LMFDB - Embedded newform 175.3.c.d.174.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,3,Mod(174,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("175.174");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	175.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.76840462631$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			174.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	175.174
Dual form			175.3.c.d.174.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +4.47214 q^{3} +3.00000 q^{4} -4.47214i q^{6} +7.00000i q^{7} -7.00000i q^{8} +11.0000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +4.47214 q^{3} +3.00000 q^{4} -4.47214i q^{6} +7.00000i q^{7} -7.00000i q^{8} +11.0000 q^{9} +2.00000 q^{11} +13.4164 q^{12} -13.4164 q^{13} +7.00000 q^{14} +5.00000 q^{16} -26.8328 q^{17} -11.0000i q^{18} +13.4164i q^{19} +31.3050i q^{21} -2.00000i q^{22} -26.0000i q^{23} -31.3050i q^{24} +13.4164i q^{26} +8.94427 q^{27} +21.0000i q^{28} +22.0000 q^{29} -53.6656i q^{31} -33.0000i q^{32} +8.94427 q^{33} +26.8328i q^{34} +33.0000 q^{36} +14.0000i q^{37} +13.4164 q^{38} -60.0000 q^{39} -26.8328i q^{41} +31.3050 q^{42} +34.0000i q^{43} +6.00000 q^{44} -26.0000 q^{46} +26.8328 q^{47} +22.3607 q^{48} -49.0000 q^{49} -120.000 q^{51} -40.2492 q^{52} +34.0000i q^{53} -8.94427i q^{54} +49.0000 q^{56} +60.0000i q^{57} -22.0000i q^{58} +40.2492i q^{59} +93.9149i q^{61} -53.6656 q^{62} +77.0000i q^{63} -13.0000 q^{64} -8.94427i q^{66} +14.0000i q^{67} -80.4984 q^{68} -116.276i q^{69} +62.0000 q^{71} -77.0000i q^{72} -53.6656 q^{73} +14.0000 q^{74} +40.2492i q^{76} +14.0000i q^{77} +60.0000i q^{78} -38.0000 q^{79} -59.0000 q^{81} -26.8328 q^{82} +40.2492 q^{83} +93.9149i q^{84} +34.0000 q^{86} +98.3870 q^{87} -14.0000i q^{88} +26.8328i q^{89} -93.9149i q^{91} -78.0000i q^{92} -240.000i q^{93} -26.8328i q^{94} -147.580i q^{96} +26.8328 q^{97} +49.0000i q^{98} +22.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{4} + 44 q^{9}+O(q^{10}) $ 4 * q + 12 * q^4 + 44 * q^9	$ 4 q + 12 q^{4} + 44 q^{9} + 8 q^{11} + 28 q^{14} + 20 q^{16} + 88 q^{29} + 132 q^{36} - 240 q^{39} + 24 q^{44} - 104 q^{46} - 196 q^{49} - 480 q^{51} + 196 q^{56} - 52 q^{64} + 248 q^{71} + 56 q^{74} - 152 q^{79} - 236 q^{81} + 136 q^{86} + 88 q^{99}+O(q^{100}) $ 4 * q + 12 * q^4 + 44 * q^9 + 8 * q^11 + 28 * q^14 + 20 * q^16 + 88 * q^29 + 132 * q^36 - 240 * q^39 + 24 * q^44 - 104 * q^46 - 196 * q^49 - 480 * q^51 + 196 * q^56 - 52 * q^64 + 248 * q^71 + 56 * q^74 - 152 * q^79 - 236 * q^81 + 136 * q^86 + 88 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 1.00000i	− 0.500000i	−0.968246	−	0.250000i	$-0.919569\pi$
$2$	− 1.00000i	− 0.500000i	0.968246	−	0.250000i	$-0.0804306\pi$
$3$	4.47214	1.49071	0.745356	−	0.666667i	$-0.232280\pi$
$3$	4.47214	1.49071	0.745356	+	0.666667i	$0.232280\pi$
$4$	3.00000	0.750000
$5$	0	0
$5$	0	0
$6$	− 4.47214i	− 0.745356i
$7$	7.00000i	1.00000i
$7$	7.00000i	1.00000i
$8$	− 7.00000i	− 0.875000i
$9$	11.0000	1.22222
$10$	0	0
$11$	2.00000	0.181818	0.0909091	−	0.995859i	$-0.471023\pi$
$11$	2.00000	0.181818	0.0909091	+	0.995859i	$0.471023\pi$
$12$	13.4164	1.11803
$13$	−13.4164	−1.03203	−0.516016	−	0.856579i	$-0.672585\pi$
$13$	−13.4164	−1.03203	−0.516016	+	0.856579i	$0.672585\pi$
$14$	7.00000	0.500000
$15$	0	0
$16$	5.00000	0.312500
$17$	−26.8328	−1.57840	−0.789200	−	0.614136i	$-0.789505\pi$
$17$	−26.8328	−1.57840	−0.789200	+	0.614136i	$0.789505\pi$
$18$	− 11.0000i	− 0.611111i
$19$	13.4164i	0.706127i	0.935599	+	0.353063i	$0.114860\pi$
$19$	13.4164i	0.706127i	−0.935599	+	0.353063i	$0.885140\pi$
$20$	0	0
$21$	31.3050i	1.49071i
$22$	− 2.00000i	− 0.0909091i
$23$	− 26.0000i	− 1.13043i	−0.824942	−	0.565217i	$-0.808792\pi$
$23$	− 26.0000i	− 1.13043i	0.824942	−	0.565217i	$-0.191208\pi$
$24$	− 31.3050i	− 1.30437i
$25$	0	0
$26$	13.4164i	0.516016i
$27$	8.94427	0.331269
$28$	21.0000i	0.750000i
$29$	22.0000	0.758621	0.379310	−	0.925270i	$-0.376161\pi$
$29$	22.0000	0.758621	0.379310	+	0.925270i	$0.376161\pi$
$30$	0	0
$31$	− 53.6656i	− 1.73115i	−0.500780	−	0.865575i	$-0.666953\pi$
$31$	− 53.6656i	− 1.73115i	0.500780	−	0.865575i	$-0.333047\pi$
$32$	− 33.0000i	− 1.03125i
$33$	8.94427	0.271039
$34$	26.8328i	0.789200i
$35$	0	0
$36$	33.0000	0.916667
$37$	14.0000i	0.378378i	0.981941	+	0.189189i	$0.0605859\pi$
$37$	14.0000i	0.378378i	−0.981941	+	0.189189i	$0.939414\pi$
$38$	13.4164	0.353063
$39$	−60.0000	−1.53846
$40$	0	0
$41$	− 26.8328i	− 0.654459i	−0.944945	−	0.327229i	$-0.893885\pi$
$41$	− 26.8328i	− 0.654459i	0.944945	−	0.327229i	$-0.106115\pi$
$42$	31.3050	0.745356
$43$	34.0000i	0.790698i	0.918531	+	0.395349i	$0.129376\pi$
$43$	34.0000i	0.790698i	−0.918531	+	0.395349i	$0.870624\pi$
$44$	6.00000	0.136364
$45$	0	0
$46$	−26.0000	−0.565217
$47$	26.8328	0.570911	0.285455	−	0.958392i	$-0.407855\pi$
$47$	26.8328	0.570911	0.285455	+	0.958392i	$0.407855\pi$
$48$	22.3607	0.465847
$49$	−49.0000	−1.00000
$50$	0	0
$51$	−120.000	−2.35294
$52$	−40.2492	−0.774024
$53$	34.0000i	0.641509i	0.947162	+	0.320755i	$0.103937\pi$
$53$	34.0000i	0.641509i	−0.947162	+	0.320755i	$0.896063\pi$
$54$	− 8.94427i	− 0.165635i
$55$	0	0
$56$	49.0000	0.875000
$57$	60.0000i	1.05263i
$58$	− 22.0000i	− 0.379310i
$59$	40.2492i	0.682190i	0.940029	+	0.341095i	$0.110798\pi$
$59$	40.2492i	0.682190i	−0.940029	+	0.341095i	$0.889202\pi$
$60$	0	0
$61$	93.9149i	1.53959i	0.638293	+	0.769794i	$0.279641\pi$
$61$	93.9149i	1.53959i	−0.638293	+	0.769794i	$0.720359\pi$
$62$	−53.6656	−0.865575
$63$	77.0000i	1.22222i
$64$	−13.0000	−0.203125
$65$	0	0
$66$	− 8.94427i	− 0.135519i
$67$	14.0000i	0.208955i	0.994527	+	0.104478i	$0.0333171\pi$
$67$	14.0000i	0.208955i	−0.994527	+	0.104478i	$0.966683\pi$
$68$	−80.4984	−1.18380
$69$	− 116.276i	− 1.68515i
$70$	0	0
$71$	62.0000	0.873239	0.436620	−	0.899646i	$-0.356176\pi$
$71$	62.0000	0.873239	0.436620	+	0.899646i	$0.356176\pi$
$72$	− 77.0000i	− 1.06944i
$73$	−53.6656	−0.735146	−0.367573	−	0.929995i	$-0.619811\pi$
$73$	−53.6656	−0.735146	−0.367573	+	0.929995i	$0.619811\pi$
$74$	14.0000	0.189189
$75$	0	0
$76$	40.2492i	0.529595i
$77$	14.0000i	0.181818i
$78$	60.0000i	0.769231i
$79$	−38.0000	−0.481013	−0.240506	−	0.970648i	$-0.577313\pi$
$79$	−38.0000	−0.481013	−0.240506	+	0.970648i	$0.577313\pi$
$80$	0	0
$81$	−59.0000	−0.728395
$82$	−26.8328	−0.327229
$83$	40.2492	0.484930	0.242465	−	0.970160i	$-0.422044\pi$
$83$	40.2492	0.484930	0.242465	+	0.970160i	$0.422044\pi$
$84$	93.9149i	1.11803i
$85$	0	0
$86$	34.0000	0.395349
$87$	98.3870	1.13088
$88$	− 14.0000i	− 0.159091i
$89$	26.8328i	0.301492i	0.988573	+	0.150746i	$0.0481676\pi$
$89$	26.8328i	0.301492i	−0.988573	+	0.150746i	$0.951832\pi$
$90$	0	0
$91$	− 93.9149i	− 1.03203i
$92$	− 78.0000i	− 0.847826i
$93$	− 240.000i	− 2.58065i
$94$	− 26.8328i	− 0.285455i
$95$	0	0
$96$	− 147.580i	− 1.53730i
$97$	26.8328	0.276627	0.138313	−	0.990388i	$-0.455832\pi$
$97$	26.8328	0.276627	0.138313	+	0.990388i	$0.455832\pi$
$98$	49.0000i	0.500000i
$99$	22.0000	0.222222
$100$	0	0
$101$	67.0820i	0.664179i	0.943248	+	0.332089i	$0.107754\pi$
$101$	67.0820i	0.664179i	−0.943248	+	0.332089i	$0.892246\pi$
$102$	120.000i	1.17647i
$103$	160.997	1.56308	0.781538	−	0.623857i	$-0.214435\pi$
$103$	160.997	1.56308	0.781538	+	0.623857i	$0.214435\pi$
$104$	93.9149i	0.903027i
$105$	0	0
$106$	34.0000	0.320755
$107$	− 106.000i	− 0.990654i	−0.868707	−	0.495327i	$-0.835048\pi$
$107$	− 106.000i	− 0.990654i	0.868707	−	0.495327i	$-0.164952\pi$
$108$	26.8328	0.248452
$109$	142.000	1.30275	0.651376	−	0.758755i	$-0.274192\pi$
$109$	142.000	1.30275	0.651376	+	0.758755i	$0.274192\pi$
$110$	0	0
$111$	62.6099i	0.564053i
$112$	35.0000i	0.312500i
$113$	34.0000i	0.300885i	0.988619	+	0.150442i	$0.0480698\pi$
$113$	34.0000i	0.300885i	−0.988619	+	0.150442i	$0.951930\pi$
$114$	60.0000	0.526316
$115$	0	0
$116$	66.0000	0.568966
$117$	−147.580	−1.26137
$118$	40.2492	0.341095
$119$	− 187.830i	− 1.57840i
$120$	0	0
$121$	−117.000	−0.966942
$122$	93.9149	0.769794
$123$	− 120.000i	− 0.975610i
$124$	− 160.997i	− 1.29836i
$125$	0	0
$126$	77.0000	0.611111
$127$	194.000i	1.52756i	0.645477	+	0.763780i	$0.276659\pi$
$127$	194.000i	1.52756i	−0.645477	+	0.763780i	$0.723341\pi$
$128$	− 119.000i	− 0.929688i
$129$	152.053i	1.17870i
$130$	0	0
$131$	− 120.748i	− 0.921738i	−0.887468	−	0.460869i	$-0.847538\pi$
$131$	− 120.748i	− 0.921738i	0.887468	−	0.460869i	$-0.152462\pi$
$132$	26.8328	0.203279
$133$	−93.9149	−0.706127
$134$	14.0000	0.104478
$135$	0	0
$136$	187.830i	1.38110i
$137$	− 166.000i	− 1.21168i	−0.795587	−	0.605839i	$-0.792837\pi$
$137$	− 166.000i	− 1.21168i	0.795587	−	0.605839i	$-0.207163\pi$
$138$	−116.276	−0.842576
$139$	93.9149i	0.675646i	0.941210	+	0.337823i	$0.109691\pi$
$139$	93.9149i	0.675646i	−0.941210	+	0.337823i	$0.890309\pi$
$140$	0	0
$141$	120.000	0.851064
$142$	− 62.0000i	− 0.436620i
$143$	−26.8328	−0.187642
$144$	55.0000	0.381944
$145$	0	0
$146$	53.6656i	0.367573i
$147$	−219.135	−1.49071
$148$	42.0000i	0.283784i
$149$	142.000	0.953020	0.476510	−	0.879169i	$-0.341902\pi$
$149$	142.000	0.953020	0.476510	+	0.879169i	$0.341902\pi$
$150$	0	0
$151$	2.00000	0.0132450	0.00662252	−	0.999978i	$-0.497892\pi$
$151$	2.00000	0.0132450	0.00662252	+	0.999978i	$0.497892\pi$
$152$	93.9149	0.617861
$153$	−295.161	−1.92916
$154$	14.0000	0.0909091
$155$	0	0
$156$	−180.000	−1.15385
$157$	67.0820	0.427274	0.213637	−	0.976913i	$-0.431469\pi$
$157$	67.0820	0.427274	0.213637	+	0.976913i	$0.431469\pi$
$158$	38.0000i	0.240506i
$159$	152.053i	0.956306i
$160$	0	0
$161$	182.000	1.13043
$162$	59.0000i	0.364198i
$163$	34.0000i	0.208589i	0.994546	+	0.104294i	$0.0332584\pi$
$163$	34.0000i	0.208589i	−0.994546	+	0.104294i	$0.966742\pi$
$164$	− 80.4984i	− 0.490844i
$165$	0	0
$166$	− 40.2492i	− 0.242465i
$167$	107.331	0.642702	0.321351	−	0.946960i	$-0.395863\pi$
$167$	107.331	0.642702	0.321351	+	0.946960i	$0.395863\pi$
$168$	219.135	1.30437
$169$	11.0000	0.0650888
$170$	0	0
$171$	147.580i	0.863044i
$172$	102.000i	0.593023i
$173$	147.580	0.853066	0.426533	−	0.904472i	$-0.359735\pi$
$173$	147.580	0.853066	0.426533	+	0.904472i	$0.359735\pi$
$174$	− 98.3870i	− 0.565442i
$175$	0	0
$176$	10.0000	0.0568182
$177$	180.000i	1.01695i
$178$	26.8328	0.150746
$179$	−218.000	−1.21788	−0.608939	−	0.793217i	$-0.708404\pi$
$179$	−218.000	−1.21788	−0.608939	+	0.793217i	$0.708404\pi$
$180$	0	0
$181$	− 254.912i	− 1.40835i	−0.710025	−	0.704176i	$-0.751317\pi$
$181$	− 254.912i	− 1.40835i	0.710025	−	0.704176i	$-0.248683\pi$
$182$	−93.9149	−0.516016
$183$	420.000i	2.29508i
$184$	−182.000	−0.989130
$185$	0	0
$186$	−240.000	−1.29032
$187$	−53.6656	−0.286982
$188$	80.4984	0.428183
$189$	62.6099i	0.331269i
$190$	0	0
$191$	−58.0000	−0.303665	−0.151832	−	0.988406i	$-0.548517\pi$
$191$	−58.0000	−0.303665	−0.151832	+	0.988406i	$0.548517\pi$
$192$	−58.1378	−0.302801
$193$	− 206.000i	− 1.06736i	−0.845687	−	0.533679i	$-0.820809\pi$
$193$	− 206.000i	− 1.06736i	0.845687	−	0.533679i	$-0.179191\pi$
$194$	− 26.8328i	− 0.138313i
$195$	0	0
$196$	−147.000	−0.750000
$197$	− 226.000i	− 1.14721i	−0.819133	−	0.573604i	$-0.805545\pi$
$197$	− 226.000i	− 1.14721i	0.819133	−	0.573604i	$-0.194455\pi$
$198$	− 22.0000i	− 0.111111i
$199$	− 134.164i	− 0.674191i	−0.941470	−	0.337096i	$-0.890555\pi$
$199$	− 134.164i	− 0.674191i	0.941470	−	0.337096i	$-0.109445\pi$
$200$	0	0
$201$	62.6099i	0.311492i
$202$	67.0820	0.332089
$203$	154.000i	0.758621i
$204$	−360.000	−1.76471
$205$	0	0
$206$	− 160.997i	− 0.781538i
$207$	− 286.000i	− 1.38164i
$208$	−67.0820	−0.322510
$209$	26.8328i	0.128387i
$210$	0	0
$211$	−118.000	−0.559242	−0.279621	−	0.960111i	$-0.590209\pi$
$211$	−118.000	−0.559242	−0.279621	+	0.960111i	$0.590209\pi$
$212$	102.000i	0.481132i
$213$	277.272	1.30175
$214$	−106.000	−0.495327
$215$	0	0
$216$	− 62.6099i	− 0.289861i
$217$	375.659	1.73115
$218$	− 142.000i	− 0.651376i
$219$	−240.000	−1.09589
$220$	0	0
$221$	360.000	1.62896
$222$	62.6099	0.282027
$223$	80.4984	0.360980	0.180490	−	0.983577i	$-0.442232\pi$
$223$	80.4984	0.360980	0.180490	+	0.983577i	$0.442232\pi$
$224$	231.000	1.03125
$225$	0	0
$226$	34.0000	0.150442
$227$	−254.912	−1.12296	−0.561480	−	0.827491i	$-0.689768\pi$
$227$	−254.912	−1.12296	−0.561480	+	0.827491i	$0.689768\pi$
$228$	180.000i	0.789474i
$229$	− 13.4164i	− 0.0585869i	−0.999571	−	0.0292935i	$-0.990674\pi$
$229$	− 13.4164i	− 0.0585869i	0.999571	−	0.0292935i	$-0.00932573\pi$
$230$	0	0
$231$	62.6099i	0.271039i
$232$	− 154.000i	− 0.663793i
$233$	214.000i	0.918455i	0.888319	+	0.459227i	$0.151874\pi$
$233$	214.000i	0.918455i	−0.888319	+	0.459227i	$0.848126\pi$
$234$	147.580i	0.630686i
$235$	0	0
$236$	120.748i	0.511643i
$237$	−169.941	−0.717051
$238$	−187.830	−0.789200
$239$	−98.0000	−0.410042	−0.205021	−	0.978758i	$-0.565726\pi$
$239$	−98.0000	−0.410042	−0.205021	+	0.978758i	$0.565726\pi$
$240$	0	0
$241$	− 160.997i	− 0.668037i	−0.942567	−	0.334018i	$-0.891595\pi$
$241$	− 160.997i	− 0.668037i	0.942567	−	0.334018i	$-0.108405\pi$
$242$	117.000i	0.483471i
$243$	−344.354	−1.41710
$244$	281.745i	1.15469i
$245$	0	0
$246$	−120.000	−0.487805
$247$	− 180.000i	− 0.728745i
$248$	−375.659	−1.51476
$249$	180.000	0.722892
$250$	0	0
$251$	335.410i	1.33630i	0.744029	+	0.668148i	$0.232913\pi$
$251$	335.410i	1.33630i	−0.744029	+	0.668148i	$0.767087\pi$
$252$	231.000i	0.916667i
$253$	− 52.0000i	− 0.205534i
$254$	194.000	0.763780
$255$	0	0
$256$	−171.000	−0.667969
$257$	134.164	0.522039	0.261020	−	0.965333i	$-0.415941\pi$
$257$	134.164	0.522039	0.261020	+	0.965333i	$0.415941\pi$
$258$	152.053	0.589351
$259$	−98.0000	−0.378378
$260$	0	0
$261$	242.000	0.927203
$262$	−120.748	−0.460869
$263$	34.0000i	0.129278i	0.997909	+	0.0646388i	$0.0205895\pi$
$263$	34.0000i	0.129278i	−0.997909	+	0.0646388i	$0.979410\pi$
$264$	− 62.6099i	− 0.237159i
$265$	0	0
$266$	93.9149i	0.353063i
$267$	120.000i	0.449438i
$268$	42.0000i	0.156716i
$269$	− 254.912i	− 0.947627i	−0.880625	−	0.473814i	$-0.842877\pi$
$269$	− 254.912i	− 0.947627i	0.880625	−	0.473814i	$-0.157123\pi$
$270$	0	0
$271$	321.994i	1.18817i	0.804403	+	0.594084i	$0.202486\pi$
$271$	321.994i	1.18817i	−0.804403	+	0.594084i	$0.797514\pi$
$272$	−134.164	−0.493250
$273$	− 420.000i	− 1.53846i
$274$	−166.000	−0.605839
$275$	0	0
$276$	− 348.827i	− 1.26386i
$277$	14.0000i	0.0505415i	0.999681	+	0.0252708i	$0.00804479\pi$
$277$	14.0000i	0.0505415i	−0.999681	+	0.0252708i	$0.991955\pi$
$278$	93.9149	0.337823
$279$	− 590.322i	− 2.11585i
$280$	0	0
$281$	2.00000	0.00711744	0.00355872	−	0.999994i	$-0.498867\pi$
$281$	2.00000	0.00711744	0.00355872	+	0.999994i	$0.498867\pi$
$282$	− 120.000i	− 0.425532i
$283$	−93.9149	−0.331855	−0.165927	−	0.986138i	$-0.553062\pi$
$283$	−93.9149	−0.331855	−0.165927	+	0.986138i	$0.553062\pi$
$284$	186.000	0.654930
$285$	0	0
$286$	26.8328i	0.0938210i
$287$	187.830	0.654459
$288$	− 363.000i	− 1.26042i
$289$	431.000	1.49135
$290$	0	0
$291$	120.000	0.412371
$292$	−160.997	−0.551359
$293$	335.410	1.14474	0.572372	−	0.819994i	$-0.306023\pi$
$293$	335.410	1.14474	0.572372	+	0.819994i	$0.306023\pi$
$294$	219.135i	0.745356i
$295$	0	0
$296$	98.0000	0.331081
$297$	17.8885	0.0602308
$298$	− 142.000i	− 0.476510i
$299$	348.827i	1.16664i
$300$	0	0
$301$	−238.000	−0.790698
$302$	− 2.00000i	− 0.00662252i
$303$	300.000i	0.990099i
$304$	67.0820i	0.220665i
$305$	0	0
$306$	295.161i	0.964578i
$307$	−201.246	−0.655525	−0.327762	−	0.944760i	$-0.606295\pi$
$307$	−201.246	−0.655525	−0.327762	+	0.944760i	$0.606295\pi$
$308$	42.0000i	0.136364i
$309$	720.000	2.33010
$310$	0	0
$311$	− 509.823i	− 1.63930i	−0.572862	−	0.819652i	$-0.694167\pi$
$311$	− 509.823i	− 1.63930i	0.572862	−	0.819652i	$-0.305833\pi$
$312$	420.000i	1.34615i
$313$	321.994	1.02873	0.514367	−	0.857570i	$-0.328027\pi$
$313$	321.994	1.02873	0.514367	+	0.857570i	$0.328027\pi$
$314$	− 67.0820i	− 0.213637i
$315$	0	0
$316$	−114.000	−0.360759
$317$	374.000i	1.17981i	0.807472	+	0.589905i	$0.200835\pi$
$317$	374.000i	1.17981i	−0.807472	+	0.589905i	$0.799165\pi$
$318$	152.053	0.478153
$319$	44.0000	0.137931
$320$	0	0
$321$	− 474.046i	− 1.47678i
$322$	− 182.000i	− 0.565217i
$323$	− 360.000i	− 1.11455i
$324$	−177.000	−0.546296
$325$	0	0
$326$	34.0000	0.104294
$327$	635.043	1.94203
$328$	−187.830	−0.572652
$329$	187.830i	0.570911i
$330$	0	0
$331$	482.000	1.45619	0.728097	−	0.685474i	$-0.240405\pi$
$331$	482.000	1.45619	0.728097	+	0.685474i	$0.240405\pi$
$332$	120.748	0.363698
$333$	154.000i	0.462462i
$334$	− 107.331i	− 0.321351i
$335$	0	0
$336$	156.525i	0.465847i
$337$	494.000i	1.46588i	0.680296	+	0.732938i	$0.261851\pi$
$337$	494.000i	1.46588i	−0.680296	+	0.732938i	$0.738149\pi$
$338$	− 11.0000i	− 0.0325444i
$339$	152.053i	0.448533i
$340$	0	0
$341$	− 107.331i	− 0.314754i
$342$	147.580	0.431522
$343$	− 343.000i	− 1.00000i
$344$	238.000	0.691860
$345$	0	0
$346$	− 147.580i	− 0.426533i
$347$	− 346.000i	− 0.997118i	−0.866856	−	0.498559i	$-0.833863\pi$
$347$	− 346.000i	− 0.997118i	0.866856	−	0.498559i	$-0.166137\pi$
$348$	295.161	0.848164
$349$	335.410i	0.961061i	0.876978	+	0.480530i	$0.159556\pi$
$349$	335.410i	0.961061i	−0.876978	+	0.480530i	$0.840444\pi$
$350$	0	0
$351$	−120.000	−0.341880
$352$	− 66.0000i	− 0.187500i
$353$	26.8328	0.0760136	0.0380068	−	0.999277i	$-0.487899\pi$
$353$	26.8328	0.0760136	0.0380068	+	0.999277i	$0.487899\pi$
$354$	180.000	0.508475
$355$	0	0
$356$	80.4984i	0.226119i
$357$	− 840.000i	− 2.35294i
$358$	218.000i	0.608939i
$359$	−338.000	−0.941504	−0.470752	−	0.882266i	$-0.656017\pi$
$359$	−338.000	−0.941504	−0.470752	+	0.882266i	$0.656017\pi$
$360$	0	0
$361$	181.000	0.501385
$362$	−254.912	−0.704176
$363$	−523.240	−1.44143
$364$	− 281.745i	− 0.774024i
$365$	0	0
$366$	420.000	1.14754
$367$	−295.161	−0.804253	−0.402127	−	0.915584i	$-0.631729\pi$
$367$	−295.161	−0.804253	−0.402127	+	0.915584i	$0.631729\pi$
$368$	− 130.000i	− 0.353261i
$369$	− 295.161i	− 0.799894i
$370$	0	0
$371$	−238.000	−0.641509
$372$	− 720.000i	− 1.93548i
$373$	− 86.0000i	− 0.230563i	−0.993333	−	0.115282i	$-0.963223\pi$
$373$	− 86.0000i	− 0.230563i	0.993333	−	0.115282i	$-0.0367770\pi$
$374$	53.6656i	0.143491i
$375$	0	0
$376$	− 187.830i	− 0.499547i
$377$	−295.161	−0.782920
$378$	62.6099	0.165635
$379$	262.000	0.691293	0.345646	−	0.938365i	$-0.387660\pi$
$379$	262.000	0.691293	0.345646	+	0.938365i	$0.387660\pi$
$380$	0	0
$381$	867.594i	2.27715i
$382$	58.0000i	0.151832i
$383$	−563.489	−1.47125	−0.735625	−	0.677388i	$-0.763112\pi$
$383$	−563.489	−1.47125	−0.735625	+	0.677388i	$0.763112\pi$
$384$	− 532.184i	− 1.38590i
$385$	0	0
$386$	−206.000	−0.533679
$387$	374.000i	0.966408i
$388$	80.4984	0.207470
$389$	−698.000	−1.79434	−0.897172	−	0.441681i	$-0.854382\pi$
$389$	−698.000	−1.79434	−0.897172	+	0.441681i	$0.854382\pi$
$390$	0	0
$391$	697.653i	1.78428i
$392$	343.000i	0.875000i
$393$	− 540.000i	− 1.37405i
$394$	−226.000	−0.573604
$395$	0	0
$396$	66.0000	0.166667
$397$	−308.577	−0.777273	−0.388636	−	0.921391i	$-0.627054\pi$
$397$	−308.577	−0.777273	−0.388636	+	0.921391i	$0.627054\pi$
$398$	−134.164	−0.337096
$399$	−420.000	−1.05263
$400$	0	0
$401$	−538.000	−1.34165	−0.670823	−	0.741618i	$-0.734059\pi$
$401$	−538.000	−1.34165	−0.670823	+	0.741618i	$0.734059\pi$
$402$	62.6099	0.155746
$403$	720.000i	1.78660i
$404$	201.246i	0.498134i
$405$	0	0
$406$	154.000	0.379310
$407$	28.0000i	0.0687961i
$408$	840.000i	2.05882i
$409$	− 295.161i	− 0.721665i	−0.932631	−	0.360832i	$-0.882493\pi$
$409$	− 295.161i	− 0.721665i	0.932631	−	0.360832i	$-0.117507\pi$
$410$	0	0
$411$	− 742.375i	− 1.80626i
$412$	482.991	1.17231
$413$	−281.745	−0.682190
$414$	−286.000	−0.690821
$415$	0	0
$416$	442.741i	1.06428i
$417$	420.000i	1.00719i
$418$	26.8328	0.0641933
$419$	818.401i	1.95322i	0.215009	+	0.976612i	$0.431022\pi$
$419$	818.401i	1.95322i	−0.215009	+	0.976612i	$0.568978\pi$
$420$	0	0
$421$	−118.000	−0.280285	−0.140143	−	0.990131i	$-0.544756\pi$
$421$	−118.000	−0.280285	−0.140143	+	0.990131i	$0.544756\pi$
$422$	118.000i	0.279621i
$423$	295.161	0.697780
$424$	238.000	0.561321
$425$	0	0
$426$	− 277.272i	− 0.650874i
$427$	−657.404	−1.53959
$428$	− 318.000i	− 0.742991i
$429$	−120.000	−0.279720
$430$	0	0
$431$	−718.000	−1.66589	−0.832947	−	0.553353i	$-0.813348\pi$
$431$	−718.000	−1.66589	−0.832947	+	0.553353i	$0.813348\pi$
$432$	44.7214	0.103522
$433$	509.823	1.17742	0.588711	−	0.808344i	$-0.299636\pi$
$433$	509.823	1.17742	0.588711	+	0.808344i	$0.299636\pi$
$434$	− 375.659i	− 0.865575i
$435$	0	0
$436$	426.000	0.977064
$437$	348.827	0.798230
$438$	240.000i	0.547945i
$439$	26.8328i	0.0611226i	0.999533	+	0.0305613i	$0.00972948\pi$
$439$	26.8328i	0.0611226i	−0.999533	+	0.0305613i	$0.990271\pi$
$440$	0	0
$441$	−539.000	−1.22222
$442$	− 360.000i	− 0.814480i
$443$	634.000i	1.43115i	0.698535	+	0.715576i	$0.253836\pi$
$443$	634.000i	1.43115i	−0.698535	+	0.715576i	$0.746164\pi$
$444$	187.830i	0.423040i
$445$	0	0
$446$	− 80.4984i	− 0.180490i
$447$	635.043	1.42068
$448$	− 91.0000i	− 0.203125i
$449$	−338.000	−0.752784	−0.376392	−	0.926461i	$-0.622835\pi$
$449$	−338.000	−0.752784	−0.376392	+	0.926461i	$0.622835\pi$
$450$	0	0
$451$	− 53.6656i	− 0.118993i
$452$	102.000i	0.225664i
$453$	8.94427	0.0197445
$454$	254.912i	0.561480i
$455$	0	0
$456$	420.000	0.921053
$457$	− 466.000i	− 1.01969i	−0.860265	−	0.509847i	$-0.829702\pi$
$457$	− 466.000i	− 1.01969i	0.860265	−	0.509847i	$-0.170298\pi$
$458$	−13.4164	−0.0292935
$459$	−240.000	−0.522876
$460$	0	0
$461$	− 442.741i	− 0.960394i	−0.877161	−	0.480197i	$-0.840565\pi$
$461$	− 442.741i	− 0.960394i	0.877161	−	0.480197i	$-0.159435\pi$
$462$	62.6099	0.135519
$463$	− 206.000i	− 0.444924i	−0.974941	−	0.222462i	$-0.928591\pi$
$463$	− 206.000i	− 0.444924i	0.974941	−	0.222462i	$-0.0714094\pi$
$464$	110.000	0.237069
$465$	0	0
$466$	214.000	0.459227
$467$	−362.243	−0.775681	−0.387840	−	0.921727i	$-0.626779\pi$
$467$	−362.243	−0.775681	−0.387840	+	0.921727i	$0.626779\pi$
$468$	−442.741	−0.946029
$469$	−98.0000	−0.208955
$470$	0	0
$471$	300.000	0.636943
$472$	281.745	0.596916
$473$	68.0000i	0.143763i
$474$	169.941i	0.358526i
$475$	0	0
$476$	− 563.489i	− 1.18380i
$477$	374.000i	0.784067i
$478$	98.0000i	0.205021i
$479$	− 214.663i	− 0.448147i	−0.974572	−	0.224074i	$-0.928064\pi$
$479$	− 214.663i	− 0.448147i	0.974572	−	0.224074i	$-0.0719356\pi$
$480$	0	0
$481$	− 187.830i	− 0.390498i
$482$	−160.997	−0.334018
$483$	813.929	1.68515
$484$	−351.000	−0.725207
$485$	0	0
$486$	344.354i	0.708548i
$487$	− 166.000i	− 0.340862i	−0.985370	−	0.170431i	$-0.945484\pi$
$487$	− 166.000i	− 0.340862i	0.985370	−	0.170431i	$-0.0545161\pi$
$488$	657.404	1.34714
$489$	152.053i	0.310946i
$490$	0	0
$491$	−838.000	−1.70672	−0.853360	−	0.521321i	$-0.825439\pi$
$491$	−838.000	−1.70672	−0.853360	+	0.521321i	$0.825439\pi$
$492$	− 360.000i	− 0.731707i
$493$	−590.322	−1.19741
$494$	−180.000	−0.364372
$495$	0	0
$496$	− 268.328i	− 0.540984i
$497$	434.000i	0.873239i
$498$	− 180.000i	− 0.361446i
$499$	262.000	0.525050	0.262525	−	0.964925i	$-0.415445\pi$
$499$	262.000	0.525050	0.262525	+	0.964925i	$0.415445\pi$
$500$	0	0
$501$	480.000	0.958084
$502$	335.410	0.668148
$503$	429.325	0.853529	0.426764	−	0.904363i	$-0.359653\pi$
$503$	429.325	0.853529	0.426764	+	0.904363i	$0.359653\pi$
$504$	539.000	1.06944
$505$	0	0
$506$	−52.0000	−0.102767
$507$	49.1935	0.0970286
$508$	582.000i	1.14567i
$509$	− 898.899i	− 1.76601i	−0.469363	−	0.883005i	$-0.655516\pi$
$509$	− 898.899i	− 1.76601i	0.469363	−	0.883005i	$-0.344484\pi$
$510$	0	0
$511$	− 375.659i	− 0.735146i
$512$	− 305.000i	− 0.595703i
$513$	120.000i	0.233918i
$514$	− 134.164i	− 0.261020i
$515$	0	0
$516$	456.158i	0.884027i
$517$	53.6656	0.103802
$518$	98.0000i	0.189189i
$519$	660.000	1.27168
$520$	0	0
$521$	724.486i	1.39057i	0.718735	+	0.695284i	$0.244721\pi$
$521$	724.486i	1.39057i	−0.718735	+	0.695284i	$0.755279\pi$
$522$	− 242.000i	− 0.463602i
$523$	−523.240	−1.00046	−0.500229	−	0.865893i	$-0.666751\pi$
$523$	−523.240	−1.00046	−0.500229	+	0.865893i	$0.666751\pi$
$524$	− 362.243i	− 0.691303i
$525$	0	0
$526$	34.0000	0.0646388
$527$	1440.00i	2.73245i
$528$	44.7214	0.0846995
$529$	−147.000	−0.277883
$530$	0	0
$531$	442.741i	0.833788i
$532$	−281.745	−0.529595
$533$	360.000i	0.675422i
$534$	120.000	0.224719
$535$	0	0
$536$	98.0000	0.182836
$537$	−974.926	−1.81550
$538$	−254.912	−0.473814
$539$	−98.0000	−0.181818
$540$	0	0
$541$	842.000	1.55638	0.778189	−	0.628031i	$-0.216139\pi$
$541$	842.000	1.55638	0.778189	+	0.628031i	$0.216139\pi$
$542$	321.994	0.594084
$543$	− 1140.00i	− 2.09945i
$544$	885.483i	1.62773i
$545$	0	0
$546$	−420.000	−0.769231
$547$	134.000i	0.244973i	0.992470	+	0.122486i	$0.0390868\pi$
$547$	134.000i	0.244973i	−0.992470	+	0.122486i	$0.960913\pi$
$548$	− 498.000i	− 0.908759i
$549$	1033.06i	1.88172i
$550$	0	0
$551$	295.161i	0.535682i
$552$	−813.929	−1.47451
$553$	− 266.000i	− 0.481013i
$554$	14.0000	0.0252708
$555$	0	0
$556$	281.745i	0.506735i
$557$	− 706.000i	− 1.26750i	−0.773536	−	0.633752i	$-0.781514\pi$
$557$	− 706.000i	− 1.26750i	0.773536	−	0.633752i	$-0.218486\pi$
$558$	−590.322	−1.05792
$559$	− 456.158i	− 0.816025i
$560$	0	0
$561$	−240.000	−0.427807
$562$	− 2.00000i	− 0.00355872i
$563$	−13.4164	−0.0238302	−0.0119151	−	0.999929i	$-0.503793\pi$
$563$	−13.4164	−0.0238302	−0.0119151	+	0.999929i	$0.503793\pi$
$564$	360.000	0.638298
$565$	0	0
$566$	93.9149i	0.165927i
$567$	− 413.000i	− 0.728395i
$568$	− 434.000i	− 0.764085i
$569$	82.0000	0.144112	0.0720562	−	0.997401i	$-0.477044\pi$
$569$	82.0000	0.144112	0.0720562	+	0.997401i	$0.477044\pi$
$570$	0	0
$571$	−118.000	−0.206655	−0.103327	−	0.994647i	$-0.532949\pi$
$571$	−118.000	−0.206655	−0.103327	+	0.994647i	$0.532949\pi$
$572$	−80.4984	−0.140732
$573$	−259.384	−0.452677
$574$	− 187.830i	− 0.327229i
$575$	0	0
$576$	−143.000	−0.248264
$577$	885.483	1.53463	0.767316	−	0.641269i	$-0.221592\pi$
$577$	885.483	1.53463	0.767316	+	0.641269i	$0.221592\pi$
$578$	− 431.000i	− 0.745675i
$579$	− 921.260i	− 1.59112i
$580$	0	0
$581$	281.745i	0.484930i
$582$	− 120.000i	− 0.206186i
$583$	68.0000i	0.116638i
$584$	375.659i	0.643252i
$585$	0	0
$586$	− 335.410i	− 0.572372i
$587$	791.568	1.34850	0.674249	−	0.738504i	$-0.264468\pi$
$587$	791.568	1.34850	0.674249	+	0.738504i	$0.264468\pi$
$588$	−657.404	−1.11803
$589$	720.000	1.22241
$590$	0	0
$591$	− 1010.70i	− 1.71016i
$592$	70.0000i	0.118243i
$593$	−134.164	−0.226246	−0.113123	−	0.993581i	$-0.536085\pi$
$593$	−134.164	−0.226246	−0.113123	+	0.993581i	$0.536085\pi$
$594$	− 17.8885i	− 0.0301154i
$595$	0	0
$596$	426.000	0.714765
$597$	− 600.000i	− 1.00503i
$598$	348.827	0.583322
$599$	−398.000	−0.664441	−0.332220	−	0.943202i	$-0.607798\pi$
$599$	−398.000	−0.664441	−0.332220	+	0.943202i	$0.607798\pi$
$600$	0	0
$601$	134.164i	0.223235i	0.993751	+	0.111617i	$0.0356031\pi$
$601$	134.164i	0.223235i	−0.993751	+	0.111617i	$0.964397\pi$
$602$	238.000i	0.395349i
$603$	154.000i	0.255390i
$604$	6.00000	0.00993377
$605$	0	0
$606$	300.000	0.495050
$607$	939.149	1.54720	0.773598	−	0.633676i	$-0.218455\pi$
$607$	939.149	1.54720	0.773598	+	0.633676i	$0.218455\pi$
$608$	442.741	0.728193
$609$	688.709i	1.13088i
$610$	0	0
$611$	−360.000	−0.589198
$612$	−885.483	−1.44687
$613$	− 206.000i	− 0.336052i	−0.985783	−	0.168026i	$-0.946261\pi$
$613$	− 206.000i	− 0.336052i	0.985783	−	0.168026i	$-0.0537393\pi$
$614$	201.246i	0.327762i
$615$	0	0
$616$	98.0000	0.159091
$617$	494.000i	0.800648i	0.916374	+	0.400324i	$0.131102\pi$
$617$	494.000i	0.800648i	−0.916374	+	0.400324i	$0.868898\pi$
$618$	− 720.000i	− 1.16505i
$619$	− 120.748i	− 0.195069i	−0.995232	−	0.0975345i	$-0.968904\pi$
$619$	− 120.748i	− 0.195069i	0.995232	−	0.0975345i	$-0.0310956\pi$
$620$	0	0
$621$	− 232.551i	− 0.374478i
$622$	−509.823	−0.819652
$623$	−187.830	−0.301492
$624$	−300.000	−0.480769
$625$	0	0
$626$	− 321.994i	− 0.514367i
$627$	120.000i	0.191388i
$628$	201.246	0.320456
$629$	− 375.659i	− 0.597233i
$630$	0	0
$631$	542.000	0.858954	0.429477	−	0.903078i	$-0.358698\pi$
$631$	542.000	0.858954	0.429477	+	0.903078i	$0.358698\pi$
$632$	266.000i	0.420886i
$633$	−527.712	−0.833668
$634$	374.000	0.589905
$635$	0	0
$636$	456.158i	0.717229i
$637$	657.404	1.03203
$638$	− 44.0000i	− 0.0689655i
$639$	682.000	1.06729
$640$	0	0
$641$	−298.000	−0.464899	−0.232449	−	0.972609i	$-0.574674\pi$
$641$	−298.000	−0.464899	−0.232449	+	0.972609i	$0.574674\pi$
$642$	−474.046	−0.738390
$643$	1006.23	1.56490	0.782450	−	0.622714i	$-0.213970\pi$
$643$	1006.23	1.56490	0.782450	+	0.622714i	$0.213970\pi$
$644$	546.000	0.847826
$645$	0	0
$646$	−360.000	−0.557276
$647$	−643.988	−0.995344	−0.497672	−	0.867365i	$-0.665812\pi$
$647$	−643.988	−0.995344	−0.497672	+	0.867365i	$0.665812\pi$
$648$	413.000i	0.637346i
$649$	80.4984i	0.124035i
$650$	0	0
$651$	1680.00	2.58065
$652$	102.000i	0.156442i
$653$	154.000i	0.235835i	0.993023	+	0.117917i	$0.0376218\pi$
$653$	154.000i	0.235835i	−0.993023	+	0.117917i	$0.962378\pi$
$654$	− 635.043i	− 0.971014i
$655$	0	0
$656$	− 134.164i	− 0.204518i
$657$	−590.322	−0.898511
$658$	187.830	0.285455
$659$	−338.000	−0.512898	−0.256449	−	0.966558i	$-0.582553\pi$
$659$	−338.000	−0.512898	−0.256449	+	0.966558i	$0.582553\pi$
$660$	0	0
$661$	− 576.906i	− 0.872777i	−0.899758	−	0.436388i	$-0.856257\pi$
$661$	− 576.906i	− 0.872777i	0.899758	−	0.436388i	$-0.143743\pi$
$662$	− 482.000i	− 0.728097i
$663$	1609.97	2.42831
$664$	− 281.745i	− 0.424314i
$665$	0	0
$666$	154.000	0.231231
$667$	− 572.000i	− 0.857571i
$668$	321.994	0.482027
$669$	360.000	0.538117
$670$	0	0
$671$	187.830i	0.279925i
$672$	1033.06	1.53730
$673$	814.000i	1.20951i	0.796412	+	0.604755i	$0.206729\pi$
$673$	814.000i	1.20951i	−0.796412	+	0.604755i	$0.793271\pi$
$674$	494.000	0.732938
$675$	0	0
$676$	33.0000	0.0488166
$677$	684.237	1.01069	0.505345	−	0.862918i	$-0.331365\pi$
$677$	684.237	1.01069	0.505345	+	0.862918i	$0.331365\pi$
$678$	152.053	0.224266
$679$	187.830i	0.276627i
$680$	0	0
$681$	−1140.00	−1.67401
$682$	−107.331	−0.157377
$683$	− 926.000i	− 1.35578i	−0.735162	−	0.677892i	$-0.762894\pi$
$683$	− 926.000i	− 1.35578i	0.735162	−	0.677892i	$-0.237106\pi$
$684$	442.741i	0.647283i
$685$	0	0
$686$	−343.000	−0.500000
$687$	− 60.0000i	− 0.0873362i
$688$	170.000i	0.247093i
$689$	− 456.158i	− 0.662058i
$690$	0	0
$691$	576.906i	0.834885i	0.908703	+	0.417443i	$0.137073\pi$
$691$	576.906i	0.834885i	−0.908703	+	0.417443i	$0.862927\pi$
$692$	442.741	0.639800
$693$	154.000i	0.222222i
$694$	−346.000	−0.498559
$695$	0	0
$696$	− 688.709i	− 0.989524i
$697$	720.000i	1.03300i
$698$	335.410	0.480530
$699$	957.037i	1.36915i
$700$	0	0
$701$	362.000	0.516405	0.258203	−	0.966091i	$-0.416870\pi$
$701$	362.000	0.516405	0.258203	+	0.966091i	$0.416870\pi$
$702$	120.000i	0.170940i
$703$	−187.830	−0.267183
$704$	−26.0000	−0.0369318
$705$	0	0
$706$	− 26.8328i	− 0.0380068i
$707$	−469.574	−0.664179
$708$	540.000i	0.762712i
$709$	−1058.00	−1.49224	−0.746121	−	0.665810i	$-0.768086\pi$
$709$	−1058.00	−1.49224	−0.746121	+	0.665810i	$0.768086\pi$
$710$	0	0
$711$	−418.000	−0.587904
$712$	187.830	0.263806
$713$	−1395.31	−1.95695
$714$	−840.000	−1.17647
$715$	0	0
$716$	−654.000	−0.913408
$717$	−438.269	−0.611254
$718$	338.000i	0.470752i
$719$	482.991i	0.671753i	0.941906	+	0.335877i	$0.109033\pi$
$719$	482.991i	0.671753i	−0.941906	+	0.335877i	$0.890967\pi$
$720$	0	0
$721$	1126.98i	1.56308i
$722$	− 181.000i	− 0.250693i
$723$	− 720.000i	− 0.995851i
$724$	− 764.735i	− 1.05626i
$725$	0	0
$726$	523.240i	0.720716i
$727$	−1126.98	−1.55018	−0.775088	−	0.631853i	$-0.782295\pi$
$727$	−1126.98	−1.55018	−0.775088	+	0.631853i	$0.782295\pi$
$728$	−657.404	−0.903027
$729$	−1009.00	−1.38409
$730$	0	0
$731$	− 912.316i	− 1.24804i
$732$	1260.00i	1.72131i
$733$	−1301.39	−1.77543	−0.887716	−	0.460392i	$-0.847709\pi$
$733$	−1301.39	−1.77543	−0.887716	+	0.460392i	$0.847709\pi$
$734$	295.161i	0.402127i
$735$	0	0
$736$	−858.000	−1.16576
$737$	28.0000i	0.0379919i
$738$	−295.161	−0.399947
$739$	982.000	1.32882	0.664411	−	0.747367i	$-0.268682\pi$
$739$	982.000	1.32882	0.664411	+	0.747367i	$0.268682\pi$
$740$	0	0
$741$	− 804.984i	− 1.08635i
$742$	238.000i	0.320755i
$743$	694.000i	0.934051i	0.884244	+	0.467026i	$0.154674\pi$
$743$	694.000i	0.934051i	−0.884244	+	0.467026i	$0.845326\pi$
$744$	−1680.00	−2.25806
$745$	0	0
$746$	−86.0000	−0.115282
$747$	442.741	0.592693
$748$	−160.997	−0.215236
$749$	742.000	0.990654
$750$	0	0
$751$	242.000	0.322237	0.161119	−	0.986935i	$-0.448490\pi$
$751$	242.000	0.322237	0.161119	+	0.986935i	$0.448490\pi$
$752$	134.164	0.178410
$753$	1500.00i	1.99203i
$754$	295.161i	0.391460i
$755$	0	0
$756$	187.830i	0.248452i
$757$	− 106.000i	− 0.140026i	−0.997546	−	0.0700132i	$-0.977696\pi$
$757$	− 106.000i	− 0.140026i	0.997546	−	0.0700132i	$-0.0223041\pi$
$758$	− 262.000i	− 0.345646i
$759$	− 232.551i	− 0.306391i
$760$	0	0
$761$	1100.15i	1.44566i	0.691027	+	0.722829i	$0.257158\pi$
$761$	1100.15i	1.44566i	−0.691027	+	0.722829i	$0.742842\pi$
$762$	867.594	1.13858
$763$	994.000i	1.30275i
$764$	−174.000	−0.227749
$765$	0	0
$766$	563.489i	0.735625i
$767$	− 540.000i	− 0.704042i
$768$	−764.735	−0.995749
$769$	− 1126.98i	− 1.46551i	−0.680492	−	0.732756i	$-0.738234\pi$
$769$	− 1126.98i	− 1.46551i	0.680492	−	0.732756i	$-0.261766\pi$
$770$	0	0
$771$	600.000	0.778210
$772$	− 618.000i	− 0.800518i
$773$	818.401	1.05873	0.529367	−	0.848393i	$-0.322430\pi$
$773$	818.401	1.05873	0.529367	+	0.848393i	$0.322430\pi$
$774$	374.000	0.483204
$775$	0	0
$776$	− 187.830i	− 0.242049i
$777$	−438.269	−0.564053
$778$	698.000i	0.897172i
$779$	360.000	0.462131
$780$	0	0
$781$	124.000	0.158771
$782$	697.653	0.892140
$783$	196.774	0.251308
$784$	−245.000	−0.312500
$785$	0	0
$786$	−540.000	−0.687023
$787$	−684.237	−0.869424	−0.434712	−	0.900569i	$-0.643150\pi$
$787$	−684.237	−0.869424	−0.434712	+	0.900569i	$0.643150\pi$
$788$	− 678.000i	− 0.860406i
$789$	152.053i	0.192716i
$790$	0	0
$791$	−238.000	−0.300885
$792$	− 154.000i	− 0.194444i
$793$	− 1260.00i	− 1.58890i
$794$	308.577i	0.388636i
$795$	0	0
$796$	− 402.492i	− 0.505644i
$797$	−308.577	−0.387174	−0.193587	−	0.981083i	$-0.562012\pi$
$797$	−308.577	−0.387174	−0.193587	+	0.981083i	$0.562012\pi$
$798$	420.000i	0.526316i
$799$	−720.000	−0.901126
$800$	0	0
$801$	295.161i	0.368491i
$802$	538.000i	0.670823i
$803$	−107.331	−0.133663
$804$	187.830i	0.233619i
$805$	0	0
$806$	720.000	0.893300
$807$	− 1140.00i	− 1.41264i
$808$	469.574	0.581156
$809$	−1358.00	−1.67862	−0.839308	−	0.543657i	$-0.817039\pi$
$809$	−1358.00	−1.67862	−0.839308	+	0.543657i	$0.817039\pi$
$810$	0	0
$811$	− 308.577i	− 0.380490i	−0.981737	−	0.190245i	$-0.939072\pi$
$811$	− 308.577i	− 0.380490i	0.981737	−	0.190245i	$-0.0609282\pi$
$812$	462.000i	0.568966i
$813$	1440.00i	1.77122i
$814$	28.0000	0.0343980
$815$	0	0
$816$	−600.000	−0.735294
$817$	−456.158	−0.558333
$818$	−295.161	−0.360832
$819$	− 1033.06i	− 1.26137i
$820$	0	0
$821$	482.000	0.587089	0.293544	−	0.955945i	$-0.405165\pi$
$821$	482.000	0.587089	0.293544	+	0.955945i	$0.405165\pi$
$822$	−742.375	−0.903132
$823$	− 926.000i	− 1.12515i	−0.826746	−	0.562576i	$-0.809810\pi$
$823$	− 926.000i	− 1.12515i	0.826746	−	0.562576i	$-0.190190\pi$
$824$	− 1126.98i	− 1.36769i
$825$	0	0
$826$	281.745i	0.341095i
$827$	− 226.000i	− 0.273277i	−0.990621	−	0.136638i	$-0.956370\pi$
$827$	− 226.000i	− 0.273277i	0.990621	−	0.136638i	$-0.0436299\pi$
$828$	− 858.000i	− 1.03623i
$829$	1462.39i	1.76404i	0.471213	+	0.882020i	$0.343816\pi$
$829$	1462.39i	1.76404i	−0.471213	+	0.882020i	$0.656184\pi$
$830$	0	0
$831$	62.6099i	0.0753428i
$832$	174.413	0.209631
$833$	1314.81	1.57840
$834$	420.000	0.503597
$835$	0	0
$836$	80.4984i	0.0962900i
$837$	− 480.000i	− 0.573477i
$838$	818.401	0.976612
$839$	831.817i	0.991439i	0.868483	+	0.495719i	$0.165096\pi$
$839$	831.817i	0.991439i	−0.868483	+	0.495719i	$0.834904\pi$
$840$	0	0
$841$	−357.000	−0.424495
$842$	118.000i	0.140143i
$843$	8.94427	0.0106100
$844$	−354.000	−0.419431
$845$	0	0
$846$	− 295.161i	− 0.348890i
$847$	− 819.000i	− 0.966942i
$848$	170.000i	0.200472i
$849$	−420.000	−0.494700
$850$	0	0
$851$	364.000	0.427732
$852$	831.817	0.976311
$853$	−40.2492	−0.0471855	−0.0235927	−	0.999722i	$-0.507511\pi$
$853$	−40.2492	−0.0471855	−0.0235927	+	0.999722i	$0.507511\pi$
$854$	657.404i	0.769794i
$855$	0	0
$856$	−742.000	−0.866822
$857$	−268.328	−0.313102	−0.156551	−	0.987670i	$-0.550038\pi$
$857$	−268.328	−0.313102	−0.156551	+	0.987670i	$0.550038\pi$
$858$	120.000i	0.139860i
$859$	308.577i	0.359229i	0.983737	+	0.179614i	$0.0574850\pi$
$859$	308.577i	0.359229i	−0.983737	+	0.179614i	$0.942515\pi$
$860$	0	0
$861$	840.000	0.975610
$862$	718.000i	0.832947i
$863$	514.000i	0.595597i	0.954629	+	0.297798i	$0.0962523\pi$
$863$	514.000i	0.595597i	−0.954629	+	0.297798i	$0.903748\pi$
$864$	− 295.161i	− 0.341621i
$865$	0	0
$866$	− 509.823i	− 0.588711i
$867$	1927.49	2.22317
$868$	1126.98	1.29836
$869$	−76.0000	−0.0874568
$870$	0	0
$871$	− 187.830i	− 0.215648i
$872$	− 994.000i	− 1.13991i
$873$	295.161	0.338100
$874$	− 348.827i	− 0.399115i
$875$	0	0
$876$	−720.000	−0.821918
$877$	− 1306.00i	− 1.48917i	−0.667529	−	0.744584i	$-0.732648\pi$
$877$	− 1306.00i	− 1.48917i	0.667529	−	0.744584i	$-0.267352\pi$
$878$	26.8328	0.0305613
$879$	1500.00	1.70648
$880$	0	0
$881$	− 1126.98i	− 1.27920i	−0.768706	−	0.639602i	$-0.779099\pi$
$881$	− 1126.98i	− 1.27920i	0.768706	−	0.639602i	$-0.220901\pi$
$882$	539.000i	0.611111i
$883$	− 1526.00i	− 1.72820i	−0.503321	−	0.864100i	$-0.667889\pi$
$883$	− 1526.00i	− 1.72820i	0.503321	−	0.864100i	$-0.332111\pi$
$884$	1080.00	1.22172
$885$	0	0
$886$	634.000	0.715576
$887$	−1556.30	−1.75457	−0.877285	−	0.479970i	$-0.840648\pi$
$887$	−1556.30	−1.75457	−0.877285	+	0.479970i	$0.840648\pi$
$888$	438.269	0.493547
$889$	−1358.00	−1.52756
$890$	0	0
$891$	−118.000	−0.132435
$892$	241.495	0.270735
$893$	360.000i	0.403135i
$894$	− 635.043i	− 0.710339i
$895$	0	0
$896$	833.000	0.929688
$897$	1560.00i	1.73913i
$898$	338.000i	0.376392i
$899$	− 1180.64i	− 1.31329i
$900$	0	0
$901$	− 912.316i	− 1.01256i
$902$	−53.6656	−0.0594963
$903$	−1064.37	−1.17870
$904$	238.000	0.263274
$905$	0	0
$906$	− 8.94427i	− 0.00987226i
$907$	734.000i	0.809261i	0.914480	+	0.404631i	$0.132600\pi$
$907$	734.000i	0.809261i	−0.914480	+	0.404631i	$0.867400\pi$
$908$	−764.735	−0.842219
$909$	737.902i	0.811774i
$910$	0	0
$911$	1202.00	1.31943	0.659715	−	0.751516i	$-0.270677\pi$
$911$	1202.00	1.31943	0.659715	+	0.751516i	$0.270677\pi$
$912$	300.000i	0.328947i
$913$	80.4984	0.0881692
$914$	−466.000	−0.509847
$915$	0	0
$916$	− 40.2492i	− 0.0439402i
$917$	845.234	0.921738
$918$	240.000i	0.261438i
$919$	1282.00	1.39499	0.697497	−	0.716587i	$-0.254297\pi$
$919$	1282.00	1.39499	0.697497	+	0.716587i	$0.254297\pi$
$920$	0	0
$921$	−900.000	−0.977199
$922$	−442.741	−0.480197
$923$	−831.817	−0.901210
$924$	187.830i	0.203279i
$925$	0	0
$926$	−206.000	−0.222462
$927$	1770.97	1.91043
$928$	− 726.000i	− 0.782328i
$929$	1126.98i	1.21311i	0.795042	+	0.606554i	$0.207449\pi$
$929$	1126.98i	1.21311i	−0.795042	+	0.606554i	$0.792551\pi$
$930$	0	0
$931$	− 657.404i	− 0.706127i
$932$	642.000i	0.688841i
$933$	− 2280.00i	− 2.44373i
$934$	362.243i	0.387840i
$935$	0	0
$936$	1033.06i	1.10370i
$937$	−214.663	−0.229096	−0.114548	−	0.993418i	$-0.536542\pi$
$937$	−214.663	−0.229096	−0.114548	+	0.993418i	$0.536542\pi$
$938$	98.0000i	0.104478i
$939$	1440.00	1.53355
$940$	0	0
$941$	845.234i	0.898229i	0.893474	+	0.449115i	$0.148261\pi$
$941$	845.234i	0.898229i	−0.893474	+	0.449115i	$0.851739\pi$
$942$	− 300.000i	− 0.318471i
$943$	−697.653	−0.739823
$944$	201.246i	0.213184i
$945$	0	0
$946$	68.0000	0.0718816
$947$	734.000i	0.775079i	0.921853	+	0.387540i	$0.126675\pi$
$947$	734.000i	0.775079i	−0.921853	+	0.387540i	$0.873325\pi$
$948$	−509.823	−0.537789
$949$	720.000	0.758693
$950$	0	0
$951$	1672.58i	1.75876i
$952$	−1314.81	−1.38110
$953$	934.000i	0.980063i	0.871705	+	0.490031i	$0.163015\pi$
$953$	934.000i	0.980063i	−0.871705	+	0.490031i	$0.836985\pi$
$954$	374.000	0.392034
$955$	0	0
$956$	−294.000	−0.307531
$957$	196.774	0.205615
$958$	−214.663	−0.224074
$959$	1162.00	1.21168
$960$	0	0
$961$	−1919.00	−1.99688
$962$	−187.830	−0.195249
$963$	− 1166.00i	− 1.21080i
$964$	− 482.991i	− 0.501028i
$965$	0	0
$966$	− 813.929i	− 0.842576i
$967$	314.000i	0.324716i	0.986732	+	0.162358i	$0.0519099\pi$
$967$	314.000i	0.324716i	−0.986732	+	0.162358i	$0.948090\pi$
$968$	819.000i	0.846074i
$969$	− 1609.97i	− 1.66147i
$970$	0	0
$971$	− 147.580i	− 0.151988i	−0.997108	−	0.0759941i	$-0.975787\pi$
$971$	− 147.580i	− 0.151988i	0.997108	−	0.0759941i	$-0.0242130\pi$
$972$	−1033.06	−1.06282
$973$	−657.404	−0.675646
$974$	−166.000	−0.170431
$975$	0	0
$976$	469.574i	0.481121i
$977$	− 1486.00i	− 1.52098i	−0.649348	−	0.760491i	$-0.724958\pi$
$977$	− 1486.00i	− 1.52098i	0.649348	−	0.760491i	$-0.275042\pi$
$978$	152.053	0.155473
$979$	53.6656i	0.0548168i
$980$	0	0
$981$	1562.00	1.59225
$982$	838.000i	0.853360i
$983$	−965.981	−0.982687	−0.491344	−	0.870966i	$-0.663494\pi$
$983$	−965.981	−0.982687	−0.491344	+	0.870966i	$0.663494\pi$
$984$	−840.000	−0.853659
$985$	0	0
$986$	590.322i	0.598704i
$987$	840.000i	0.851064i
$988$	− 540.000i	− 0.546559i
$989$	884.000	0.893832
$990$	0	0
$991$	−58.0000	−0.0585267	−0.0292634	−	0.999572i	$-0.509316\pi$
$991$	−58.0000	−0.0585267	−0.0292634	+	0.999572i	$0.509316\pi$
$992$	−1770.97	−1.78525
$993$	2155.57	2.17076
$994$	434.000	0.436620
$995$	0	0
$996$	540.000	0.542169
$997$	630.571	0.632469	0.316234	−	0.948681i	$-0.397581\pi$
$997$	630.571	0.632469	0.316234	+	0.948681i	$0.397581\pi$
$998$	− 262.000i	− 0.262525i
$999$	125.220i	0.125345i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.3.c.d.174.2		4
5.2	odd	4		175.3.d.f.76.1		2
5.3	odd	4		35.3.d.a.6.2	yes	2
5.4	even	2	inner	175.3.c.d.174.3		4
7.6	odd	2	inner	175.3.c.d.174.1		4
15.8	even	4		315.3.h.b.181.1		2
20.3	even	4		560.3.f.a.321.1		2
35.3	even	12		245.3.h.b.166.1		4
35.13	even	4		35.3.d.a.6.1	✓	2
35.18	odd	12		245.3.h.b.166.2		4
35.23	odd	12		245.3.h.b.31.1		4
35.27	even	4		175.3.d.f.76.2		2
35.33	even	12		245.3.h.b.31.2		4
35.34	odd	2	inner	175.3.c.d.174.4		4
105.83	odd	4		315.3.h.b.181.2		2
140.83	odd	4		560.3.f.a.321.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.d.a.6.1	✓	2	35.13	even	4
35.3.d.a.6.2	yes	2	5.3	odd	4
175.3.c.d.174.1		4	7.6	odd	2	inner
175.3.c.d.174.2		4	1.1	even	1	trivial
175.3.c.d.174.3		4	5.4	even	2	inner
175.3.c.d.174.4		4	35.34	odd	2	inner
175.3.d.f.76.1		2	5.2	odd	4
175.3.d.f.76.2		2	35.27	even	4
245.3.h.b.31.1		4	35.23	odd	12
245.3.h.b.31.2		4	35.33	even	12
245.3.h.b.166.1		4	35.3	even	12
245.3.h.b.166.2		4	35.18	odd	12
315.3.h.b.181.1		2	15.8	even	4
315.3.h.b.181.2		2	105.83	odd	4
560.3.f.a.321.1		2	20.3	even	4
560.3.f.a.321.2		2	140.83	odd	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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +4.47214 q^{3} +3.00000 q^{4} -4.47214i q^{6} +7.00000i q^{7} -7.00000i q^{8} +11.0000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +4.47214 q^{3} +3.00000 q^{4} -4.47214i q^{6} +7.00000i q^{7} -7.00000i q^{8} +11.0000 q^{9} +2.00000 q^{11} +13.4164 q^{12} -13.4164 q^{13} +7.00000 q^{14} +5.00000 q^{16} -26.8328 q^{17} -11.0000i q^{18} +13.4164i q^{19} +31.3050i q^{21} -2.00000i q^{22} -26.0000i q^{23} -31.3050i q^{24} +13.4164i q^{26} +8.94427 q^{27} +21.0000i q^{28} +22.0000 q^{29} -53.6656i q^{31} -33.0000i q^{32} +8.94427 q^{33} +26.8328i q^{34} +33.0000 q^{36} +14.0000i q^{37} +13.4164 q^{38} -60.0000 q^{39} -26.8328i q^{41} +31.3050 q^{42} +34.0000i q^{43} +6.00000 q^{44} -26.0000 q^{46} +26.8328 q^{47} +22.3607 q^{48} -49.0000 q^{49} -120.000 q^{51} -40.2492 q^{52} +34.0000i q^{53} -8.94427i q^{54} +49.0000 q^{56} +60.0000i q^{57} -22.0000i q^{58} +40.2492i q^{59} +93.9149i q^{61} -53.6656 q^{62} +77.0000i q^{63} -13.0000 q^{64} -8.94427i q^{66} +14.0000i q^{67} -80.4984 q^{68} -116.276i q^{69} +62.0000 q^{71} -77.0000i q^{72} -53.6656 q^{73} +14.0000 q^{74} +40.2492i q^{76} +14.0000i q^{77} +60.0000i q^{78} -38.0000 q^{79} -59.0000 q^{81} -26.8328 q^{82} +40.2492 q^{83} +93.9149i q^{84} +34.0000 q^{86} +98.3870 q^{87} -14.0000i q^{88} +26.8328i q^{89} -93.9149i q^{91} -78.0000i q^{92} -240.000i q^{93} -26.8328i q^{94} -147.580i q^{96} +26.8328 q^{97} +49.0000i q^{98} +22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{4} + 44 q^{9}+O(q^{10}) \) 4 * q + 12 * q^4 + 44 * q^9	\( 4 q + 12 q^{4} + 44 q^{9} + 8 q^{11} + 28 q^{14} + 20 q^{16} + 88 q^{29} + 132 q^{36} - 240 q^{39} + 24 q^{44} - 104 q^{46} - 196 q^{49} - 480 q^{51} + 196 q^{56} - 52 q^{64} + 248 q^{71} + 56 q^{74} - 152 q^{79} - 236 q^{81} + 136 q^{86} + 88 q^{99}+O(q^{100}) \) 4 * q + 12 * q^4 + 44 * q^9 + 8 * q^11 + 28 * q^14 + 20 * q^16 + 88 * q^29 + 132 * q^36 - 240 * q^39 + 24 * q^44 - 104 * q^46 - 196 * q^49 - 480 * q^51 + 196 * q^56 - 52 * q^64 + 248 * q^71 + 56 * q^74 - 152 * q^79 - 236 * q^81 + 136 * q^86 + 88 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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