LMFDB - Embedded newform 175.6.b.c.99.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,6,Mod(99,175)]

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, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	175.b (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:	$28.0671684673$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{57})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 196 $ x^4 + 29*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.4
Root			$3.27492i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.6.b.c.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.27492i q^{2} +25.6495i q^{3} -36.4743 q^{4} -212.248 q^{6} +49.0000i q^{7} -37.0241i q^{8} -414.897 q^{9} +O(q^{10})$	\(q+8.27492i q^{2} +25.6495i q^{3} -36.4743 q^{4} -212.248 q^{6} +49.0000i q^{7} -37.0241i q^{8} -414.897 q^{9} -270.090 q^{11} -935.547i q^{12} -300.640i q^{13} -405.471 q^{14} -860.805 q^{16} +613.106i q^{17} -3433.24i q^{18} +1700.95 q^{19} -1256.83 q^{21} -2234.97i q^{22} -3188.15i q^{23} +949.650 q^{24} +2487.77 q^{26} -4409.07i q^{27} -1787.24i q^{28} -4299.28 q^{29} +2028.46 q^{31} -8307.86i q^{32} -6927.67i q^{33} -5073.40 q^{34} +15133.1 q^{36} +5154.46i q^{37} +14075.2i q^{38} +7711.26 q^{39} -7146.21 q^{41} -10400.1i q^{42} +19584.3i q^{43} +9851.32 q^{44} +26381.7 q^{46} +19998.4i q^{47} -22079.2i q^{48} -2401.00 q^{49} -15725.9 q^{51} +10965.6i q^{52} -3948.82i q^{53} +36484.7 q^{54} +1814.18 q^{56} +43628.5i q^{57} -35576.2i q^{58} +29707.6 q^{59} -50519.3 q^{61} +16785.3i q^{62} -20330.0i q^{63} +41201.1 q^{64} +57325.9 q^{66} +5053.56i q^{67} -22362.6i q^{68} +81774.5 q^{69} +32853.3 q^{71} +15361.2i q^{72} +11115.0i q^{73} -42652.7 q^{74} -62040.8 q^{76} -13234.4i q^{77} +63810.0i q^{78} -81889.4 q^{79} +12270.6 q^{81} -59134.3i q^{82} -118234. i q^{83} +45841.8 q^{84} -162058. q^{86} -110274. i q^{87} +9999.83i q^{88} +41695.4 q^{89} +14731.3 q^{91} +116286. i q^{92} +52028.9i q^{93} -165485. q^{94} +213092. q^{96} +43682.8i q^{97} -19868.1i q^{98} +112059. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4} - 396 q^{6} - 1116 q^{9}+O(q^{10}) $ 4 * q - 10 * q^4 - 396 * q^6 - 1116 * q^9	$ 4 q - 10 q^{4} - 396 q^{6} - 1116 q^{9} + 792 q^{11} - 882 q^{14} + 226 q^{16} + 6532 q^{19} - 588 q^{21} + 3708 q^{24} + 4032 q^{26} - 13392 q^{29} - 40 q^{31} - 11868 q^{34} + 21258 q^{36} + 40992 q^{39} - 12096 q^{41} + 61632 q^{44} + 51168 q^{46} - 9604 q^{49} + 15192 q^{51} + 75816 q^{54} - 882 q^{56} + 87876 q^{59} - 129508 q^{61} + 141566 q^{64} + 133632 q^{66} + 206784 q^{69} + 194832 q^{71} - 86868 q^{74} - 25564 q^{76} - 102512 q^{79} - 122148 q^{81} + 152292 q^{84} - 300096 q^{86} - 168552 q^{89} - 34300 q^{91} - 318936 q^{94} + 511812 q^{96} + 33480 q^{99}+O(q^{100}) $ 4 * q - 10 * q^4 - 396 * q^6 - 1116 * q^9 + 792 * q^11 - 882 * q^14 + 226 * q^16 + 6532 * q^19 - 588 * q^21 + 3708 * q^24 + 4032 * q^26 - 13392 * q^29 - 40 * q^31 - 11868 * q^34 + 21258 * q^36 + 40992 * q^39 - 12096 * q^41 + 61632 * q^44 + 51168 * q^46 - 9604 * q^49 + 15192 * q^51 + 75816 * q^54 - 882 * q^56 + 87876 * q^59 - 129508 * q^61 + 141566 * q^64 + 133632 * q^66 + 206784 * q^69 + 194832 * q^71 - 86868 * q^74 - 25564 * q^76 - 102512 * q^79 - 122148 * q^81 + 152292 * q^84 - 300096 * q^86 - 168552 * q^89 - 34300 * q^91 - 318936 * q^94 + 511812 * q^96 + 33480 * 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 $

$2$	8.27492i	1.46281i	0.681942	+	0.731406i	$0.261136\pi$
$2$	8.27492i	1.46281i	−0.681942	+	0.731406i	$0.738864\pi$
$3$	25.6495i	1.64542i	0.568464	+	0.822708i	$0.307538\pi$
$3$	25.6495i	1.64542i	−0.568464	+	0.822708i	$0.692462\pi$
$4$	−36.4743	−1.13982
$5$	0	0
$5$	0	0
$6$	−212.248	−2.40694
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	− 37.0241i	− 0.204531i
$9$	−414.897	−1.70740
$10$	0	0
$11$	−270.090	−0.673018	−0.336509	−	0.941680i	$-0.609246\pi$
$11$	−270.090	−0.673018	−0.336509	+	0.941680i	$0.609246\pi$
$12$	− 935.547i	− 1.87548i
$13$	− 300.640i	− 0.493387i	−0.969094	−	0.246694i	$-0.920656\pi$
$13$	− 300.640i	− 0.493387i	0.969094	−	0.246694i	$-0.0793441\pi$
$14$	−405.471	−0.552891
$15$	0	0
$16$	−860.805	−0.840630
$17$	613.106i	0.514533i	0.966340	+	0.257267i	$0.0828219\pi$
$17$	613.106i	0.514533i	−0.966340	+	0.257267i	$0.917178\pi$
$18$	− 3433.24i	− 2.49760i
$19$	1700.95	1.08095	0.540477	−	0.841359i	$-0.318244\pi$
$19$	1700.95	1.08095	0.540477	+	0.841359i	$0.318244\pi$
$20$	0	0
$21$	−1256.83	−0.621909
$22$	− 2234.97i	− 0.984498i
$23$	− 3188.15i	− 1.25667i	−0.777945	−	0.628333i	$-0.783738\pi$
$23$	− 3188.15i	− 1.25667i	0.777945	−	0.628333i	$-0.216262\pi$
$24$	949.650	0.336539
$25$	0	0
$26$	2487.77	0.721733
$27$	− 4409.07i	− 1.16396i
$28$	− 1787.24i	− 0.430812i
$29$	−4299.28	−0.949294	−0.474647	−	0.880176i	$-0.657424\pi$
$29$	−4299.28	−0.949294	−0.474647	+	0.880176i	$0.657424\pi$
$30$	0	0
$31$	2028.46	0.379106	0.189553	−	0.981870i	$-0.439296\pi$
$31$	2028.46	0.379106	0.189553	+	0.981870i	$0.439296\pi$
$32$	− 8307.86i	− 1.43421i
$33$	− 6927.67i	− 1.10739i
$34$	−5073.40	−0.752666
$35$	0	0
$36$	15133.1	1.94612
$37$	5154.46i	0.618983i	0.950902	+	0.309491i	$0.100159\pi$
$37$	5154.46i	0.618983i	−0.950902	+	0.309491i	$0.899841\pi$
$38$	14075.2i	1.58123i
$39$	7711.26	0.811827
$40$	0	0
$41$	−7146.21	−0.663921	−0.331960	−	0.943293i	$-0.607710\pi$
$41$	−7146.21	−0.663921	−0.331960	+	0.943293i	$0.607710\pi$
$42$	− 10400.1i	− 0.909736i
$43$	19584.3i	1.61524i	0.589703	+	0.807620i	$0.299245\pi$
$43$	19584.3i	1.61524i	−0.589703	+	0.807620i	$0.700755\pi$
$44$	9851.32	0.767119
$45$	0	0
$46$	26381.7	1.83827
$47$	19998.4i	1.32054i	0.751030	+	0.660268i	$0.229557\pi$
$47$	19998.4i	1.32054i	−0.751030	+	0.660268i	$0.770443\pi$
$48$	− 22079.2i	− 1.38319i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−15725.9	−0.846621
$52$	10965.6i	0.562373i
$53$	− 3948.82i	− 0.193098i	−0.995328	−	0.0965489i	$-0.969220\pi$
$53$	− 3948.82i	− 0.193098i	0.995328	−	0.0965489i	$-0.0307804\pi$
$54$	36484.7	1.70265
$55$	0	0
$56$	1814.18	0.0773055
$57$	43628.5i	1.77862i
$58$	− 35576.2i	− 1.38864i
$59$	29707.6	1.11106	0.555530	−	0.831497i	$-0.312516\pi$
$59$	29707.6	1.11106	0.555530	+	0.831497i	$0.312516\pi$
$60$	0	0
$61$	−50519.3	−1.73833	−0.869165	−	0.494522i	$-0.835343\pi$
$61$	−50519.3	−1.73833	−0.869165	+	0.494522i	$0.835343\pi$
$62$	16785.3i	0.554562i
$63$	− 20330.0i	− 0.645335i
$64$	41201.1	1.25736
$65$	0	0
$66$	57325.9	1.61991
$67$	5053.56i	0.137534i	0.997633	+	0.0687671i	$0.0219065\pi$
$67$	5053.56i	0.137534i	−0.997633	+	0.0687671i	$0.978093\pi$
$68$	− 22362.6i	− 0.586476i
$69$	81774.5	2.06774
$70$	0	0
$71$	32853.3	0.773453	0.386726	−	0.922195i	$-0.373606\pi$
$71$	32853.3	0.773453	0.386726	+	0.922195i	$0.373606\pi$
$72$	15361.2i	0.349215i
$73$	11115.0i	0.244119i	0.992523	+	0.122059i	$0.0389498\pi$
$73$	11115.0i	0.244119i	−0.992523	+	0.122059i	$0.961050\pi$
$74$	−42652.7	−0.905456
$75$	0	0
$76$	−62040.8	−1.23209
$77$	− 13234.4i	− 0.254377i
$78$	63810.0i	1.18755i
$79$	−81889.4	−1.47625	−0.738125	−	0.674664i	$-0.764288\pi$
$79$	−81889.4	−1.47625	−0.738125	+	0.674664i	$0.764288\pi$
$80$	0	0
$81$	12270.6	0.207803
$82$	− 59134.3i	− 0.971191i
$83$	− 118234.i	− 1.88385i	−0.335819	−	0.941926i	$-0.609013\pi$
$83$	− 118234.i	− 1.88385i	0.335819	−	0.941926i	$-0.390987\pi$
$84$	45841.8	0.708865
$85$	0	0
$86$	−162058.	−2.36279
$87$	− 110274.i	− 1.56198i
$88$	9999.83i	0.137653i
$89$	41695.4	0.557972	0.278986	−	0.960295i	$-0.410002\pi$
$89$	41695.4	0.557972	0.278986	+	0.960295i	$0.410002\pi$
$90$	0	0
$91$	14731.3	0.186483
$92$	116286.i	1.43237i
$93$	52028.9i	0.623788i
$94$	−165485.	−1.93170
$95$	0	0
$96$	213092.	2.35988
$97$	43682.8i	0.471391i	0.971827	+	0.235695i	$0.0757367\pi$
$97$	43682.8i	0.471391i	−0.971827	+	0.235695i	$0.924263\pi$
$98$	− 19868.1i	− 0.208973i
$99$	112059.	1.14911
$100$	0	0
$101$	25648.1	0.250179	0.125090	−	0.992145i	$-0.460078\pi$
$101$	25648.1	0.250179	0.125090	+	0.992145i	$0.460078\pi$
$102$	− 130130.i	− 1.23845i
$103$	14320.0i	0.133000i	0.997786	+	0.0664999i	$0.0211832\pi$
$103$	14320.0i	0.133000i	−0.997786	+	0.0664999i	$0.978817\pi$
$104$	−11130.9	−0.100913
$105$	0	0
$106$	32676.1	0.282466
$107$	17201.8i	0.145249i	0.997359	+	0.0726247i	$0.0231375\pi$
$107$	17201.8i	0.145249i	−0.997359	+	0.0726247i	$0.976862\pi$
$108$	160818.i	1.32670i
$109$	86017.6	0.693459	0.346730	−	0.937965i	$-0.387292\pi$
$109$	86017.6	0.693459	0.346730	+	0.937965i	$0.387292\pi$
$110$	0	0
$111$	−132209.	−1.01848
$112$	− 42179.4i	− 0.317728i
$113$	− 137568.i	− 1.01349i	−0.862096	−	0.506745i	$-0.830848\pi$
$113$	− 137568.i	− 1.01349i	0.862096	−	0.506745i	$-0.169152\pi$
$114$	−361022.	−2.60179
$115$	0	0
$116$	156813.	1.08202
$117$	124734.i	0.842407i
$118$	245828.i	1.62527i
$119$	−30042.2	−0.194475
$120$	0	0
$121$	−88102.5	−0.547047
$122$	− 418043.i	− 2.54285i
$123$	− 183297.i	− 1.09243i
$124$	−73986.4	−0.432113
$125$	0	0
$126$	168229.	0.944004
$127$	− 70567.1i	− 0.388233i	−0.980978	−	0.194117i	$-0.937816\pi$
$127$	− 70567.1i	− 0.388233i	0.980978	−	0.194117i	$-0.0621840\pi$
$128$	75084.2i	0.405064i
$129$	−502328.	−2.65774
$130$	0	0
$131$	−173712.	−0.884408	−0.442204	−	0.896914i	$-0.645803\pi$
$131$	−173712.	−0.884408	−0.442204	+	0.896914i	$0.645803\pi$
$132$	252682.i	1.26223i
$133$	83346.5i	0.408562i
$134$	−41817.8	−0.201187
$135$	0	0
$136$	22699.7	0.105238
$137$	− 1989.94i	− 0.00905813i	−0.999990	−	0.00452907i	$-0.998558\pi$
$137$	− 1989.94i	− 0.00905813i	0.999990	−	0.00452907i	$-0.00144165\pi$
$138$	676678.i	3.02471i
$139$	−366409.	−1.60853	−0.804264	−	0.594272i	$-0.797440\pi$
$139$	−366409.	−1.60853	−0.804264	+	0.594272i	$0.797440\pi$
$140$	0	0
$141$	−512949.	−2.17283
$142$	271859.i	1.13142i
$143$	81199.7i	0.332058i
$144$	357145.	1.43529
$145$	0	0
$146$	−91975.4	−0.357100
$147$	− 61584.5i	− 0.235059i
$148$	− 188005.i	− 0.705529i
$149$	−140719.	−0.519261	−0.259631	−	0.965708i	$-0.583601\pi$
$149$	−140719.	−0.519261	−0.259631	+	0.965708i	$0.583601\pi$
$150$	0	0
$151$	50064.6	0.178685	0.0893425	−	0.996001i	$-0.471523\pi$
$151$	50064.6	0.178685	0.0893425	+	0.996001i	$0.471523\pi$
$152$	− 62976.1i	− 0.221089i
$153$	− 254376.i	− 0.878512i
$154$	109514.	0.372105
$155$	0	0
$156$	−281262.	−0.925337
$157$	− 89794.6i	− 0.290738i	−0.989378	−	0.145369i	$-0.953563\pi$
$157$	− 89794.6i	− 0.290738i	0.989378	−	0.145369i	$-0.0464369\pi$
$158$	− 677628.i	− 2.15948i
$159$	101285.	0.317726
$160$	0	0
$161$	156219.	0.474975
$162$	101538.i	0.303977i
$163$	481230.i	1.41868i	0.704867	+	0.709339i	$0.251006\pi$
$163$	481230.i	1.41868i	−0.704867	+	0.709339i	$0.748994\pi$
$164$	260653.	0.756750
$165$	0	0
$166$	978376.	2.75572
$167$	− 86572.7i	− 0.240209i	−0.992761	−	0.120105i	$-0.961677\pi$
$167$	− 86572.7i	− 0.240209i	0.992761	−	0.120105i	$-0.0383230\pi$
$168$	46532.8i	0.127200i
$169$	280909.	0.756569
$170$	0	0
$171$	−705718.	−1.84562
$172$	− 714323.i	− 1.84108i
$173$	58137.4i	0.147686i	0.997270	+	0.0738432i	$0.0235264\pi$
$173$	58137.4i	0.147686i	−0.997270	+	0.0738432i	$0.976474\pi$
$174$	912511.	2.28489
$175$	0	0
$176$	232495.	0.565759
$177$	761985.i	1.82816i
$178$	345026.i	0.816209i
$179$	209380.	0.488431	0.244215	−	0.969721i	$-0.421470\pi$
$179$	209380.	0.488431	0.244215	+	0.969721i	$0.421470\pi$
$180$	0	0
$181$	278996.	0.632996	0.316498	−	0.948593i	$-0.397493\pi$
$181$	278996.	0.632996	0.316498	+	0.948593i	$0.397493\pi$
$182$	121901.i	0.272789i
$183$	− 1.29579e6i	− 2.86028i
$184$	−118038.	−0.257027
$185$	0	0
$186$	−430535.	−0.912485
$187$	− 165594.i	− 0.346290i
$188$	− 729426.i	− 1.50517i
$189$	216045.	0.439935
$190$	0	0
$191$	−445132.	−0.882888	−0.441444	−	0.897289i	$-0.645534\pi$
$191$	−445132.	−0.882888	−0.441444	+	0.897289i	$0.645534\pi$
$192$	1.05679e6i	2.06888i
$193$	726811.i	1.40452i	0.711920	+	0.702260i	$0.247826\pi$
$193$	726811.i	1.40452i	−0.711920	+	0.702260i	$0.752174\pi$
$194$	−361471.	−0.689556
$195$	0	0
$196$	87574.7	0.162831
$197$	− 364897.i	− 0.669892i	−0.942237	−	0.334946i	$-0.891282\pi$
$197$	− 364897.i	− 0.669892i	0.942237	−	0.334946i	$-0.108718\pi$
$198$	927282.i	1.68093i
$199$	−289307.	−0.517877	−0.258938	−	0.965894i	$-0.583373\pi$
$199$	−289307.	−0.517877	−0.258938	+	0.965894i	$0.583373\pi$
$200$	0	0
$201$	−129621.	−0.226301
$202$	212236.i	0.365965i
$203$	− 210665.i	− 0.358799i
$204$	573589.	0.964997
$205$	0	0
$206$	−118497.	−0.194554
$207$	1.32276e6i	2.14562i
$208$	258792.i	0.414756i
$209$	−459409.	−0.727501
$210$	0	0
$211$	750147.	1.15995	0.579976	−	0.814633i	$-0.303062\pi$
$211$	750147.	1.15995	0.579976	+	0.814633i	$0.303062\pi$
$212$	144030.i	0.220097i
$213$	842672.i	1.27265i
$214$	−142343.	−0.212473
$215$	0	0
$216$	−163242.	−0.238066
$217$	99394.3i	0.143289i
$218$	711788.i	1.01440i
$219$	−285093.	−0.401677
$220$	0	0
$221$	184324.	0.253864
$222$	− 1.09402e6i	− 1.48985i
$223$	− 534398.i	− 0.719619i	−0.933026	−	0.359810i	$-0.882842\pi$
$223$	− 534398.i	− 0.719619i	0.933026	−	0.359810i	$-0.117158\pi$
$224$	407085.	0.542082
$225$	0	0
$226$	1.13836e6	1.48255
$227$	− 410624.i	− 0.528907i	−0.964398	−	0.264453i	$-0.914808\pi$
$227$	− 410624.i	− 0.528907i	0.964398	−	0.264453i	$-0.0851916\pi$
$228$	− 1.59132e6i	− 2.02731i
$229$	−1.03036e6	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$229$	−1.03036e6	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$230$	0	0
$231$	339456.	0.418556
$232$	159177.i	0.194160i
$233$	119211.i	0.143856i	0.997410	+	0.0719278i	$0.0229151\pi$
$233$	119211.i	0.143856i	−0.997410	+	0.0719278i	$0.977085\pi$
$234$	−1.03217e6	−1.23228
$235$	0	0
$236$	−1.08356e6	−1.26641
$237$	− 2.10042e6i	− 2.42905i
$238$	− 248597.i	− 0.284481i
$239$	254090.	0.287735	0.143868	−	0.989597i	$-0.454046\pi$
$239$	254090.	0.287735	0.143868	+	0.989597i	$0.454046\pi$
$240$	0	0
$241$	1.41251e6	1.56656	0.783282	−	0.621667i	$-0.213544\pi$
$241$	1.41251e6	1.56656	0.783282	+	0.621667i	$0.213544\pi$
$242$	− 729041.i	− 0.800228i
$243$	− 756671.i	− 0.822037i
$244$	1.84265e6	1.98138
$245$	0	0
$246$	1.51677e6	1.59801
$247$	− 511372.i	− 0.533329i
$248$	− 75101.7i	− 0.0775391i
$249$	3.03264e6	3.09972
$250$	0	0
$251$	−1.67542e6	−1.67857	−0.839286	−	0.543690i	$-0.817027\pi$
$251$	−1.67542e6	−1.67857	−0.839286	+	0.543690i	$0.817027\pi$
$252$	741520.i	0.735566i
$253$	861087.i	0.845758i
$254$	583937.	0.567913
$255$	0	0
$256$	697120.	0.664825
$257$	726996.i	0.686593i	0.939227	+	0.343296i	$0.111544\pi$
$257$	726996.i	0.686593i	−0.939227	+	0.343296i	$0.888456\pi$
$258$	− 4.15672e6i	− 3.88778i
$259$	−252568.	−0.233953
$260$	0	0
$261$	1.78376e6	1.62082
$262$	− 1.43746e6i	− 1.29372i
$263$	225880.i	0.201367i	0.994918	+	0.100684i	$0.0321030\pi$
$263$	225880.i	0.201367i	−0.994918	+	0.100684i	$0.967897\pi$
$264$	−256491.	−0.226497
$265$	0	0
$266$	−689685.	−0.597650
$267$	1.06947e6i	0.918097i
$268$	− 184325.i	− 0.156764i
$269$	−1.80527e6	−1.52111	−0.760557	−	0.649272i	$-0.775074\pi$
$269$	−1.80527e6	−1.52111	−0.760557	+	0.649272i	$0.775074\pi$
$270$	0	0
$271$	−1.71380e6	−1.41754	−0.708771	−	0.705439i	$-0.750750\pi$
$271$	−1.71380e6	−1.41754	−0.708771	+	0.705439i	$0.750750\pi$
$272$	− 527765.i	− 0.432532i
$273$	377852.i	0.306842i
$274$	16466.6	0.0132504
$275$	0	0
$276$	−2.98267e6	−2.35685
$277$	2.23055e6i	1.74668i	0.487115	+	0.873338i	$0.338049\pi$
$277$	2.23055e6i	1.74668i	−0.487115	+	0.873338i	$0.661951\pi$
$278$	− 3.03200e6i	− 2.35298i
$279$	−841600.	−0.647285
$280$	0	0
$281$	1.67140e6	1.26274	0.631371	−	0.775481i	$-0.282493\pi$
$281$	1.67140e6	1.26274	0.631371	+	0.775481i	$0.282493\pi$
$282$	− 4.24461e6i	− 3.17845i
$283$	396152.i	0.294033i	0.989134	+	0.147016i	$0.0469670\pi$
$283$	396152.i	0.294033i	−0.989134	+	0.147016i	$0.953033\pi$
$284$	−1.19830e6	−0.881597
$285$	0	0
$286$	−671920.	−0.485739
$287$	− 350164.i	− 0.250938i
$288$	3.44691e6i	2.44877i
$289$	1.04396e6	0.735256
$290$	0	0
$291$	−1.12044e6	−0.775634
$292$	− 405410.i	− 0.278251i
$293$	929465.i	0.632505i	0.948675	+	0.316252i	$0.102425\pi$
$293$	929465.i	0.632505i	−0.948675	+	0.316252i	$0.897575\pi$
$294$	509606.	0.343848
$295$	0	0
$296$	190839.	0.126601
$297$	1.19085e6i	0.783365i
$298$	− 1.16443e6i	− 0.759582i
$299$	−958485.	−0.620022
$300$	0	0
$301$	−959631.	−0.610503
$302$	414280.i	0.261383i
$303$	657860.i	0.411649i
$304$	−1.46418e6	−0.908682
$305$	0	0
$306$	2.10494e6	1.28510
$307$	1.83295e6i	1.10995i	0.831866	+	0.554976i	$0.187273\pi$
$307$	1.83295e6i	1.10995i	−0.831866	+	0.554976i	$0.812727\pi$
$308$	482715.i	0.289944i
$309$	−367302.	−0.218840
$310$	0	0
$311$	−2.29685e6	−1.34658	−0.673289	−	0.739379i	$-0.735119\pi$
$311$	−2.29685e6	−1.34658	−0.673289	+	0.739379i	$0.735119\pi$
$312$	− 285502.i	− 0.166044i
$313$	3.42470e6i	1.97589i	0.154817	+	0.987943i	$0.450521\pi$
$313$	3.42470e6i	1.97589i	−0.154817	+	0.987943i	$0.549479\pi$
$314$	743043.	0.425295
$315$	0	0
$316$	2.98685e6	1.68266
$317$	2.94305e6i	1.64494i	0.568808	+	0.822470i	$0.307405\pi$
$317$	2.94305e6i	1.64494i	−0.568808	+	0.822470i	$0.692595\pi$
$318$	838127.i	0.464774i
$319$	1.16119e6	0.638891
$320$	0	0
$321$	−441217.	−0.238996
$322$	1.29270e6i	0.694799i
$323$	1.04286e6i	0.556187i
$324$	−447560.	−0.236858
$325$	0	0
$326$	−3.98214e6	−2.07526
$327$	2.20631e6i	1.14103i
$328$	264582.i	0.135792i
$329$	−979921.	−0.499116
$330$	0	0
$331$	966164.	0.484709	0.242354	−	0.970188i	$-0.422080\pi$
$331$	966164.	0.484709	0.242354	+	0.970188i	$0.422080\pi$
$332$	4.31250e6i	2.14725i
$333$	− 2.13857e6i	− 1.05685i
$334$	716382.	0.351381
$335$	0	0
$336$	1.08188e6	0.522795
$337$	136417.i	0.0654327i	0.999465	+	0.0327163i	$0.0104158\pi$
$337$	136417.i	0.0654327i	−0.999465	+	0.0327163i	$0.989584\pi$
$338$	2.32450e6i	1.10672i
$339$	3.52854e6	1.66761
$340$	0	0
$341$	−547865.	−0.255145
$342$	− 5.83976e6i	− 2.69979i
$343$	− 117649.i	− 0.0539949i
$344$	725091.	0.330367
$345$	0	0
$346$	−481082.	−0.216038
$347$	355408.i	0.158454i	0.996857	+	0.0792270i	$0.0252452\pi$
$347$	355408.i	0.158454i	−0.996857	+	0.0792270i	$0.974755\pi$
$348$	4.02218e6i	1.78038i
$349$	140128.	0.0615830	0.0307915	−	0.999526i	$-0.490197\pi$
$349$	140128.	0.0615830	0.0307915	+	0.999526i	$0.490197\pi$
$350$	0	0
$351$	−1.32554e6	−0.574283
$352$	2.24387e6i	0.965252i
$353$	− 3.48141e6i	− 1.48703i	−0.668721	−	0.743514i	$-0.733158\pi$
$353$	− 3.48141e6i	− 1.48703i	0.668721	−	0.743514i	$-0.266842\pi$
$354$	−6.30536e6	−2.67425
$355$	0	0
$356$	−1.52081e6	−0.635988
$357$	− 770568.i	− 0.319993i
$358$	1.73260e6i	0.714482i
$359$	−1.75285e6	−0.717810	−0.358905	−	0.933374i	$-0.616850\pi$
$359$	−1.75285e6	−0.717810	−0.358905	+	0.933374i	$0.616850\pi$
$360$	0	0
$361$	417127.	0.168461
$362$	2.30867e6i	0.925955i
$363$	− 2.25979e6i	− 0.900121i
$364$	−537315.	−0.212557
$365$	0	0
$366$	1.07226e7	4.18405
$367$	− 1.76939e6i	− 0.685738i	−0.939383	−	0.342869i	$-0.888601\pi$
$367$	− 1.76939e6i	− 0.685738i	0.939383	−	0.342869i	$-0.111399\pi$
$368$	2.74438e6i	1.05639i
$369$	2.96494e6	1.13357
$370$	0	0
$371$	193492.	0.0729841
$372$	− 1.89771e6i	− 0.711006i
$373$	4.16212e6i	1.54897i	0.632592	+	0.774485i	$0.281991\pi$
$373$	4.16212e6i	1.54897i	−0.632592	+	0.774485i	$0.718009\pi$
$374$	1.37027e6	0.506557
$375$	0	0
$376$	740422.	0.270091
$377$	1.29253e6i	0.468369i
$378$	1.78775e6i	0.643543i
$379$	−618163.	−0.221057	−0.110529	−	0.993873i	$-0.535254\pi$
$379$	−618163.	−0.221057	−0.110529	+	0.993873i	$0.535254\pi$
$380$	0	0
$381$	1.81001e6	0.638805
$382$	− 3.68343e6i	− 1.29150i
$383$	4.11163e6i	1.43225i	0.697974	+	0.716123i	$0.254085\pi$
$383$	4.11163e6i	1.43225i	−0.697974	+	0.716123i	$0.745915\pi$
$384$	−1.92587e6	−0.666498
$385$	0	0
$386$	−6.01430e6	−2.05455
$387$	− 8.12547e6i	− 2.75785i
$388$	− 1.59330e6i	− 0.537301i
$389$	−4.62076e6	−1.54824	−0.774122	−	0.633037i	$-0.781808\pi$
$389$	−4.62076e6	−1.54824	−0.774122	+	0.633037i	$0.781808\pi$
$390$	0	0
$391$	1.95468e6	0.646596
$392$	88894.8i	0.0292187i
$393$	− 4.45564e6i	− 1.45522i
$394$	3.01949e6	0.979926
$395$	0	0
$396$	−4.08728e6	−1.30978
$397$	5.07349e6i	1.61559i	0.589465	+	0.807794i	$0.299339\pi$
$397$	5.07349e6i	1.61559i	−0.589465	+	0.807794i	$0.700661\pi$
$398$	− 2.39399e6i	− 0.757557i
$399$	−2.13780e6	−0.672255
$400$	0	0
$401$	−1.48056e6	−0.459795	−0.229898	−	0.973215i	$-0.573839\pi$
$401$	−1.48056e6	−0.459795	−0.229898	+	0.973215i	$0.573839\pi$
$402$	− 1.07261e6i	− 0.331036i
$403$	− 609834.i	− 0.187046i
$404$	−935495.	−0.285160
$405$	0	0
$406$	1.74323e6	0.524856
$407$	− 1.39217e6i	− 0.416586i
$408$	582236.i	0.173160i
$409$	4.53379e6	1.34015	0.670075	−	0.742294i	$-0.266262\pi$
$409$	4.53379e6	1.34015	0.670075	+	0.742294i	$0.266262\pi$
$410$	0	0
$411$	51041.0	0.0149044
$412$	− 522313.i	− 0.151596i
$413$	1.45567e6i	0.419941i
$414$	−1.09457e7	−3.13865
$415$	0	0
$416$	−2.49767e6	−0.707623
$417$	− 9.39820e6i	− 2.64670i
$418$	− 3.80157e6i	− 1.06420i
$419$	−111026.	−0.0308952	−0.0154476	−	0.999881i	$-0.504917\pi$
$419$	−111026.	−0.0308952	−0.0154476	+	0.999881i	$0.504917\pi$
$420$	0	0
$421$	−1.41151e6	−0.388132	−0.194066	−	0.980988i	$-0.562168\pi$
$421$	−1.41151e6	−0.388132	−0.194066	+	0.980988i	$0.562168\pi$
$422$	6.20740e6i	1.69679i
$423$	− 8.29727e6i	− 2.25468i
$424$	−146201.	−0.0394945
$425$	0	0
$426$	−6.97304e6	−1.86165
$427$	− 2.47544e6i	− 0.657027i
$428$	− 627422.i	− 0.165558i
$429$	−2.08273e6	−0.546374
$430$	0	0
$431$	1.07640e6	0.279113	0.139557	−	0.990214i	$-0.455432\pi$
$431$	1.07640e6	0.279113	0.139557	+	0.990214i	$0.455432\pi$
$432$	3.79535e6i	0.978459i
$433$	310172.i	0.0795029i	0.999210	+	0.0397515i	$0.0126566\pi$
$433$	310172.i	0.0795029i	−0.999210	+	0.0397515i	$0.987343\pi$
$434$	−822480.	−0.209605
$435$	0	0
$436$	−3.13743e6	−0.790419
$437$	− 5.42288e6i	− 1.35840i
$438$	− 2.35912e6i	− 0.587578i
$439$	−5.67650e6	−1.40579	−0.702893	−	0.711296i	$-0.748109\pi$
$439$	−5.67650e6	−1.40579	−0.702893	+	0.711296i	$0.748109\pi$
$440$	0	0
$441$	996168.	0.243914
$442$	1.52527e6i	0.371356i
$443$	− 4.05966e6i	− 0.982834i	−0.870924	−	0.491417i	$-0.836479\pi$
$443$	− 4.05966e6i	− 0.982834i	0.870924	−	0.491417i	$-0.163521\pi$
$444$	4.82223e6	1.16089
$445$	0	0
$446$	4.42210e6	1.05267
$447$	− 3.60936e6i	− 0.854401i
$448$	2.01885e6i	0.475237i
$449$	6.96544e6	1.63054	0.815272	−	0.579078i	$-0.196587\pi$
$449$	6.96544e6	1.63054	0.815272	+	0.579078i	$0.196587\pi$
$450$	0	0
$451$	1.93012e6	0.446830
$452$	5.01767e6i	1.15520i
$453$	1.28413e6i	0.294011i
$454$	3.39788e6	0.773692
$455$	0	0
$456$	1.61530e6	0.363783
$457$	1.79523e6i	0.402096i	0.979581	+	0.201048i	$0.0644347\pi$
$457$	1.79523e6i	0.402096i	−0.979581	+	0.201048i	$0.935565\pi$
$458$	− 8.52616e6i	− 1.89928i
$459$	2.70323e6	0.598896
$460$	0	0
$461$	−2.11294e6	−0.463058	−0.231529	−	0.972828i	$-0.574373\pi$
$461$	−2.11294e6	−0.463058	−0.231529	+	0.972828i	$0.574373\pi$
$462$	2.80897e6i	0.612268i
$463$	− 1.26223e6i	− 0.273643i	−0.990596	−	0.136822i	$-0.956311\pi$
$463$	− 1.26223e6i	− 0.273643i	0.990596	−	0.136822i	$-0.0436887\pi$
$464$	3.70084e6	0.798005
$465$	0	0
$466$	−986462.	−0.210434
$467$	− 3.58926e6i	− 0.761576i	−0.924662	−	0.380788i	$-0.875653\pi$
$467$	− 3.58926e6i	− 0.761576i	0.924662	−	0.380788i	$-0.124347\pi$
$468$	− 4.54960e6i	− 0.960192i
$469$	−247624.	−0.0519830
$470$	0	0
$471$	2.30319e6	0.478384
$472$	− 1.09990e6i	− 0.227246i
$473$	− 5.28952e6i	− 1.08708i
$474$	1.73808e7	3.55324
$475$	0	0
$476$	1.09577e6	0.221667
$477$	1.63835e6i	0.329694i
$478$	2.10257e6i	0.420903i
$479$	2.41693e6	0.481311	0.240655	−	0.970611i	$-0.422638\pi$
$479$	2.41693e6	0.481311	0.240655	+	0.970611i	$0.422638\pi$
$480$	0	0
$481$	1.54963e6	0.305398
$482$	1.16884e7i	2.29159i
$483$	4.00695e6i	0.781531i
$484$	3.21347e6	0.623536
$485$	0	0
$486$	6.26139e6	1.20249
$487$	− 5.19403e6i	− 0.992388i	−0.868212	−	0.496194i	$-0.834730\pi$
$487$	− 5.19403e6i	− 0.992388i	0.868212	−	0.496194i	$-0.165270\pi$
$488$	1.87043e6i	0.355543i
$489$	−1.23433e7	−2.33432
$490$	0	0
$491$	5.38961e6	1.00891	0.504456	−	0.863437i	$-0.331693\pi$
$491$	5.38961e6	1.00891	0.504456	+	0.863437i	$0.331693\pi$
$492$	6.68561e6i	1.24517i
$493$	− 2.63592e6i	− 0.488443i
$494$	4.23156e6	0.780160
$495$	0	0
$496$	−1.74610e6	−0.318688
$497$	1.60981e6i	0.292338i
$498$	2.50949e7i	4.53431i
$499$	3.29606e6	0.592576	0.296288	−	0.955099i	$-0.404251\pi$
$499$	3.29606e6	0.592576	0.296288	+	0.955099i	$0.404251\pi$
$500$	0	0
$501$	2.22055e6	0.395244
$502$	− 1.38640e7i	− 2.45544i
$503$	1.06512e7i	1.87706i	0.345204	+	0.938528i	$0.387810\pi$
$503$	1.06512e7i	1.87706i	−0.345204	+	0.938528i	$0.612190\pi$
$504$	−752698.	−0.131991
$505$	0	0
$506$	−7.12543e6	−1.23718
$507$	7.20517e6i	1.24487i
$508$	2.57388e6i	0.442516i
$509$	2.74268e6	0.469225	0.234612	−	0.972089i	$-0.424618\pi$
$509$	2.74268e6	0.469225	0.234612	+	0.972089i	$0.424618\pi$
$510$	0	0
$511$	−544633.	−0.0922682
$512$	8.17130e6i	1.37758i
$513$	− 7.49961e6i	− 1.25819i
$514$	−6.01583e6	−1.00436
$515$	0	0
$516$	1.83220e7	3.02935
$517$	− 5.40136e6i	− 0.888744i
$518$	− 2.08998e6i	− 0.342230i
$519$	−1.49120e6	−0.243006
$520$	0	0
$521$	4.97077e6	0.802286	0.401143	−	0.916015i	$-0.368613\pi$
$521$	4.97077e6	0.802286	0.401143	+	0.916015i	$0.368613\pi$
$522$	1.47605e7i	2.37096i
$523$	− 2.41579e6i	− 0.386193i	−0.981180	−	0.193096i	$-0.938147\pi$
$523$	− 2.41579e6i	− 0.386193i	0.981180	−	0.193096i	$-0.0618530\pi$
$524$	6.33603e6	1.00807
$525$	0	0
$526$	−1.86914e6	−0.294563
$527$	1.24366e6i	0.195063i
$528$	5.96337e6i	0.930908i
$529$	−3.72798e6	−0.579207
$530$	0	0
$531$	−1.23256e7	−1.89702
$532$	− 3.04000e6i	− 0.465688i
$533$	2.14843e6i	0.327570i
$534$	−8.84974e6	−1.34300
$535$	0	0
$536$	187103.	0.0281300
$537$	5.37050e6i	0.803672i
$538$	− 1.49385e7i	− 2.22510i
$539$	648485.	0.0961454
$540$	0	0
$541$	472165.	0.0693587	0.0346794	−	0.999398i	$-0.488959\pi$
$541$	472165.	0.0693587	0.0346794	+	0.999398i	$0.488959\pi$
$542$	− 1.41815e7i	− 2.07360i
$543$	7.15610e6i	1.04154i
$544$	5.09360e6	0.737951
$545$	0	0
$546$	−3.12669e6	−0.448852
$547$	7.63716e6i	1.09135i	0.837997	+	0.545675i	$0.183727\pi$
$547$	7.63716e6i	1.09135i	−0.837997	+	0.545675i	$0.816273\pi$
$548$	72581.6i	0.0103246i
$549$	2.09603e7	2.96802
$550$	0	0
$551$	−7.31285e6	−1.02614
$552$	− 3.02763e6i	− 0.422917i
$553$	− 4.01258e6i	− 0.557970i
$554$	−1.84576e7	−2.55506
$555$	0	0
$556$	1.33645e7	1.83343
$557$	− 4.48807e6i	− 0.612946i	−0.951879	−	0.306473i	$-0.900851\pi$
$557$	− 4.48807e6i	− 0.612946i	0.951879	−	0.306473i	$-0.0991489\pi$
$558$	− 6.96417e6i	− 0.946856i
$559$	5.88782e6	0.796938
$560$	0	0
$561$	4.24740e6	0.569791
$562$	1.38307e7i	1.84715i
$563$	2.16500e6i	0.287864i	0.989588	+	0.143932i	$0.0459746\pi$
$563$	2.16500e6i	0.287864i	−0.989588	+	0.143932i	$0.954025\pi$
$564$	1.87094e7	2.47664
$565$	0	0
$566$	−3.27812e6	−0.430115
$567$	601258.i	0.0785422i
$568$	− 1.21637e6i	− 0.158195i
$569$	1.13325e7	1.46739	0.733696	−	0.679478i	$-0.237794\pi$
$569$	1.13325e7	1.46739	0.733696	+	0.679478i	$0.237794\pi$
$570$	0	0
$571$	−843773.	−0.108302	−0.0541509	−	0.998533i	$-0.517245\pi$
$571$	−843773.	−0.108302	−0.0541509	+	0.998533i	$0.517245\pi$
$572$	− 2.96170e6i	− 0.378487i
$573$	− 1.14174e7i	− 1.45272i
$574$	2.89758e6	0.367076
$575$	0	0
$576$	−1.70942e7	−2.14681
$577$	− 2.23784e6i	− 0.279827i	−0.990164	−	0.139914i	$-0.955318\pi$
$577$	− 2.23784e6i	− 0.279827i	0.990164	−	0.139914i	$-0.0446825\pi$
$578$	8.63866e6i	1.07554i
$579$	−1.86423e7	−2.31102
$580$	0	0
$581$	5.79346e6	0.712029
$582$	− 9.27156e6i	− 1.13461i
$583$	1.06653e6i	0.129958i
$584$	411521.	0.0499299
$585$	0	0
$586$	−7.69124e6	−0.925236
$587$	1.21190e7i	1.45168i	0.687864	+	0.725839i	$0.258548\pi$
$587$	1.21190e7i	1.45168i	−0.687864	+	0.725839i	$0.741452\pi$
$588$	2.24625e6i	0.267926i
$589$	3.45030e6	0.409797
$590$	0	0
$591$	9.35942e6	1.10225
$592$	− 4.43698e6i	− 0.520335i
$593$	− 8.00167e6i	− 0.934424i	−0.884145	−	0.467212i	$-0.845258\pi$
$593$	− 8.00167e6i	− 0.934424i	0.884145	−	0.467212i	$-0.154742\pi$
$594$	−9.85415e6	−1.14592
$595$	0	0
$596$	5.13261e6	0.591865
$597$	− 7.42058e6i	− 0.852123i
$598$	− 7.93138e6i	− 0.906976i
$599$	−1.45899e7	−1.66144	−0.830719	−	0.556692i	$-0.812070\pi$
$599$	−1.45899e7	−1.66144	−0.830719	+	0.556692i	$0.812070\pi$
$600$	0	0
$601$	−8.67178e6	−0.979314	−0.489657	−	0.871915i	$-0.662878\pi$
$601$	−8.67178e6	−0.979314	−0.489657	+	0.871915i	$0.662878\pi$
$602$	− 7.94087e6i	− 0.893052i
$603$	− 2.09671e6i	− 0.234825i
$604$	−1.82607e6	−0.203669
$605$	0	0
$606$	−5.44374e6	−0.602166
$607$	− 1.33059e7i	− 1.46580i	−0.680339	−	0.732898i	$-0.738167\pi$
$607$	− 1.33059e7i	− 1.46580i	0.680339	−	0.732898i	$-0.261833\pi$
$608$	− 1.41312e7i	− 1.55032i
$609$	5.40344e6	0.590374
$610$	0	0
$611$	6.01231e6	0.651536
$612$	9.27817e6i	1.00135i
$613$	− 2.35101e6i	− 0.252699i	−0.991986	−	0.126350i	$-0.959674\pi$
$613$	− 2.35101e6i	− 0.252699i	0.991986	−	0.126350i	$-0.0403261\pi$
$614$	−1.51675e7	−1.62365
$615$	0	0
$616$	−489991.	−0.0520280
$617$	9.63523e6i	1.01894i	0.860488	+	0.509470i	$0.170159\pi$
$617$	9.63523e6i	1.01894i	−0.860488	+	0.509470i	$0.829841\pi$
$618$	− 3.03939e6i	− 0.320122i
$619$	4.86148e6	0.509967	0.254983	−	0.966945i	$-0.417930\pi$
$619$	4.86148e6	0.509967	0.254983	+	0.966945i	$0.417930\pi$
$620$	0	0
$621$	−1.40568e7	−1.46271
$622$	− 1.90062e7i	− 1.96979i
$623$	2.04307e6i	0.210894i
$624$	−6.63789e6	−0.682446
$625$	0	0
$626$	−2.83391e7	−2.89035
$627$	− 1.17836e7i	− 1.19704i
$628$	3.27519e6i	0.331389i
$629$	−3.16023e6	−0.318487
$630$	0	0
$631$	−6.59770e6	−0.659659	−0.329829	−	0.944041i	$-0.606991\pi$
$631$	−6.59770e6	−0.659659	−0.329829	+	0.944041i	$0.606991\pi$
$632$	3.03188e6i	0.301939i
$633$	1.92409e7i	1.90860i
$634$	−2.43535e7	−2.40624
$635$	0	0
$636$	−3.69430e6	−0.362151
$637$	721836.i	0.0704839i
$638$	9.60876e6i	0.934578i
$639$	−1.36308e7	−1.32059
$640$	0	0
$641$	1.44525e7	1.38930	0.694651	−	0.719347i	$-0.255559\pi$
$641$	1.44525e7	1.38930	0.694651	+	0.719347i	$0.255559\pi$
$642$	− 3.65104e6i	− 0.349606i
$643$	1.54720e7i	1.47577i	0.674926	+	0.737886i	$0.264176\pi$
$643$	1.54720e7i	1.47577i	−0.674926	+	0.737886i	$0.735824\pi$
$644$	−5.69799e6	−0.541386
$645$	0	0
$646$	−8.62960e6	−0.813597
$647$	1.66647e7i	1.56508i	0.622601	+	0.782540i	$0.286076\pi$
$647$	1.66647e7i	1.56508i	−0.622601	+	0.782540i	$0.713924\pi$
$648$	− 454306.i	− 0.0425022i
$649$	−8.02371e6	−0.747762
$650$	0	0
$651$	−2.54941e6	−0.235770
$652$	− 1.75525e7i	− 1.61704i
$653$	1.33451e7i	1.22472i	0.790578	+	0.612361i	$0.209780\pi$
$653$	1.33451e7i	1.22472i	−0.790578	+	0.612361i	$0.790220\pi$
$654$	−1.82570e7	−1.66911
$655$	0	0
$656$	6.15149e6	0.558111
$657$	− 4.61157e6i	− 0.416807i
$658$	− 8.10877e6i	− 0.730113i
$659$	4.00667e6	0.359393	0.179697	−	0.983722i	$-0.442488\pi$
$659$	4.00667e6	0.359393	0.179697	+	0.983722i	$0.442488\pi$
$660$	0	0
$661$	1.08005e7	0.961478	0.480739	−	0.876864i	$-0.340368\pi$
$661$	1.08005e7	0.961478	0.480739	+	0.876864i	$0.340368\pi$
$662$	7.99493e6i	0.709038i
$663$	4.72782e6i	0.417712i
$664$	−4.37750e6	−0.385307
$665$	0	0
$666$	1.76965e7	1.54597
$667$	1.37068e7i	1.19294i
$668$	3.15767e6i	0.273795i
$669$	1.37070e7	1.18407
$670$	0	0
$671$	1.36447e7	1.16993
$672$	1.04415e7i	0.891951i
$673$	− 1.09119e7i	− 0.928676i	−0.885658	−	0.464338i	$-0.846292\pi$
$673$	− 1.09119e7i	− 0.928676i	0.885658	−	0.464338i	$-0.153708\pi$
$674$	−1.12884e6	−0.0957158
$675$	0	0
$676$	−1.02459e7	−0.862353
$677$	− 1.35765e7i	− 1.13846i	−0.822179	−	0.569229i	$-0.807242\pi$
$677$	− 1.35765e7i	− 1.13846i	0.822179	−	0.569229i	$-0.192758\pi$
$678$	2.91984e7i	2.43941i
$679$	−2.14046e6	−0.178169
$680$	0	0
$681$	1.05323e7	0.870272
$682$	− 4.53354e6i	− 0.373230i
$683$	1.26726e7i	1.03948i	0.854326	+	0.519738i	$0.173970\pi$
$683$	1.26726e7i	1.03948i	−0.854326	+	0.519738i	$0.826030\pi$
$684$	2.57406e7	2.10367
$685$	0	0
$686$	973536.	0.0789845
$687$	− 2.64283e7i	− 2.13637i
$688$	− 1.68583e7i	− 1.35782i
$689$	−1.18717e6	−0.0952720
$690$	0	0
$691$	7.11964e6	0.567235	0.283617	−	0.958938i	$-0.408465\pi$
$691$	7.11964e6	0.567235	0.283617	+	0.958938i	$0.408465\pi$
$692$	− 2.12052e6i	− 0.168336i
$693$	5.49091e6i	0.434322i
$694$	−2.94097e6	−0.231789
$695$	0	0
$696$	−4.08281e6	−0.319474
$697$	− 4.38139e6i	− 0.341609i
$698$	1.15955e6i	0.0900844i
$699$	−3.05770e6	−0.236702
$700$	0	0
$701$	−1.00155e7	−0.769803	−0.384902	−	0.922958i	$-0.625765\pi$
$701$	−1.00155e7	−0.769803	−0.384902	+	0.922958i	$0.625765\pi$
$702$	− 1.09687e7i	− 0.840068i
$703$	8.76746e6i	0.669092i
$704$	−1.11280e7	−0.846224
$705$	0	0
$706$	2.88084e7	2.17524
$707$	1.25676e6i	0.0945589i
$708$	− 2.77928e7i	− 2.08377i
$709$	8.84454e6	0.660784	0.330392	−	0.943844i	$-0.392819\pi$
$709$	8.84454e6	0.660784	0.330392	+	0.943844i	$0.392819\pi$
$710$	0	0
$711$	3.39757e7	2.52054
$712$	− 1.54373e6i	− 0.114123i
$713$	− 6.46703e6i	− 0.476410i
$714$	6.37638e6	0.468090
$715$	0	0
$716$	−7.63698e6	−0.556723
$717$	6.51728e6i	0.473444i
$718$	− 1.45047e7i	− 1.05002i
$719$	−6.58086e6	−0.474745	−0.237373	−	0.971419i	$-0.576286\pi$
$719$	−6.58086e6	−0.474745	−0.237373	+	0.971419i	$0.576286\pi$
$720$	0	0
$721$	−701682.	−0.0502692
$722$	3.45169e6i	0.246427i
$723$	3.62301e7i	2.57765i
$724$	−1.01762e7	−0.721502
$725$	0	0
$726$	1.86995e7	1.31671
$727$	1.88401e7i	1.32205i	0.750365	+	0.661023i	$0.229878\pi$
$727$	1.88401e7i	1.32205i	−0.750365	+	0.661023i	$0.770122\pi$
$728$	− 545414.i	− 0.0381415i
$729$	2.23900e7	1.56040
$730$	0	0
$731$	−1.20073e7	−0.831095
$732$	4.72631e7i	3.26020i
$733$	2.78330e6i	0.191337i	0.995413	+	0.0956687i	$0.0304989\pi$
$733$	2.78330e6i	0.191337i	−0.995413	+	0.0956687i	$0.969501\pi$
$734$	1.46416e7	1.00311
$735$	0	0
$736$	−2.64867e7	−1.80233
$737$	− 1.36491e6i	− 0.0925629i
$738$	2.45346e7i	1.65821i
$739$	2.48970e7	1.67701	0.838505	−	0.544894i	$-0.183430\pi$
$739$	2.48970e7	1.67701	0.838505	+	0.544894i	$0.183430\pi$
$740$	0	0
$741$	1.31164e7	0.877548
$742$	1.60113e6i	0.106762i
$743$	3.86085e6i	0.256573i	0.991737	+	0.128286i	$0.0409477\pi$
$743$	3.86085e6i	0.256573i	−0.991737	+	0.128286i	$0.959052\pi$
$744$	1.92632e6	0.127584
$745$	0	0
$746$	−3.44412e7	−2.26585
$747$	4.90549e7i	3.21648i
$748$	6.03991e6i	0.394708i
$749$	−842888.	−0.0548991
$750$	0	0
$751$	6.72737e6	0.435257	0.217628	−	0.976032i	$-0.430168\pi$
$751$	6.72737e6	0.435257	0.217628	+	0.976032i	$0.430168\pi$
$752$	− 1.72147e7i	− 1.11008i
$753$	− 4.29737e7i	− 2.76195i
$754$	−1.06956e7	−0.685136
$755$	0	0
$756$	−7.88007e6	−0.501447
$757$	2.17782e7i	1.38128i	0.723197	+	0.690642i	$0.242672\pi$
$757$	2.17782e7i	1.38128i	−0.723197	+	0.690642i	$0.757328\pi$
$758$	− 5.11525e6i	− 0.323366i
$759$	−2.20865e7	−1.39162
$760$	0	0
$761$	−2.57074e7	−1.60915	−0.804575	−	0.593851i	$-0.797607\pi$
$761$	−2.57074e7	−1.60915	−0.804575	+	0.593851i	$0.797607\pi$
$762$	1.49777e7i	0.934453i
$763$	4.21486e6i	0.262103i
$764$	1.62359e7	1.00633
$765$	0	0
$766$	−3.40234e7	−2.09511
$767$	− 8.93127e6i	− 0.548182i
$768$	1.78808e7i	1.09391i
$769$	1.34375e7	0.819413	0.409706	−	0.912217i	$-0.365631\pi$
$769$	1.34375e7	0.819413	0.409706	+	0.912217i	$0.365631\pi$
$770$	0	0
$771$	−1.86471e7	−1.12973
$772$	− 2.65099e7i	− 1.60090i
$773$	− 3.05572e7i	− 1.83935i	−0.392682	−	0.919674i	$-0.628453\pi$
$773$	− 3.05572e7i	− 1.83935i	0.392682	−	0.919674i	$-0.371547\pi$
$774$	6.72376e7	4.03422
$775$	0	0
$776$	1.61731e6	0.0964140
$777$	− 6.47825e6i	− 0.384951i
$778$	− 3.82364e7i	− 2.26479i
$779$	−1.21553e7	−0.717667
$780$	0	0
$781$	−8.87335e6	−0.520547
$782$	1.61748e7i	0.945849i
$783$	1.89558e7i	1.10494i
$784$	2.06679e6	0.120090
$785$	0	0
$786$	3.68700e7	2.12871
$787$	− 2.07672e6i	− 0.119520i	−0.998213	−	0.0597602i	$-0.980966\pi$
$787$	− 2.07672e6i	− 0.119520i	0.998213	−	0.0597602i	$-0.0190336\pi$
$788$	1.33093e7i	0.763556i
$789$	−5.79372e6	−0.331333
$790$	0	0
$791$	6.74081e6	0.383064
$792$	− 4.14890e6i	− 0.235028i
$793$	1.51881e7i	0.857670i
$794$	−4.19827e7	−2.36330
$795$	0	0
$796$	1.05523e7	0.590286
$797$	− 5.98563e6i	− 0.333783i	−0.985975	−	0.166892i	$-0.946627\pi$
$797$	− 5.98563e6i	− 0.333783i	0.985975	−	0.166892i	$-0.0533730\pi$
$798$	− 1.76901e7i	− 0.983383i
$799$	−1.22611e7	−0.679460
$800$	0	0
$801$	−1.72993e7	−0.952679
$802$	− 1.22515e7i	− 0.672594i
$803$	− 3.00204e6i	− 0.164296i
$804$	4.72784e6	0.257942
$805$	0	0
$806$	5.04633e6	0.273614
$807$	− 4.63043e7i	− 2.50286i
$808$	− 949597.i	− 0.0511695i
$809$	−1.96864e7	−1.05754	−0.528769	−	0.848766i	$-0.677346\pi$
$809$	−1.96864e7	−1.05754	−0.528769	+	0.848766i	$0.677346\pi$
$810$	0	0
$811$	8.50101e6	0.453856	0.226928	−	0.973912i	$-0.427132\pi$
$811$	8.50101e6	0.453856	0.226928	+	0.973912i	$0.427132\pi$
$812$	7.68384e6i	0.408967i
$813$	− 4.39580e7i	− 2.33245i
$814$	1.15201e7	0.609387
$815$	0	0
$816$	1.35369e7	0.711695
$817$	3.33119e7i	1.74600i
$818$	3.75168e7i	1.96039i
$819$	−6.11199e6	−0.318400
$820$	0	0
$821$	−1.36199e6	−0.0705204	−0.0352602	−	0.999378i	$-0.511226\pi$
$821$	−1.36199e6	−0.0705204	−0.0352602	+	0.999378i	$0.511226\pi$
$822$	422360.i	0.0218023i
$823$	1.35934e6i	0.0699566i	0.999388	+	0.0349783i	$0.0111362\pi$
$823$	1.35934e6i	0.0699566i	−0.999388	+	0.0349783i	$0.988864\pi$
$824$	530186.	0.0272026
$825$	0	0
$826$	−1.20456e7	−0.614295
$827$	1.00727e7i	0.512132i	0.966659	+	0.256066i	$0.0824264\pi$
$827$	1.00727e7i	0.512132i	−0.966659	+	0.256066i	$0.917574\pi$
$828$	− 4.82465e7i	− 2.44563i
$829$	5.63984e6	0.285023	0.142512	−	0.989793i	$-0.454482\pi$
$829$	5.63984e6	0.285023	0.142512	+	0.989793i	$0.454482\pi$
$830$	0	0
$831$	−5.72125e7	−2.87401
$832$	− 1.23867e7i	− 0.620364i
$833$	− 1.47207e6i	− 0.0735048i
$834$	7.77693e7	3.87163
$835$	0	0
$836$	1.67566e7	0.829220
$837$	− 8.94361e6i	− 0.441265i
$838$	− 918733.i	− 0.0451938i
$839$	1.16351e7	0.570642	0.285321	−	0.958432i	$-0.407900\pi$
$839$	1.16351e7	0.570642	0.285321	+	0.958432i	$0.407900\pi$
$840$	0	0
$841$	−2.02735e6	−0.0988413
$842$	− 1.16801e7i	− 0.567764i
$843$	4.28706e7i	2.07774i
$844$	−2.73610e7	−1.32214
$845$	0	0
$846$	6.86592e7	3.29817
$847$	− 4.31702e6i	− 0.206764i
$848$	3.39916e6i	0.162324i
$849$	−1.01611e7	−0.483806
$850$	0	0
$851$	1.64332e7	0.777854
$852$	− 3.07358e7i	− 1.45059i
$853$	− 2.85205e7i	− 1.34210i	−0.741413	−	0.671049i	$-0.765844\pi$
$853$	− 2.85205e7i	− 1.34210i	0.741413	−	0.671049i	$-0.234156\pi$
$854$	2.04841e7	0.961108
$855$	0	0
$856$	636880.	0.0297080
$857$	− 9.95725e6i	− 0.463113i	−0.972821	−	0.231557i	$-0.925618\pi$
$857$	− 9.95725e6i	− 0.463113i	0.972821	−	0.231557i	$-0.0743819\pi$
$858$	− 1.72344e7i	− 0.799243i
$859$	1.49322e7	0.690463	0.345232	−	0.938517i	$-0.387800\pi$
$859$	1.49322e7	0.690463	0.345232	+	0.938517i	$0.387800\pi$
$860$	0	0
$861$	8.98154e6	0.412898
$862$	8.90711e6i	0.408290i
$863$	− 3.84933e7i	− 1.75937i	−0.475553	−	0.879687i	$-0.657752\pi$
$863$	− 3.84933e7i	− 1.75937i	0.475553	−	0.879687i	$-0.342248\pi$
$864$	−3.66300e7	−1.66937
$865$	0	0
$866$	−2.56665e6	−0.116298
$867$	2.67770e7i	1.20980i
$868$	− 3.62533e6i	− 0.163323i
$869$	2.21175e7	0.993542
$870$	0	0
$871$	1.51930e6	0.0678576
$872$	− 3.18472e6i	− 0.141834i
$873$	− 1.81239e7i	− 0.804850i
$874$	4.48739e7	1.98708
$875$	0	0
$876$	1.03986e7	0.457839
$877$	9.40311e6i	0.412831i	0.978464	+	0.206416i	$0.0661799\pi$
$877$	9.40311e6i	0.412831i	−0.978464	+	0.206416i	$0.933820\pi$
$878$	− 4.69726e7i	− 2.05640i
$879$	−2.38403e7	−1.04073
$880$	0	0
$881$	1.10395e6	0.0479194	0.0239597	−	0.999713i	$-0.492373\pi$
$881$	1.10395e6	0.0479194	0.0239597	+	0.999713i	$0.492373\pi$
$882$	8.24321e6i	0.356800i
$883$	− 8.06579e6i	− 0.348133i	−0.984734	−	0.174067i	$-0.944309\pi$
$883$	− 8.06579e6i	− 0.348133i	0.984734	−	0.174067i	$-0.0556908\pi$
$884$	−6.72308e6	−0.289359
$885$	0	0
$886$	3.35933e7	1.43770
$887$	− 1.49902e7i	− 0.639732i	−0.947463	−	0.319866i	$-0.896362\pi$
$887$	− 1.49902e7i	− 0.639732i	0.947463	−	0.319866i	$-0.103638\pi$
$888$	4.89493e6i	0.208312i
$889$	3.45779e6	0.146738
$890$	0	0
$891$	−3.31415e6	−0.139855
$892$	1.94918e7i	0.820237i
$893$	3.40162e7i	1.42744i
$894$	2.98672e7	1.24983
$895$	0	0
$896$	−3.67912e6	−0.153100
$897$	− 2.45847e7i	− 1.02019i
$898$	5.76384e7i	2.38518i
$899$	−8.72090e6	−0.359883
$900$	0	0
$901$	2.42104e6	0.0993553
$902$	1.59716e7i	0.653629i
$903$	− 2.46141e7i	− 1.00453i
$904$	−5.09331e6	−0.207290
$905$	0	0
$906$	−1.06261e7	−0.430083
$907$	4.12622e6i	0.166546i	0.996527	+	0.0832730i	$0.0265374\pi$
$907$	4.12622e6i	0.166546i	−0.996527	+	0.0832730i	$0.973463\pi$
$908$	1.49772e7i	0.602859i
$909$	−1.06413e7	−0.427155
$910$	0	0
$911$	4.04272e7	1.61391	0.806953	−	0.590616i	$-0.201115\pi$
$911$	4.04272e7	1.61391	0.806953	+	0.590616i	$0.201115\pi$
$912$	− 3.75556e7i	− 1.49516i
$913$	3.19338e7i	1.26787i
$914$	−1.48554e7	−0.588191
$915$	0	0
$916$	3.75817e7	1.47992
$917$	− 8.51191e6i	− 0.334275i
$918$	2.23690e7i	0.876072i
$919$	−2.18546e7	−0.853600	−0.426800	−	0.904346i	$-0.640359\pi$
$919$	−2.18546e7	−0.853600	−0.426800	+	0.904346i	$0.640359\pi$
$920$	0	0
$921$	−4.70142e7	−1.82633
$922$	− 1.74844e7i	− 0.677366i
$923$	− 9.87702e6i	− 0.381612i
$924$	−1.23814e7	−0.477078
$925$	0	0
$926$	1.04448e7	0.400289
$927$	− 5.94134e6i	− 0.227083i
$928$	3.57178e7i	1.36149i
$929$	−1.06843e7	−0.406169	−0.203085	−	0.979161i	$-0.565097\pi$
$929$	−1.06843e7	−0.406169	−0.203085	+	0.979161i	$0.565097\pi$
$930$	0	0
$931$	−4.08398e6	−0.154422
$932$	− 4.34813e6i	− 0.163970i
$933$	− 5.89130e7i	− 2.21568i
$934$	2.97009e7	1.11404
$935$	0	0
$936$	4.61818e6	0.172298
$937$	− 3.99105e7i	− 1.48504i	−0.669823	−	0.742521i	$-0.733630\pi$
$937$	− 3.99105e7i	− 1.48504i	0.669823	−	0.742521i	$-0.266370\pi$
$938$	− 2.04907e6i	− 0.0760414i
$939$	−8.78419e7	−3.25116
$940$	0	0
$941$	1.32350e6	0.0487248	0.0243624	−	0.999703i	$-0.492244\pi$
$941$	1.32350e6	0.0487248	0.0243624	+	0.999703i	$0.492244\pi$
$942$	1.90587e7i	0.699787i
$943$	2.27832e7i	0.834326i
$944$	−2.55724e7	−0.933990
$945$	0	0
$946$	4.37703e7	1.59020
$947$	− 2.76322e7i	− 1.00124i	−0.865666	−	0.500622i	$-0.833105\pi$
$947$	− 2.76322e7i	− 1.00124i	0.865666	−	0.500622i	$-0.166895\pi$
$948$	7.66113e7i	2.76868i
$949$	3.34160e6	0.120445
$950$	0	0
$951$	−7.54879e7	−2.70661
$952$	1.11229e6i	0.0397763i
$953$	3.07901e7i	1.09819i	0.835759	+	0.549096i	$0.185028\pi$
$953$	3.07901e7i	1.09819i	−0.835759	+	0.549096i	$0.814972\pi$
$954$	−1.35572e7	−0.482281
$955$	0	0
$956$	−9.26775e6	−0.327966
$957$	2.97840e7i	1.05124i
$958$	1.99999e7i	0.704068i
$959$	97507.1	0.00342365
$960$	0	0
$961$	−2.45145e7	−0.856278
$962$	1.28231e7i	0.446740i
$963$	− 7.13697e6i	− 0.247998i
$964$	−5.15202e7	−1.78560
$965$	0	0
$966$	−3.31572e7	−1.14323
$967$	2.92557e6i	0.100611i	0.998734	+	0.0503055i	$0.0160195\pi$
$967$	2.92557e6i	0.100611i	−0.998734	+	0.0503055i	$0.983981\pi$
$968$	3.26192e6i	0.111888i
$969$	−2.67489e7	−0.915159
$970$	0	0
$971$	2.78109e6	0.0946601	0.0473301	−	0.998879i	$-0.484929\pi$
$971$	2.78109e6	0.0946601	0.0473301	+	0.998879i	$0.484929\pi$
$972$	2.75990e7i	0.936975i
$973$	− 1.79540e7i	− 0.607967i
$974$	4.29801e7	1.45168
$975$	0	0
$976$	4.34872e7	1.46129
$977$	7.48673e6i	0.250932i	0.992098	+	0.125466i	$0.0400426\pi$
$977$	7.48673e6i	0.250932i	−0.992098	+	0.125466i	$0.959957\pi$
$978$	− 1.02140e8i	− 3.41467i
$979$	−1.12615e7	−0.375525
$980$	0	0
$981$	−3.56884e7	−1.18401
$982$	4.45985e7i	1.47585i
$983$	− 1.79815e7i	− 0.593528i	−0.954951	−	0.296764i	$-0.904093\pi$
$983$	− 1.79815e7i	− 0.593528i	0.954951	−	0.296764i	$-0.0959075\pi$
$984$	−6.78639e6	−0.223435
$985$	0	0
$986$	2.18120e7	0.714501
$987$	− 2.51345e7i	− 0.821253i
$988$	1.86519e7i	0.607899i
$989$	6.24378e7	2.02982
$990$	0	0
$991$	3.72778e7	1.20578	0.602888	−	0.797826i	$-0.294017\pi$
$991$	3.72778e7	1.20578	0.602888	+	0.797826i	$0.294017\pi$
$992$	− 1.68521e7i	− 0.543720i
$993$	2.47816e7i	0.797548i
$994$	−1.33211e7	−0.427635
$995$	0	0
$996$	−1.10613e8	−3.53313
$997$	4.87422e7i	1.55298i	0.630128	+	0.776492i	$0.283003\pi$
$997$	4.87422e7i	1.55298i	−0.630128	+	0.776492i	$0.716997\pi$
$998$	2.72747e7i	0.866828i
$999$	2.27264e7	0.720471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.6.b.c.99.4		4
5.2	odd	4		175.6.a.c.1.1		2
5.3	odd	4		7.6.a.b.1.2	✓	2
5.4	even	2	inner	175.6.b.c.99.1		4
15.8	even	4		63.6.a.f.1.1		2
20.3	even	4		112.6.a.h.1.2		2
35.3	even	12		49.6.c.d.30.1		4
35.13	even	4		49.6.a.f.1.2		2
35.18	odd	12		49.6.c.e.30.1		4
35.23	odd	12		49.6.c.e.18.1		4
35.33	even	12		49.6.c.d.18.1		4
40.3	even	4		448.6.a.u.1.1		2
40.13	odd	4		448.6.a.w.1.2		2
55.43	even	4		847.6.a.c.1.1		2
60.23	odd	4		1008.6.a.bq.1.1		2
105.83	odd	4		441.6.a.l.1.1		2
140.83	odd	4		784.6.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.2	✓	2	5.3	odd	4
49.6.a.f.1.2		2	35.13	even	4
49.6.c.d.18.1		4	35.33	even	12
49.6.c.d.30.1		4	35.3	even	12
49.6.c.e.18.1		4	35.23	odd	12
49.6.c.e.30.1		4	35.18	odd	12
63.6.a.f.1.1		2	15.8	even	4
112.6.a.h.1.2		2	20.3	even	4
175.6.a.c.1.1		2	5.2	odd	4
175.6.b.c.99.1		4	5.4	even	2	inner
175.6.b.c.99.4		4	1.1	even	1	trivial
441.6.a.l.1.1		2	105.83	odd	4
448.6.a.u.1.1		2	40.3	even	4
448.6.a.w.1.2		2	40.13	odd	4
784.6.a.v.1.1		2	140.83	odd	4
847.6.a.c.1.1		2	55.43	even	4
1008.6.a.bq.1.1		2	60.23	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.27492i q^{2} +25.6495i q^{3} -36.4743 q^{4} -212.248 q^{6} +49.0000i q^{7} -37.0241i q^{8} -414.897 q^{9} +O(q^{10})\)	\(q+8.27492i q^{2} +25.6495i q^{3} -36.4743 q^{4} -212.248 q^{6} +49.0000i q^{7} -37.0241i q^{8} -414.897 q^{9} -270.090 q^{11} -935.547i q^{12} -300.640i q^{13} -405.471 q^{14} -860.805 q^{16} +613.106i q^{17} -3433.24i q^{18} +1700.95 q^{19} -1256.83 q^{21} -2234.97i q^{22} -3188.15i q^{23} +949.650 q^{24} +2487.77 q^{26} -4409.07i q^{27} -1787.24i q^{28} -4299.28 q^{29} +2028.46 q^{31} -8307.86i q^{32} -6927.67i q^{33} -5073.40 q^{34} +15133.1 q^{36} +5154.46i q^{37} +14075.2i q^{38} +7711.26 q^{39} -7146.21 q^{41} -10400.1i q^{42} +19584.3i q^{43} +9851.32 q^{44} +26381.7 q^{46} +19998.4i q^{47} -22079.2i q^{48} -2401.00 q^{49} -15725.9 q^{51} +10965.6i q^{52} -3948.82i q^{53} +36484.7 q^{54} +1814.18 q^{56} +43628.5i q^{57} -35576.2i q^{58} +29707.6 q^{59} -50519.3 q^{61} +16785.3i q^{62} -20330.0i q^{63} +41201.1 q^{64} +57325.9 q^{66} +5053.56i q^{67} -22362.6i q^{68} +81774.5 q^{69} +32853.3 q^{71} +15361.2i q^{72} +11115.0i q^{73} -42652.7 q^{74} -62040.8 q^{76} -13234.4i q^{77} +63810.0i q^{78} -81889.4 q^{79} +12270.6 q^{81} -59134.3i q^{82} -118234. i q^{83} +45841.8 q^{84} -162058. q^{86} -110274. i q^{87} +9999.83i q^{88} +41695.4 q^{89} +14731.3 q^{91} +116286. i q^{92} +52028.9i q^{93} -165485. q^{94} +213092. q^{96} +43682.8i q^{97} -19868.1i q^{98} +112059. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} - 396 q^{6} - 1116 q^{9}+O(q^{10}) \) 4 * q - 10 * q^4 - 396 * q^6 - 1116 * q^9	\( 4 q - 10 q^{4} - 396 q^{6} - 1116 q^{9} + 792 q^{11} - 882 q^{14} + 226 q^{16} + 6532 q^{19} - 588 q^{21} + 3708 q^{24} + 4032 q^{26} - 13392 q^{29} - 40 q^{31} - 11868 q^{34} + 21258 q^{36} + 40992 q^{39} - 12096 q^{41} + 61632 q^{44} + 51168 q^{46} - 9604 q^{49} + 15192 q^{51} + 75816 q^{54} - 882 q^{56} + 87876 q^{59} - 129508 q^{61} + 141566 q^{64} + 133632 q^{66} + 206784 q^{69} + 194832 q^{71} - 86868 q^{74} - 25564 q^{76} - 102512 q^{79} - 122148 q^{81} + 152292 q^{84} - 300096 q^{86} - 168552 q^{89} - 34300 q^{91} - 318936 q^{94} + 511812 q^{96} + 33480 q^{99}+O(q^{100}) \) 4 * q - 10 * q^4 - 396 * q^6 - 1116 * q^9 + 792 * q^11 - 882 * q^14 + 226 * q^16 + 6532 * q^19 - 588 * q^21 + 3708 * q^24 + 4032 * q^26 - 13392 * q^29 - 40 * q^31 - 11868 * q^34 + 21258 * q^36 + 40992 * q^39 - 12096 * q^41 + 61632 * q^44 + 51168 * q^46 - 9604 * q^49 + 15192 * q^51 + 75816 * q^54 - 882 * q^56 + 87876 * q^59 - 129508 * q^61 + 141566 * q^64 + 133632 * q^66 + 206784 * q^69 + 194832 * q^71 - 86868 * q^74 - 25564 * q^76 - 102512 * q^79 - 122148 * q^81 + 152292 * q^84 - 300096 * q^86 - 168552 * q^89 - 34300 * q^91 - 318936 * q^94 + 511812 * q^96 + 33480 * q^99

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