+	0.493792i	$0.835610\pi$
$840$		0			0
$841$	14529.4			0.595737
$842$		0			0
$843$	21181.8	+	8602.44i	0.865409	+	0.351464i
$844$		0			0
$845$		−	7147.06i		−	0.290966i
$846$		0			0
$847$		−	22338.8i		−	0.906224i
$848$		0			0
$849$	2433.99	+	988.501i	0.0983913	+	0.0399591i
$850$		0			0
$851$	−18251.5			−0.735199
$852$		0			0
$853$	39949.7			1.60358			0.801789	−	0.597607i	$-0.203882\pi$
$853$	39949.7			1.60358			0.801789	+	0.597607i	$0.203882\pi$
$854$		0			0
$855$	−12379.7	−	12041.5i	−0.495177	−	0.481649i
$856$		0			0
$857$			35527.4i			1.41609i	0.706165	+	0.708047i	$0.250424\pi$
$857$			35527.4i			1.41609i	−0.706165	+	0.708047i	$0.749576\pi$
$858$		0			0
$859$			15149.2i			0.601728i	0.953667	+	0.300864i	$0.0972750\pi$
$859$			15149.2i			0.601728i	−0.953667	+	0.300864i	$0.902725\pi$
$860$		0			0
$861$	11661.9	−	28715.1i	0.461599	−	1.13660i
$862$		0			0
$863$	3598.72			0.141949			0.0709744	−	0.997478i	$-0.477389\pi$
$863$	3598.72			0.141949			0.0709744	+	0.997478i	$0.477389\pi$
$864$		0			0
$865$	22180.5			0.871862
$866$		0			0
$867$	−7124.36	+	17542.3i	−0.279073	+	0.687161i
$868$		0			0
$869$		−	1694.40i		−	0.0661436i
$870$		0			0
$871$			8300.66i			0.322913i
$872$		0			0
$873$	505.449	+	491.640i	0.0195955	+	0.0190601i
$874$		0			0
$875$	2545.53			0.0983482
$876$		0			0
$877$	47342.6			1.82286			0.911428	−	0.411459i	$-0.134981\pi$
$877$	47342.6			1.82286			0.911428	+	0.411459i	$0.134981\pi$
$878$		0			0
$879$	−30647.2	−	12446.6i	−1.17600	−	0.477603i
$880$		0			0
$881$		−	5694.16i		−	0.217754i	−0.994055	−	0.108877i	$-0.965275\pi$
$881$		−	5694.16i		−	0.217754i	0.994055	−	0.108877i	$-0.0347254\pi$
$882$		0			0
$883$		−	1161.20i		−	0.0442555i	−0.999755	−	0.0221278i	$-0.992956\pi$
$883$		−	1161.20i		−	0.0442555i	0.999755	−	0.0221278i	$-0.00704406\pi$
$884$		0			0
$885$	−7117.61	−	2890.63i	−0.270346	−	0.109794i
$886$		0			0
$887$	1024.65			0.0387874			0.0193937	−	0.999812i	$-0.493826\pi$
$887$	1024.65			0.0387874			0.0193937	+	0.999812i	$0.493826\pi$
$888$		0			0
$889$	−21785.3			−0.821885
$890$		0			0
$891$	308.854	+	11148.1i	0.0116128	+	0.419166i
$892$		0			0
$893$			73371.5i			2.74948i
$894$		0			0
$895$		−	13046.1i		−	0.487245i
$896$		0			0
$897$	−2772.27	+	6826.17i	−0.103192	+	0.254090i
$898$		0			0
$899$	2562.56			0.0950680
$900$		0			0
$901$	28284.2			1.04582
$902$		0			0
$903$	−20763.1	+	51125.0i	−0.765174	+	1.88409i
$904$		0			0
$905$			7642.05i			0.280697i
$906$		0			0
$907$			6399.77i			0.234290i	0.993115	+	0.117145i	$0.0373742\pi$
$907$			6399.77i			0.234290i	−0.993115	+	0.117145i	$0.962626\pi$
$908$		0			0
$909$	−3690.69	+	3794.35i	−0.134667	+	0.138450i
$910$		0			0
$911$	36922.8			1.34282			0.671410	−	0.741086i	$-0.265689\pi$
$911$	36922.8			1.34282			0.671410	+	0.741086i	$0.265689\pi$
$912$		0			0
$913$	−22638.0			−0.820600
$914$		0			0
$915$	−7858.06	−	3191.35i	−0.283912	−	0.115304i
$916$		0			0
$917$			14828.0i			0.533986i
$918$		0			0
$919$		−	3415.26i		−	0.122589i	−0.998120	−	0.0612944i	$-0.980477\pi$
$919$		−	3415.26i		−	0.122589i	0.998120	−	0.0612944i	$-0.0195229\pi$
$920$		0			0
$921$	35575.3	+	14448.0i	1.27280	+	0.516913i
$922$		0			0
$923$	18114.9			0.646000
$924$		0			0
$925$	8915.73			0.316916
$926$		0			0
$927$	12903.0	−	13265.4i	0.457162	−	0.470003i
$928$		0			0
$929$			25200.8i			0.890001i	0.895530	+	0.445001i	$0.146797\pi$
$929$			25200.8i			0.890001i	−0.895530	+	0.445001i	$0.853203\pi$
$930$		0			0
$931$			9172.62i			0.322901i
$932$		0			0
$933$	3187.92	−	7849.62i	0.111863	−	0.275440i
$934$		0			0
$935$	−7075.67			−0.247486
$936$		0			0
$937$	−17484.4			−0.609595			−0.304797	−	0.952417i	$-0.598589\pi$
$937$	−17484.4			−0.609595			−0.304797	+	0.952417i	$0.598589\pi$
$938$		0			0
$939$	11578.9	−	28510.8i	0.402411	−	0.990858i
$940$		0			0
$941$		−	10865.5i		−	0.376412i	−0.982130	−	0.188206i	$-0.939733\pi$
$941$		−	10865.5i		−	0.376412i	0.982130	−	0.188206i	$-0.0602672\pi$
$942$		0			0
$943$		−	14989.7i		−	0.517638i
$944$		0			0
$945$	13081.3	−	5739.64i	0.450302	−	0.197577i
$946$		0			0
$947$	6102.06			0.209388			0.104694	−	0.994504i	$-0.466614\pi$
$947$	6102.06			0.209388			0.104694	+	0.994504i	$0.466614\pi$
$948$		0			0
$949$	−13968.8			−0.477814
$950$		0			0
$951$	−37591.6	−	15266.9i	−1.28180	−	0.520570i
$952$		0			0
$953$			11131.8i			0.378377i	0.981941	+	0.189189i	$0.0605858\pi$
$953$			11131.8i			0.378377i	−0.981941	+	0.189189i	$0.939414\pi$
$954$		0			0
$955$		−	4160.30i		−	0.140968i
$956$		0			0
$957$	−7313.09	−	2970.02i	−0.247021	−	0.100321i
$958$		0			0
$959$	−27195.1			−0.915719
$960$		0			0
$961$	29125.0			0.977643
$962$		0			0
$963$	9750.89	+	9484.49i	0.326291	+	0.317376i
$964$		0			0
$965$			3296.76i			0.109976i
$966$		0			0
$967$		−	17597.4i		−	0.585207i	−0.956234	−	0.292603i	$-0.905478\pi$
$967$		−	17597.4i		−	0.585207i	0.956234	−	0.292603i	$-0.0945215\pi$
$968$		0			0
$969$	23136.9	−	56970.0i	0.767043	−	1.88869i
$970$		0			0
$971$	742.846			0.0245510			0.0122755	−	0.999925i	$-0.496092\pi$
$971$	742.846			0.0245510			0.0122755	+	0.999925i	$0.496092\pi$
$972$		0			0
$973$	−40867.8			−1.34652
$974$		0			0
$975$	1354.23	−	3334.53i	0.0444822	−	0.109529i
$976$		0			0
$977$		−	41139.6i		−	1.34716i	−0.739116	−	0.673578i	$-0.764756\pi$
$977$		−	41139.6i		−	1.34716i	0.739116	−	0.673578i	$-0.235244\pi$
$978$		0			0
$979$		−	11824.5i		−	0.386020i
$980$		0			0
$981$	−17972.7	−	17481.7i	−0.584939	−	0.568958i
$982$		0			0
$983$	−23649.8			−0.767356			−0.383678	−	0.923467i	$-0.625343\pi$
$983$	−23649.8			−0.767356			−0.383678	+	0.923467i	$0.625343\pi$
$984$		0			0
$985$	−4816.68			−0.155809
$986$		0			0
$987$	−56229.9	−	22836.3i	−1.81339	−	0.736462i
$988$		0			0
$989$			26688.0i			0.858068i
$990$		0			0
$991$			38069.9i			1.22031i	0.792280	+	0.610157i	$0.208894\pi$
$991$			38069.9i			1.22031i	−0.792280	+	0.610157i	$0.791106\pi$
$992$		0			0
$993$	7553.83	+	3067.79i	0.241403	+	0.0980397i
$994$		0			0
$995$	4352.44			0.138675
$996$		0			0
$997$	2800.44			0.0889576			0.0444788	−	0.999010i	$-0.485837\pi$
$997$	2800.44			0.0889576			0.0444788	+	0.999010i	$0.485837\pi$
$998$		0			0
$999$	45817.4	−	20103.1i	1.45105	−	0.636671i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.h.d.191.15		24
3.2	odd	2	inner	960.4.h.d.191.9		24
4.3	odd	2	inner	960.4.h.d.191.10		24
8.3	odd	2		60.4.e.a.11.10	yes	24
8.5	even	2		60.4.e.a.11.16	yes	24
12.11	even	2	inner	960.4.h.d.191.16		24
24.5	odd	2		60.4.e.a.11.9	✓	24
24.11	even	2		60.4.e.a.11.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.e.a.11.9	✓	24	24.5	odd	2
60.4.e.a.11.10	yes	24	8.3	odd	2
60.4.e.a.11.15	yes	24	24.11	even	2
60.4.e.a.11.16	yes	24	8.5	even	2
960.4.h.d.191.9		24	3.2	odd	2	inner
960.4.h.d.191.10		24	4.3	odd	2	inner
960.4.h.d.191.15		24	1.1	even	1	trivial
960.4.h.d.191.16		24	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.95519 - 4.81427i) q^{3} +5.00000i q^{5} +20.3642i q^{7} +(-19.3544 - 18.8257i) q^{9} +O(q^{10})\)	\(q+(1.95519 - 4.81427i) q^{3} +5.00000i q^{5} +20.3642i q^{7} +(-19.3544 - 18.8257i) q^{9} +15.2982 q^{11} -27.7054 q^{13} +(24.0714 + 9.77596i) q^{15} +92.5031i q^{17} -127.926i q^{19} +(98.0390 + 39.8160i) q^{21} +51.1779 q^{23} -25.0000 q^{25} +(-128.474 + 56.3698i) q^{27} -99.2953i q^{29} +25.8075i q^{31} +(29.9110 - 73.6499i) q^{33} -101.821 q^{35} -356.629 q^{37} +(-54.1693 + 133.381i) q^{39} -292.895i q^{41} +521.476i q^{43} +(94.1283 - 96.7722i) q^{45} -573.546 q^{47} -71.7025 q^{49} +(445.335 + 180.861i) q^{51} -305.765i q^{53} +76.4912i q^{55} +(-615.871 - 250.120i) q^{57} -295.688 q^{59} -326.449 q^{61} +(383.370 - 394.139i) q^{63} -138.527i q^{65} -299.605i q^{67} +(100.063 - 246.384i) q^{69} -653.840 q^{71} +504.190 q^{73} +(-48.8798 + 120.357i) q^{75} +311.537i q^{77} -110.758i q^{79} +(20.1889 + 728.720i) q^{81} -1479.78 q^{83} -462.515 q^{85} +(-478.035 - 194.141i) q^{87} -772.934i q^{89} -564.199i q^{91} +(124.244 + 50.4585i) q^{93} +639.631 q^{95} -26.1154 q^{97} +(-296.089 - 288.000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{9}+O(q^{10}) \) 24 * q + 20 * q^9	\( 24 q + 20 q^{9} - 72 q^{13} + 68 q^{21} - 600 q^{25} + 848 q^{33} - 504 q^{37} + 220 q^{45} - 2256 q^{49} + 1416 q^{57} - 1992 q^{61} + 1548 q^{69} - 2304 q^{73} + 3840 q^{81} - 240 q^{85} + 4384 q^{93} - 2448 q^{97}+O(q^{100}) \) 24 * q + 20 * q^9 - 72 * q^13 + 68 * q^21 - 600 * q^25 + 848 * q^33 - 504 * q^37 + 220 * q^45 - 2256 * q^49 + 1416 * q^57 - 1992 * q^61 + 1548 * q^69 - 2304 * q^73 + 3840 * q^81 - 240 * q^85 + 4384 * q^93 - 2448 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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