LMFDB - Embedded newform 175.6.b.d.99.2

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:	$\Q(i, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 33x^{2} + 256 $ x^4 + 33*x^2 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$-3.53113i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.6.b.d.99.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.53113i q^{2} -13.5934i q^{3} +19.5311 q^{4} -48.0000 q^{6} -49.0000i q^{7} -181.963i q^{8} +58.2198 q^{9} +O(q^{10})$	\(q-3.53113i q^{2} -13.5934i q^{3} +19.5311 q^{4} -48.0000 q^{6} -49.0000i q^{7} -181.963i q^{8} +58.2198 q^{9} -691.520 q^{11} -265.494i q^{12} +502.150i q^{13} -173.025 q^{14} -17.5389 q^{16} -991.313i q^{17} -205.582i q^{18} -661.677 q^{19} -666.076 q^{21} +2441.84i q^{22} -3415.08i q^{23} -2473.49 q^{24} +1773.16 q^{26} -4094.60i q^{27} -957.025i q^{28} -6751.92 q^{29} -3922.76 q^{31} -5760.89i q^{32} +9400.09i q^{33} -3500.46 q^{34} +1137.10 q^{36} +627.222i q^{37} +2336.47i q^{38} +6825.92 q^{39} +16277.9 q^{41} +2352.00i q^{42} +17277.7i q^{43} -13506.2 q^{44} -12059.1 q^{46} -4295.47i q^{47} +238.413i q^{48} -2401.00 q^{49} -13475.3 q^{51} +9807.55i q^{52} +25960.9i q^{53} -14458.6 q^{54} -8916.19 q^{56} +8994.43i q^{57} +23841.9i q^{58} -8902.63 q^{59} -48924.6 q^{61} +13851.8i q^{62} -2852.77i q^{63} -20903.7 q^{64} +33192.9 q^{66} -4257.80i q^{67} -19361.5i q^{68} -46422.5 q^{69} +18990.9 q^{71} -10593.9i q^{72} -10132.5i q^{73} +2214.80 q^{74} -12923.3 q^{76} +33884.5i q^{77} -24103.2i q^{78} +96986.5 q^{79} -41512.0 q^{81} -57479.2i q^{82} -70732.1i q^{83} -13009.2 q^{84} +61009.8 q^{86} +91781.4i q^{87} +125831. i q^{88} -4241.12 q^{89} +24605.3 q^{91} -66700.3i q^{92} +53323.6i q^{93} -15167.9 q^{94} -78309.9 q^{96} -104376. i q^{97} +8478.24i q^{98} -40260.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9}+O(q^{10}) $ 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9	$ 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9} - 1202 q^{11} + 98 q^{14} - 1086 q^{16} - 1260 q^{19} - 294 q^{21} - 9120 q^{24} + 2868 q^{26} - 11770 q^{29} - 792 q^{31} - 16356 q^{34} + 5274 q^{36} + 12066 q^{39} + 3548 q^{41} - 24936 q^{44} - 59072 q^{46} - 9604 q^{49} - 48822 q^{51} - 64800 q^{54} + 1470 q^{56} - 119200 q^{59} - 103692 q^{61} - 110978 q^{64} + 57696 q^{66} - 174564 q^{69} + 161488 q^{71} + 90804 q^{74} - 25120 q^{76} + 103590 q^{79} - 103356 q^{81} - 14112 q^{84} + 32808 q^{86} + 75300 q^{89} + 56546 q^{91} + 123664 q^{94} - 239712 q^{96} - 56844 q^{99}+O(q^{100}) $ 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9 - 1202 * q^11 + 98 * q^14 - 1086 * q^16 - 1260 * q^19 - 294 * q^21 - 9120 * q^24 + 2868 * q^26 - 11770 * q^29 - 792 * q^31 - 16356 * q^34 + 5274 * q^36 + 12066 * q^39 + 3548 * q^41 - 24936 * q^44 - 59072 * q^46 - 9604 * q^49 - 48822 * q^51 - 64800 * q^54 + 1470 * q^56 - 119200 * q^59 - 103692 * q^61 - 110978 * q^64 + 57696 * q^66 - 174564 * q^69 + 161488 * q^71 + 90804 * q^74 - 25120 * q^76 + 103590 * q^79 - 103356 * q^81 - 14112 * q^84 + 32808 * q^86 + 75300 * q^89 + 56546 * q^91 + 123664 * q^94 - 239712 * q^96 - 56844 * 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 $

$2$	− 3.53113i	− 0.624221i	−0.950046	−	0.312111i	$-0.898964\pi$
$2$	− 3.53113i	− 0.624221i	0.950046	−	0.312111i	$-0.101036\pi$
$3$	− 13.5934i	− 0.872016i	−0.899943	−	0.436008i	$-0.856392\pi$
$3$	− 13.5934i	− 0.872016i	0.899943	−	0.436008i	$-0.143608\pi$
$4$	19.5311	0.610348
$5$	0	0
$5$	0	0
$6$	−48.0000	−0.544331
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	− 181.963i	− 1.00521i
$9$	58.2198	0.239588
$10$	0	0
$11$	−691.520	−1.72315	−0.861574	−	0.507632i	$-0.830521\pi$
$11$	−691.520	−1.72315	−0.861574	+	0.507632i	$0.830521\pi$
$12$	− 265.494i	− 0.532233i
$13$	502.150i	0.824091i	0.911163	+	0.412045i	$0.135185\pi$
$13$	502.150i	0.824091i	−0.911163	+	0.412045i	$0.864815\pi$
$14$	−173.025	−0.235933
$15$	0	0
$16$	−17.5389	−0.0171278
$17$	− 991.313i	− 0.831934i	−0.909380	−	0.415967i	$-0.863443\pi$
$17$	− 991.313i	− 0.831934i	0.909380	−	0.415967i	$-0.136557\pi$
$18$	− 205.582i	− 0.149556i
$19$	−661.677	−0.420496	−0.210248	−	0.977648i	$-0.567427\pi$
$19$	−661.677	−0.420496	−0.210248	+	0.977648i	$0.567427\pi$
$20$	0	0
$21$	−666.076	−0.329591
$22$	2441.84i	1.07563i
$23$	− 3415.08i	− 1.34611i	−0.739592	−	0.673056i	$-0.764981\pi$
$23$	− 3415.08i	− 1.34611i	0.739592	−	0.673056i	$-0.235019\pi$
$24$	−2473.49	−0.876562
$25$	0	0
$26$	1773.16	0.514415
$27$	− 4094.60i	− 1.08094i
$28$	− 957.025i	− 0.230690i
$29$	−6751.92	−1.49084	−0.745422	−	0.666593i	$-0.767752\pi$
$29$	−6751.92	−1.49084	−0.745422	+	0.666593i	$0.767752\pi$
$30$	0	0
$31$	−3922.76	−0.733142	−0.366571	−	0.930390i	$-0.619468\pi$
$31$	−3922.76	−0.733142	−0.366571	+	0.930390i	$0.619468\pi$
$32$	− 5760.89i	− 0.994522i
$33$	9400.09i	1.50261i
$34$	−3500.46	−0.519311
$35$	0	0
$36$	1137.10	0.146232
$37$	627.222i	0.0753212i	0.999291	+	0.0376606i	$0.0119906\pi$
$37$	627.222i	0.0753212i	−0.999291	+	0.0376606i	$0.988009\pi$
$38$	2336.47i	0.262483i
$39$	6825.92	0.718620
$40$	0	0
$41$	16277.9	1.51230	0.756149	−	0.654399i	$-0.227078\pi$
$41$	16277.9	1.51230	0.756149	+	0.654399i	$0.227078\pi$
$42$	2352.00i	0.205738i
$43$	17277.7i	1.42500i	0.701672	+	0.712500i	$0.252437\pi$
$43$	17277.7i	1.42500i	−0.701672	+	0.712500i	$0.747563\pi$
$44$	−13506.2	−1.05172
$45$	0	0
$46$	−12059.1	−0.840272
$47$	− 4295.47i	− 0.283639i	−0.989893	−	0.141820i	$-0.954705\pi$
$47$	− 4295.47i	− 0.283639i	0.989893	−	0.141820i	$-0.0452953\pi$
$48$	238.413i	0.0149357i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−13475.3	−0.725460
$52$	9807.55i	0.502982i
$53$	25960.9i	1.26949i	0.772721	+	0.634745i	$0.218895\pi$
$53$	25960.9i	1.26949i	−0.772721	+	0.634745i	$0.781105\pi$
$54$	−14458.6	−0.674746
$55$	0	0
$56$	−8916.19	−0.379935
$57$	8994.43i	0.366679i
$58$	23841.9i	0.930616i
$59$	−8902.63	−0.332957	−0.166479	−	0.986045i	$-0.553240\pi$
$59$	−8902.63	−0.332957	−0.166479	+	0.986045i	$0.553240\pi$
$60$	0	0
$61$	−48924.6	−1.68346	−0.841730	−	0.539898i	$-0.818463\pi$
$61$	−48924.6	−1.68346	−0.841730	+	0.539898i	$0.818463\pi$
$62$	13851.8i	0.457643i
$63$	− 2852.77i	− 0.0905557i
$64$	−20903.7	−0.637930
$65$	0	0
$66$	33192.9	0.937963
$67$	− 4257.80i	− 0.115877i	−0.998320	−	0.0579387i	$-0.981547\pi$
$67$	− 4257.80i	− 0.115877i	0.998320	−	0.0579387i	$-0.0184528\pi$
$68$	− 19361.5i	− 0.507769i
$69$	−46422.5	−1.17383
$70$	0	0
$71$	18990.9	0.447095	0.223547	−	0.974693i	$-0.428236\pi$
$71$	18990.9	0.447095	0.223547	+	0.974693i	$0.428236\pi$
$72$	− 10593.9i	− 0.240837i
$73$	− 10132.5i	− 0.222541i	−0.993790	−	0.111270i	$-0.964508\pi$
$73$	− 10132.5i	− 0.222541i	0.993790	−	0.111270i	$-0.0354919\pi$
$74$	2214.80	0.0470171
$75$	0	0
$76$	−12923.3	−0.256649
$77$	33884.5i	0.651289i
$78$	− 24103.2i	− 0.448578i
$79$	96986.5	1.74841	0.874205	−	0.485557i	$-0.161383\pi$
$79$	96986.5	1.74841	0.874205	+	0.485557i	$0.161383\pi$
$80$	0	0
$81$	−41512.0	−0.703010
$82$	− 57479.2i	− 0.944009i
$83$	− 70732.1i	− 1.12699i	−0.826118	−	0.563497i	$-0.809456\pi$
$83$	− 70732.1i	− 1.12699i	0.826118	−	0.563497i	$-0.190544\pi$
$84$	−13009.2	−0.201165
$85$	0	0
$86$	61009.8	0.889515
$87$	91781.4i	1.30004i
$88$	125831.i	1.73213i
$89$	−4241.12	−0.0567552	−0.0283776	−	0.999597i	$-0.509034\pi$
$89$	−4241.12	−0.0567552	−0.0283776	+	0.999597i	$0.509034\pi$
$90$	0	0
$91$	24605.3	0.311477
$92$	− 66700.3i	− 0.821596i
$93$	53323.6i	0.639311i
$94$	−15167.9	−0.177054
$95$	0	0
$96$	−78309.9	−0.867239
$97$	− 104376.i	− 1.12634i	−0.826341	−	0.563170i	$-0.809582\pi$
$97$	− 104376.i	− 1.12634i	0.826341	−	0.563170i	$-0.190418\pi$
$98$	8478.24i	0.0891745i
$99$	−40260.2	−0.412845
$100$	0	0
$101$	−45715.1	−0.445919	−0.222959	−	0.974828i	$-0.571572\pi$
$101$	−45715.1	−0.445919	−0.222959	+	0.974828i	$0.571572\pi$
$102$	47583.0i	0.452847i
$103$	− 89278.1i	− 0.829186i	−0.910007	−	0.414593i	$-0.863924\pi$
$103$	− 89278.1i	− 0.829186i	0.910007	−	0.414593i	$-0.136076\pi$
$104$	91372.7	0.828387
$105$	0	0
$106$	91671.2	0.792443
$107$	− 106330.i	− 0.897834i	−0.893573	−	0.448917i	$-0.851810\pi$
$107$	− 106330.i	− 0.897834i	0.893573	−	0.448917i	$-0.148190\pi$
$108$	− 79972.1i	− 0.659750i
$109$	49816.5	0.401613	0.200806	−	0.979631i	$-0.435644\pi$
$109$	49816.5	0.401613	0.200806	+	0.979631i	$0.435644\pi$
$110$	0	0
$111$	8526.08	0.0656813
$112$	859.405i	0.00647370i
$113$	37160.7i	0.273771i	0.990587	+	0.136886i	$0.0437093\pi$
$113$	37160.7i	0.273771i	−0.990587	+	0.136886i	$0.956291\pi$
$114$	31760.5	0.228889
$115$	0	0
$116$	−131873.	−0.909933
$117$	29235.1i	0.197442i
$118$	31436.3i	0.207839i
$119$	−48574.4	−0.314441
$120$	0	0
$121$	317148.	1.96924
$122$	172759.i	1.05085i
$123$	− 221271.i	− 1.31875i
$124$	−76616.0	−0.447471
$125$	0	0
$126$	−10073.5	−0.0565268
$127$	− 46510.2i	− 0.255882i	−0.991782	−	0.127941i	$-0.959163\pi$
$127$	− 46510.2i	− 0.255882i	0.991782	−	0.127941i	$-0.0408367\pi$
$128$	− 110535.i	− 0.596313i
$129$	234862.	1.24262
$130$	0	0
$131$	381771.	1.94368	0.971839	−	0.235646i	$-0.0757206\pi$
$131$	381771.	1.94368	0.971839	+	0.235646i	$0.0757206\pi$
$132$	183594.i	0.917117i
$133$	32422.2i	0.158933i
$134$	−15034.9	−0.0723331
$135$	0	0
$136$	−180382.	−0.836271
$137$	− 1894.54i	− 0.00862389i	−0.999991	−	0.00431194i	$-0.998627\pi$
$137$	− 1894.54i	− 0.00862389i	0.999991	−	0.00431194i	$-0.00137254\pi$
$138$	163924.i	0.732730i
$139$	−201798.	−0.885889	−0.442944	−	0.896549i	$-0.646066\pi$
$139$	−201798.	−0.885889	−0.442944	+	0.896549i	$0.646066\pi$
$140$	0	0
$141$	−58390.0	−0.247338
$142$	− 67059.3i	− 0.279086i
$143$	− 347246.i	− 1.42003i
$144$	−1021.11	−0.00410362
$145$	0	0
$146$	−35779.1	−0.138915
$147$	32637.7i	0.124574i
$148$	12250.4i	0.0459721i
$149$	466237.	1.72045	0.860224	−	0.509917i	$-0.170324\pi$
$149$	466237.	1.72045	0.860224	+	0.509917i	$0.170324\pi$
$150$	0	0
$151$	−122212.	−0.436185	−0.218093	−	0.975928i	$-0.569983\pi$
$151$	−122212.	−0.436185	−0.218093	+	0.975928i	$0.569983\pi$
$152$	120401.i	0.422688i
$153$	− 57714.1i	− 0.199321i
$154$	119650.	0.406548
$155$	0	0
$156$	133318.	0.438608
$157$	− 410638.i	− 1.32957i	−0.747036	−	0.664784i	$-0.768524\pi$
$157$	− 410638.i	− 1.32957i	0.747036	−	0.664784i	$-0.231476\pi$
$158$	− 342472.i	− 1.09139i
$159$	352896.	1.10702
$160$	0	0
$161$	−167339.	−0.508782
$162$	146584.i	0.438834i
$163$	− 78525.4i	− 0.231495i	−0.993279	−	0.115747i	$-0.963074\pi$
$163$	− 78525.4i	− 0.231495i	0.993279	−	0.115747i	$-0.0369263\pi$
$164$	317925.	0.923028
$165$	0	0
$166$	−249764.	−0.703494
$167$	− 597714.i	− 1.65845i	−0.558916	−	0.829224i	$-0.688783\pi$
$167$	− 597714.i	− 1.65845i	0.558916	−	0.829224i	$-0.311217\pi$
$168$	121201.i	0.331309i
$169$	119139.	0.320875
$170$	0	0
$171$	−38522.7	−0.100746
$172$	337453.i	0.869745i
$173$	59874.0i	0.152098i	0.997104	+	0.0760490i	$0.0242305\pi$
$173$	59874.0i	0.152098i	−0.997104	+	0.0760490i	$0.975769\pi$
$174$	324092.	0.811512
$175$	0	0
$176$	12128.5	0.0295138
$177$	121017.i	0.290344i
$178$	14975.9i	0.0354278i
$179$	−616812.	−1.43887	−0.719433	−	0.694562i	$-0.755598\pi$
$179$	−616812.	−1.43887	−0.719433	+	0.694562i	$0.755598\pi$
$180$	0	0
$181$	−37287.0	−0.0845981	−0.0422990	−	0.999105i	$-0.513468\pi$
$181$	−37287.0	−0.0845981	−0.0422990	+	0.999105i	$0.513468\pi$
$182$	− 86884.6i	− 0.194431i
$183$	665051.i	1.46800i
$184$	−621418.	−1.35313
$185$	0	0
$186$	188293.	0.399072
$187$	685513.i	1.43355i
$188$	− 83895.4i	− 0.173119i
$189$	−200635.	−0.408557
$190$	0	0
$191$	326760.	0.648106	0.324053	−	0.946039i	$-0.394954\pi$
$191$	326760.	0.648106	0.324053	+	0.946039i	$0.394954\pi$
$192$	284152.i	0.556285i
$193$	265735.i	0.513518i	0.966475	+	0.256759i	$0.0826547\pi$
$193$	265735.i	0.513518i	−0.966475	+	0.256759i	$0.917345\pi$
$194$	−368563.	−0.703085
$195$	0	0
$196$	−46894.2	−0.0871925
$197$	− 517865.i	− 0.950716i	−0.879792	−	0.475358i	$-0.842318\pi$
$197$	− 517865.i	− 0.950716i	0.879792	−	0.475358i	$-0.157682\pi$
$198$	142164.i	0.257707i
$199$	148687.	0.266158	0.133079	−	0.991105i	$-0.457514\pi$
$199$	148687.	0.266158	0.133079	+	0.991105i	$0.457514\pi$
$200$	0	0
$201$	−57878.0	−0.101047
$202$	161426.i	0.278352i
$203$	330844.i	0.563486i
$204$	−263188.	−0.442783
$205$	0	0
$206$	−315252.	−0.517595
$207$	− 198825.i	− 0.322512i
$208$	− 8807.15i	− 0.0141149i
$209$	457563.	0.724577
$210$	0	0
$211$	7443.09	0.0115093	0.00575463	−	0.999983i	$-0.498168\pi$
$211$	7443.09	0.0115093	0.00575463	+	0.999983i	$0.498168\pi$
$212$	507045.i	0.774831i
$213$	− 258151.i	− 0.389874i
$214$	−375465.	−0.560447
$215$	0	0
$216$	−745066.	−1.08658
$217$	192215.i	0.277101i
$218$	− 175909.i	− 0.250695i
$219$	−137735.	−0.194059
$220$	0	0
$221$	497788.	0.685589
$222$	− 30106.7i	− 0.0409997i
$223$	− 119157.i	− 0.160456i	−0.996777	−	0.0802280i	$-0.974435\pi$
$223$	− 119157.i	− 0.160456i	0.996777	−	0.0802280i	$-0.0255648\pi$
$224$	−282283.	−0.375894
$225$	0	0
$226$	131219.	0.170894
$227$	− 388843.i	− 0.500852i	−0.968136	−	0.250426i	$-0.919429\pi$
$227$	− 388843.i	− 0.500852i	0.968136	−	0.250426i	$-0.0805706\pi$
$228$	175671.i	0.223802i
$229$	−732622.	−0.923191	−0.461595	−	0.887091i	$-0.652723\pi$
$229$	−732622.	−0.923191	−0.461595	+	0.887091i	$0.652723\pi$
$230$	0	0
$231$	460605.	0.567934
$232$	1.22860e6i	1.49862i
$233$	1.12639e6i	1.35925i	0.733559	+	0.679626i	$0.237858\pi$
$233$	1.12639e6i	1.35925i	−0.733559	+	0.679626i	$0.762142\pi$
$234$	103233.	0.123248
$235$	0	0
$236$	−173878.	−0.203220
$237$	− 1.31837e6i	− 1.52464i
$238$	171522.i	0.196281i
$239$	−772317.	−0.874583	−0.437292	−	0.899320i	$-0.644062\pi$
$239$	−772317.	−0.874583	−0.437292	+	0.899320i	$0.644062\pi$
$240$	0	0
$241$	1.40297e6	1.55598	0.777991	−	0.628275i	$-0.216239\pi$
$241$	1.40297e6	1.55598	0.777991	+	0.628275i	$0.216239\pi$
$242$	− 1.11989e6i	− 1.22924i
$243$	− 430698.i	− 0.467905i
$244$	−955553.	−1.02750
$245$	0	0
$246$	−781337.	−0.823191
$247$	− 332261.i	− 0.346527i
$248$	713798.i	0.736964i
$249$	−961489.	−0.982757
$250$	0	0
$251$	−1.63922e6	−1.64230	−0.821151	−	0.570712i	$-0.806667\pi$
$251$	−1.63922e6	−1.64230	−0.821151	+	0.570712i	$0.806667\pi$
$252$	− 55717.9i	− 0.0552705i
$253$	2.36159e6i	2.31955i
$254$	−164234.	−0.159727
$255$	0	0
$256$	−1.05923e6	−1.01016
$257$	− 223664.i	− 0.211234i	−0.994407	−	0.105617i	$-0.966318\pi$
$257$	− 223664.i	− 0.211234i	0.994407	−	0.105617i	$-0.0336818\pi$
$258$	− 829330.i	− 0.775672i
$259$	30733.9	0.0284687
$260$	0	0
$261$	−393096.	−0.357188
$262$	− 1.34808e6i	− 1.21329i
$263$	− 299519.i	− 0.267014i	−0.991048	−	0.133507i	$-0.957376\pi$
$263$	− 299519.i	− 0.267014i	0.991048	−	0.133507i	$-0.0426239\pi$
$264$	1.71047e6	1.51045
$265$	0	0
$266$	114487.	0.0992091
$267$	57651.2i	0.0494914i
$268$	− 83159.7i	− 0.0707255i
$269$	−134341.	−0.113195	−0.0565974	−	0.998397i	$-0.518025\pi$
$269$	−134341.	−0.113195	−0.0565974	+	0.998397i	$0.518025\pi$
$270$	0	0
$271$	1.93414e6	1.59980	0.799898	−	0.600136i	$-0.204887\pi$
$271$	1.93414e6	1.59980	0.799898	+	0.600136i	$0.204887\pi$
$272$	17386.5i	0.0142492i
$273$	− 334470.i	− 0.271613i
$274$	−6689.88	−0.00538321
$275$	0	0
$276$	−906683.	−0.716445
$277$	177599.i	0.139072i	0.997579	+	0.0695362i	$0.0221519\pi$
$277$	177599.i	0.139072i	−0.997579	+	0.0695362i	$0.977848\pi$
$278$	712574.i	0.552991i
$279$	−228383.	−0.175652
$280$	0	0
$281$	1.85131e6	1.39867	0.699333	−	0.714796i	$-0.253481\pi$
$281$	1.85131e6	1.39867	0.699333	+	0.714796i	$0.253481\pi$
$282$	206183.i	0.154394i
$283$	− 2.39851e6i	− 1.78023i	−0.455737	−	0.890114i	$-0.650624\pi$
$283$	− 2.39851e6i	− 1.78023i	0.455737	−	0.890114i	$-0.349376\pi$
$284$	370914.	0.272883
$285$	0	0
$286$	−1.22617e6	−0.886413
$287$	− 797615.i	− 0.571595i
$288$	− 335398.i	− 0.238275i
$289$	437155.	0.307887
$290$	0	0
$291$	−1.41882e6	−0.982186
$292$	− 197899.i	− 0.135827i
$293$	− 2.49922e6i	− 1.70073i	−0.526196	−	0.850364i	$-0.676382\pi$
$293$	− 2.49922e6i	− 1.70073i	0.526196	−	0.850364i	$-0.323618\pi$
$294$	115248.	0.0777616
$295$	0	0
$296$	114131.	0.0757139
$297$	2.83149e6i	1.86262i
$298$	− 1.64634e6i	− 1.07394i
$299$	1.71488e6	1.10932
$300$	0	0
$301$	846607.	0.538599
$302$	431546.i	0.272276i
$303$	621422.i	0.388848i
$304$	11605.1	0.00720218
$305$	0	0
$306$	−203796.	−0.124421
$307$	3.07195e6i	1.86024i	0.367258	+	0.930119i	$0.380297\pi$
$307$	3.07195e6i	1.86024i	−0.367258	+	0.930119i	$0.619703\pi$
$308$	661802.i	0.397513i
$309$	−1.21359e6	−0.723063
$310$	0	0
$311$	661233.	0.387662	0.193831	−	0.981035i	$-0.437909\pi$
$311$	661233.	0.387662	0.193831	+	0.981035i	$0.437909\pi$
$312$	− 1.24206e6i	− 0.722367i
$313$	3.29393e6i	1.90043i	0.311588	+	0.950217i	$0.399139\pi$
$313$	3.29393e6i	1.90043i	−0.311588	+	0.950217i	$0.600861\pi$
$314$	−1.45002e6	−0.829944
$315$	0	0
$316$	1.89426e6	1.06714
$317$	639724.i	0.357556i	0.983889	+	0.178778i	$0.0572144\pi$
$317$	639724.i	0.357556i	−0.983889	+	0.178778i	$0.942786\pi$
$318$	− 1.24612e6i	− 0.691023i
$319$	4.66908e6	2.56894
$320$	0	0
$321$	−1.44538e6	−0.782926
$322$	590895.i	0.317593i
$323$	655929.i	0.349825i
$324$	−810777.	−0.429081
$325$	0	0
$326$	−277283.	−0.144504
$327$	− 677175.i	− 0.350213i
$328$	− 2.96197e6i	− 1.52018i
$329$	−210478.	−0.107206
$330$	0	0
$331$	−1.13876e6	−0.571298	−0.285649	−	0.958334i	$-0.592209\pi$
$331$	−1.13876e6	−0.571298	−0.285649	+	0.958334i	$0.592209\pi$
$332$	− 1.38148e6i	− 0.687858i
$333$	36516.8i	0.0180460i
$334$	−2.11060e6	−1.03524
$335$	0	0
$336$	11682.2	0.00564518
$337$	685493.i	0.328797i	0.986394	+	0.164399i	$0.0525684\pi$
$337$	685493.i	0.328797i	−0.986394	+	0.164399i	$0.947432\pi$
$338$	− 420694.i	− 0.200297i
$339$	505140.	0.238733
$340$	0	0
$341$	2.71267e6	1.26331
$342$	136029.i	0.0628877i
$343$	117649.i	0.0539949i
$344$	3.14390e6	1.43243
$345$	0	0
$346$	211423.	0.0949428
$347$	− 1.25151e6i	− 0.557970i	−0.960295	−	0.278985i	$-0.910002\pi$
$347$	− 1.25151e6i	− 0.557970i	0.960295	−	0.278985i	$-0.0899980\pi$
$348$	1.79259e6i	0.793476i
$349$	3.16606e6	1.39141	0.695706	−	0.718327i	$-0.255092\pi$
$349$	3.16606e6	1.39141	0.695706	+	0.718327i	$0.255092\pi$
$350$	0	0
$351$	2.05610e6	0.890793
$352$	3.98376e6i	1.71371i
$353$	2.43368e6i	1.03951i	0.854316	+	0.519754i	$0.173976\pi$
$353$	2.43368e6i	1.03951i	−0.854316	+	0.519754i	$0.826024\pi$
$354$	427326.	0.181239
$355$	0	0
$356$	−82833.8	−0.0346404
$357$	660290.i	0.274198i
$358$	2.17804e6i	0.898171i
$359$	2.13021e6	0.872341	0.436170	−	0.899864i	$-0.356335\pi$
$359$	2.13021e6	0.872341	0.436170	+	0.899864i	$0.356335\pi$
$360$	0	0
$361$	−2.03828e6	−0.823183
$362$	131665.i	0.0528079i
$363$	− 4.31112e6i	− 1.71721i
$364$	480570.	0.190109
$365$	0	0
$366$	2.34838e6	0.916360
$367$	− 3.10976e6i	− 1.20521i	−0.798041	−	0.602604i	$-0.794130\pi$
$367$	− 3.10976e6i	− 1.20521i	0.798041	−	0.602604i	$-0.205870\pi$
$368$	59896.7i	0.0230559i
$369$	947694.	0.362328
$370$	0	0
$371$	1.27208e6	0.479822
$372$	1.04147e6i	0.390202i
$373$	3.15189e6i	1.17300i	0.809948	+	0.586502i	$0.199495\pi$
$373$	3.15189e6i	1.17300i	−0.809948	+	0.586502i	$0.800505\pi$
$374$	2.42063e6	0.894849
$375$	0	0
$376$	−781617.	−0.285118
$377$	− 3.39047e6i	− 1.22859i
$378$	708469.i	0.255030i
$379$	−342350.	−0.122426	−0.0612129	−	0.998125i	$-0.519497\pi$
$379$	−342350.	−0.122426	−0.0612129	+	0.998125i	$0.519497\pi$
$380$	0	0
$381$	−632231.	−0.223133
$382$	− 1.15383e6i	− 0.404562i
$383$	− 3.69387e6i	− 1.28672i	−0.765563	−	0.643361i	$-0.777539\pi$
$383$	− 3.69387e6i	− 1.28672i	0.765563	−	0.643361i	$-0.222461\pi$
$384$	−1.50254e6	−0.519994
$385$	0	0
$386$	938345.	0.320549
$387$	1.00590e6i	0.341413i
$388$	− 2.03857e6i	− 0.687459i
$389$	−2.05313e6	−0.687928	−0.343964	−	0.938983i	$-0.611770\pi$
$389$	−2.05313e6	−0.687928	−0.343964	+	0.938983i	$0.611770\pi$
$390$	0	0
$391$	−3.38541e6	−1.11988
$392$	436893.i	0.143602i
$393$	− 5.18956e6i	− 1.69492i
$394$	−1.82865e6	−0.593457
$395$	0	0
$396$	−786326.	−0.251979
$397$	− 2.28107e6i	− 0.726377i	−0.931716	−	0.363189i	$-0.881688\pi$
$397$	− 2.28107e6i	− 0.726377i	0.931716	−	0.363189i	$-0.118312\pi$
$398$	− 525031.i	− 0.166141i
$399$	440727.	0.138592
$400$	0	0
$401$	3.32082e6	1.03130	0.515649	−	0.856800i	$-0.327551\pi$
$401$	3.32082e6	1.03130	0.515649	+	0.856800i	$0.327551\pi$
$402$	204375.i	0.0630756i
$403$	− 1.96981e6i	− 0.604175i
$404$	−892867.	−0.272166
$405$	0	0
$406$	1.16825e6	0.351740
$407$	− 433737.i	− 0.129790i
$408$	2.45201e6i	0.729242i
$409$	−5.15938e6	−1.52507	−0.762534	−	0.646948i	$-0.776045\pi$
$409$	−5.15938e6	−1.52507	−0.762534	+	0.646948i	$0.776045\pi$
$410$	0	0
$411$	−25753.3	−0.00752017
$412$	− 1.74370e6i	− 0.506092i
$413$	436229.i	0.125846i
$414$	−702078.	−0.201319
$415$	0	0
$416$	2.89283e6	0.819576
$417$	2.74311e6i	0.772509i
$418$	− 1.61571e6i	− 0.452297i
$419$	4.85187e6	1.35012	0.675062	−	0.737761i	$-0.264117\pi$
$419$	4.85187e6	1.35012	0.675062	+	0.737761i	$0.264117\pi$
$420$	0	0
$421$	−6.14767e6	−1.69046	−0.845231	−	0.534401i	$-0.820537\pi$
$421$	−6.14767e6	−1.69046	−0.845231	+	0.534401i	$0.820537\pi$
$422$	− 26282.5i	− 0.00718432i
$423$	− 250082.i	− 0.0679565i
$424$	4.72392e6	1.27611
$425$	0	0
$426$	−911563.	−0.243368
$427$	2.39731e6i	0.636288i
$428$	− 2.07674e6i	− 0.547991i
$429$	−4.72025e6	−1.23829
$430$	0	0
$431$	3.55411e6	0.921590	0.460795	−	0.887507i	$-0.347564\pi$
$431$	3.55411e6	0.921590	0.460795	+	0.887507i	$0.347564\pi$
$432$	71814.7i	0.0185141i
$433$	− 2.82650e6i	− 0.724485i	−0.932084	−	0.362243i	$-0.882011\pi$
$433$	− 2.82650e6i	− 0.724485i	0.932084	−	0.362243i	$-0.117989\pi$
$434$	678737.	0.172973
$435$	0	0
$436$	972973.	0.245123
$437$	2.25968e6i	0.566035i
$438$	486360.i	0.121136i
$439$	−4.64410e6	−1.15011	−0.575056	−	0.818114i	$-0.695020\pi$
$439$	−4.64410e6	−1.15011	−0.575056	+	0.818114i	$0.695020\pi$
$440$	0	0
$441$	−139786.	−0.0342268
$442$	− 1.75775e6i	− 0.427959i
$443$	6.15534e6i	1.49019i	0.666957	+	0.745097i	$0.267597\pi$
$443$	6.15534e6i	1.49019i	−0.666957	+	0.745097i	$0.732403\pi$
$444$	166524.	0.0400884
$445$	0	0
$446$	−420757.	−0.100160
$447$	− 6.33774e6i	− 1.50026i
$448$	1.02428e6i	0.241115i
$449$	−3.67035e6	−0.859196	−0.429598	−	0.903020i	$-0.641345\pi$
$449$	−3.67035e6	−0.859196	−0.429598	+	0.903020i	$0.641345\pi$
$450$	0	0
$451$	−1.12565e7	−2.60591
$452$	725791.i	0.167096i
$453$	1.66127e6i	0.380360i
$454$	−1.37305e6	−0.312642
$455$	0	0
$456$	1.63665e6	0.368591
$457$	− 866327.i	− 0.194040i	−0.995282	−	0.0970201i	$-0.969069\pi$
$457$	− 866327.i	− 0.194040i	0.995282	−	0.0970201i	$-0.0309311\pi$
$458$	2.58698e6i	0.576275i
$459$	−4.05903e6	−0.899271
$460$	0	0
$461$	−1.88572e6	−0.413261	−0.206631	−	0.978419i	$-0.566250\pi$
$461$	−1.88572e6	−0.413261	−0.206631	+	0.978419i	$0.566250\pi$
$462$	− 1.62645e6i	− 0.354517i
$463$	8.49017e6i	1.84062i	0.391192	+	0.920309i	$0.372063\pi$
$463$	8.49017e6i	1.84062i	−0.391192	+	0.920309i	$0.627937\pi$
$464$	118421.	0.0255349
$465$	0	0
$466$	3.97744e6	0.848474
$467$	1.82738e6i	0.387737i	0.981028	+	0.193868i	$0.0621035\pi$
$467$	1.82738e6i	0.387737i	−0.981028	+	0.193868i	$0.937896\pi$
$468$	570994.i	0.120508i
$469$	−208632.	−0.0437975
$470$	0	0
$471$	−5.58197e6	−1.15940
$472$	1.61995e6i	0.334693i
$473$	− 1.19479e7i	− 2.45549i
$474$	−4.65535e6	−0.951714
$475$	0	0
$476$	−948712.	−0.191919
$477$	1.51144e6i	0.304155i
$478$	2.72715e6i	0.545933i
$479$	−4.68744e6	−0.933463	−0.466731	−	0.884399i	$-0.654568\pi$
$479$	−4.68744e6	−0.933463	−0.466731	+	0.884399i	$0.654568\pi$
$480$	0	0
$481$	−314960.	−0.0620715
$482$	− 4.95405e6i	− 0.971277i
$483$	2.27470e6i	0.443666i
$484$	6.19426e6	1.20192
$485$	0	0
$486$	−1.52085e6	−0.292076
$487$	4.59651e6i	0.878225i	0.898432	+	0.439112i	$0.144707\pi$
$487$	4.59651e6i	0.878225i	−0.898432	+	0.439112i	$0.855293\pi$
$488$	8.90247e6i	1.69224i
$489$	−1.06743e6	−0.201867
$490$	0	0
$491$	6.62099e6	1.23942	0.619711	−	0.784830i	$-0.287250\pi$
$491$	6.62099e6	1.23942	0.619711	+	0.784830i	$0.287250\pi$
$492$	− 4.32167e6i	− 0.804895i
$493$	6.69327e6i	1.24028i
$494$	−1.17326e6	−0.216310
$495$	0	0
$496$	68800.9	0.0125571
$497$	− 930554.i	− 0.168986i
$498$	3.39514e6i	0.613458i
$499$	4.52632e6	0.813756	0.406878	−	0.913482i	$-0.366617\pi$
$499$	4.52632e6	0.813756	0.406878	+	0.913482i	$0.366617\pi$
$500$	0	0
$501$	−8.12495e6	−1.44619
$502$	5.78829e6i	1.02516i
$503$	− 3.83316e6i	− 0.675518i	−0.941233	−	0.337759i	$-0.890331\pi$
$503$	− 3.83316e6i	− 0.675518i	0.941233	−	0.337759i	$-0.109669\pi$
$504$	−519099.	−0.0910278
$505$	0	0
$506$	8.33909e6	1.44791
$507$	− 1.61950e6i	− 0.279808i
$508$	− 908397.i	− 0.156177i
$509$	3.25460e6	0.556806	0.278403	−	0.960464i	$-0.410195\pi$
$509$	3.25460e6	0.556806	0.278403	+	0.960464i	$0.410195\pi$
$510$	0	0
$511$	−496492.	−0.0841124
$512$	203165.i	0.0342511i
$513$	2.70930e6i	0.454531i
$514$	−789788.	−0.131857
$515$	0	0
$516$	4.58713e6	0.758432
$517$	2.97040e6i	0.488752i
$518$	− 108525.i	− 0.0177708i
$519$	813891.	0.132632
$520$	0	0
$521$	−1.07842e6	−0.174057	−0.0870287	−	0.996206i	$-0.527737\pi$
$521$	−1.07842e6	−0.174057	−0.0870287	+	0.996206i	$0.527737\pi$
$522$	1.38807e6i	0.222964i
$523$	− 408626.i	− 0.0653238i	−0.999466	−	0.0326619i	$-0.989602\pi$
$523$	− 408626.i	− 0.0653238i	0.999466	−	0.0326619i	$-0.0103985\pi$
$524$	7.45641e6	1.18632
$525$	0	0
$526$	−1.05764e6	−0.166676
$527$	3.88869e6i	0.609925i
$528$	− 164867.i	− 0.0257365i
$529$	−5.22642e6	−0.812016
$530$	0	0
$531$	−518310.	−0.0797724
$532$	633242.i	0.0970042i
$533$	8.17392e6i	1.24627i
$534$	203574.	0.0308936
$535$	0	0
$536$	−774763.	−0.116481
$537$	8.38456e6i	1.25471i
$538$	474374.i	0.0706586i
$539$	1.66034e6	0.246164
$540$	0	0
$541$	1.13918e7	1.67340	0.836701	−	0.547659i	$-0.184481\pi$
$541$	1.13918e7	1.67340	0.836701	+	0.547659i	$0.184481\pi$
$542$	− 6.82970e6i	− 0.998627i
$543$	506856.i	0.0737709i
$544$	−5.71084e6	−0.827376
$545$	0	0
$546$	−1.18106e6	−0.169547
$547$	− 3.44866e6i	− 0.492813i	−0.969167	−	0.246406i	$-0.920750\pi$
$547$	− 3.44866e6i	− 0.492813i	0.969167	−	0.246406i	$-0.0792498\pi$
$548$	− 37002.6i	− 0.00526357i
$549$	−2.84838e6	−0.403337
$550$	0	0
$551$	4.46759e6	0.626894
$552$	8.44718e6i	1.17995i
$553$	− 4.75234e6i	− 0.660837i
$554$	627124.	0.0868119
$555$	0	0
$556$	−3.94134e6	−0.540700
$557$	− 4.69772e6i	− 0.641577i	−0.947151	−	0.320789i	$-0.896052\pi$
$557$	− 4.69772e6i	− 0.641577i	0.947151	−	0.320789i	$-0.103948\pi$
$558$	806449.i	0.109646i
$559$	−8.67599e6	−1.17433
$560$	0	0
$561$	9.31844e6	1.25007
$562$	− 6.53722e6i	− 0.873077i
$563$	561642.i	0.0746772i	0.999303	+	0.0373386i	$0.0118880\pi$
$563$	561642.i	0.0746772i	−0.999303	+	0.0373386i	$0.988112\pi$
$564$	−1.14042e6	−0.150962
$565$	0	0
$566$	−8.46945e6	−1.11126
$567$	2.03409e6i	0.265713i
$568$	− 3.45564e6i	− 0.449426i
$569$	−5.19019e6	−0.672051	−0.336026	−	0.941853i	$-0.609083\pi$
$569$	−5.19019e6	−0.672051	−0.336026	+	0.941853i	$0.609083\pi$
$570$	0	0
$571$	−5.20274e6	−0.667793	−0.333897	−	0.942610i	$-0.608364\pi$
$571$	−5.20274e6	−0.667793	−0.333897	+	0.942610i	$0.608364\pi$
$572$	− 6.78211e6i	− 0.866712i
$573$	− 4.44178e6i	− 0.565159i
$574$	−2.81648e6	−0.356802
$575$	0	0
$576$	−1.21701e6	−0.152840
$577$	8.70662e6i	1.08870i	0.838857	+	0.544352i	$0.183225\pi$
$577$	8.70662e6i	1.08870i	−0.838857	+	0.544352i	$0.816775\pi$
$578$	− 1.54365e6i	− 0.192189i
$579$	3.61224e6	0.447796
$580$	0	0
$581$	−3.46588e6	−0.425964
$582$	5.01003e6i	0.613102i
$583$	− 1.79524e7i	− 2.18752i
$584$	−1.84374e6	−0.223701
$585$	0	0
$586$	−8.82505e6	−1.06163
$587$	− 4.35717e6i	− 0.521926i	−0.965349	−	0.260963i	$-0.915960\pi$
$587$	− 4.35717e6i	− 0.521926i	0.965349	−	0.260963i	$-0.0840401\pi$
$588$	637452.i	0.0760333i
$589$	2.59560e6	0.308283
$590$	0	0
$591$	−7.03954e6	−0.829040
$592$	− 11000.8i	− 0.00129009i
$593$	3.22991e6i	0.377184i	0.982055	+	0.188592i	$0.0603924\pi$
$593$	3.22991e6i	0.377184i	−0.982055	+	0.188592i	$0.939608\pi$
$594$	9.99837e6	1.16269
$595$	0	0
$596$	9.10614e6	1.05007
$597$	− 2.02115e6i	− 0.232094i
$598$	− 6.05547e6i	− 0.692460i
$599$	7.23988e6	0.824450	0.412225	−	0.911082i	$-0.364752\pi$
$599$	7.23988e6	0.824450	0.412225	+	0.911082i	$0.364752\pi$
$600$	0	0
$601$	−1.06837e7	−1.20652	−0.603262	−	0.797543i	$-0.706133\pi$
$601$	−1.06837e7	−1.20652	−0.603262	+	0.797543i	$0.706133\pi$
$602$	− 2.98948e6i	− 0.336205i
$603$	− 247889.i	− 0.0277628i
$604$	−2.38693e6	−0.266225
$605$	0	0
$606$	2.19432e6	0.242727
$607$	2.51528e6i	0.277086i	0.990356	+	0.138543i	$0.0442419\pi$
$607$	2.51528e6i	0.277086i	−0.990356	+	0.138543i	$0.955758\pi$
$608$	3.81185e6i	0.418193i
$609$	4.49729e6	0.491369
$610$	0	0
$611$	2.15697e6	0.233744
$612$	− 1.12722e6i	− 0.121655i
$613$	− 213999.i	− 0.0230017i	−0.999934	−	0.0115009i	$-0.996339\pi$
$613$	− 213999.i	− 0.0230017i	0.999934	−	0.0115009i	$-0.00366092\pi$
$614$	1.08475e7	1.16120
$615$	0	0
$616$	6.16572e6	0.654684
$617$	− 127951.i	− 0.0135310i	−0.999977	−	0.00676552i	$-0.997846\pi$
$617$	− 127951.i	− 0.0135310i	0.999977	−	0.00676552i	$-0.00215355\pi$
$618$	4.28535e6i	0.451352i
$619$	1.23980e7	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$619$	1.23980e7	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$620$	0	0
$621$	−1.39834e7	−1.45507
$622$	− 2.33490e6i	− 0.241987i
$623$	207815.i	0.0214514i
$624$	−119719.	−0.0123084
$625$	0	0
$626$	1.16313e7	1.18629
$627$	− 6.21983e6i	− 0.631843i
$628$	− 8.02023e6i	− 0.811499i
$629$	621774.	0.0626622
$630$	0	0
$631$	−5.87683e6	−0.587584	−0.293792	−	0.955869i	$-0.594917\pi$
$631$	−5.87683e6	−0.587584	−0.293792	+	0.955869i	$0.594917\pi$
$632$	− 1.76480e7i	− 1.75753i
$633$	− 101177.i	− 0.0100363i
$634$	2.25895e6	0.223194
$635$	0	0
$636$	6.89246e6	0.675665
$637$	− 1.20566e6i	− 0.117727i
$638$	− 1.64871e7i	− 1.60359i
$639$	1.10565e6	0.107118
$640$	0	0
$641$	−2.60122e6	−0.250053	−0.125026	−	0.992153i	$-0.539902\pi$
$641$	−2.60122e6	−0.250053	−0.125026	+	0.992153i	$0.539902\pi$
$642$	5.10384e6i	0.488719i
$643$	− 1.31345e7i	− 1.25282i	−0.779495	−	0.626408i	$-0.784524\pi$
$643$	− 1.31345e7i	− 1.25282i	0.779495	−	0.626408i	$-0.215476\pi$
$644$	−3.26832e6	−0.310534
$645$	0	0
$646$	2.31617e6	0.218368
$647$	5.54662e6i	0.520916i	0.965485	+	0.260458i	$0.0838736\pi$
$647$	5.54662e6i	0.520916i	−0.965485	+	0.260458i	$0.916126\pi$
$648$	7.55366e6i	0.706675i
$649$	6.15634e6	0.573734
$650$	0	0
$651$	2.61286e6	0.241637
$652$	− 1.53369e6i	− 0.141292i
$653$	1.54993e7i	1.42242i	0.702978	+	0.711212i	$0.251853\pi$
$653$	1.54993e7i	1.42242i	−0.702978	+	0.711212i	$0.748147\pi$
$654$	−2.39119e6	−0.218610
$655$	0	0
$656$	−285495.	−0.0259024
$657$	− 589912.i	− 0.0533180i
$658$	743225.i	0.0669200i
$659$	4.30145e6	0.385835	0.192917	−	0.981215i	$-0.438205\pi$
$659$	4.30145e6	0.385835	0.192917	+	0.981215i	$0.438205\pi$
$660$	0	0
$661$	861980.	0.0767350	0.0383675	−	0.999264i	$-0.487784\pi$
$661$	861980.	0.0767350	0.0383675	+	0.999264i	$0.487784\pi$
$662$	4.02111e6i	0.356616i
$663$	− 6.76662e6i	− 0.597844i
$664$	−1.28706e7	−1.13287
$665$	0	0
$666$	128945.	0.0112647
$667$	2.30583e7i	2.00684i
$668$	− 1.16740e7i	− 1.01223i
$669$	−1.61974e6	−0.139920
$670$	0	0
$671$	3.38323e7	2.90085
$672$	3.83719e6i	0.327786i
$673$	953818.i	0.0811760i	0.999176	+	0.0405880i	$0.0129231\pi$
$673$	953818.i	0.0811760i	−0.999176	+	0.0405880i	$0.987077\pi$
$674$	2.42056e6	0.205242
$675$	0	0
$676$	2.32691e6	0.195845
$677$	− 1.48606e7i	− 1.24613i	−0.782170	−	0.623065i	$-0.785887\pi$
$677$	− 1.48606e7i	− 1.24613i	0.782170	−	0.623065i	$-0.214113\pi$
$678$	− 1.78371e6i	− 0.149022i
$679$	−5.11440e6	−0.425716
$680$	0	0
$681$	−5.28569e6	−0.436751
$682$	− 9.57878e6i	− 0.788586i
$683$	1.57605e7i	1.29276i	0.763015	+	0.646381i	$0.223718\pi$
$683$	1.57605e7i	1.29276i	−0.763015	+	0.646381i	$0.776282\pi$
$684$	−752392.	−0.0614900
$685$	0	0
$686$	415434.	0.0337048
$687$	9.95882e6i	0.805037i
$688$	− 303032.i	− 0.0244071i
$689$	−1.30362e7	−1.04618
$690$	0	0
$691$	1.59740e7	1.27267	0.636337	−	0.771411i	$-0.280449\pi$
$691$	1.59740e7	1.27267	0.636337	+	0.771411i	$0.280449\pi$
$692$	1.16941e6i	0.0928326i
$693$	1.97275e6i	0.156041i
$694$	−4.41925e6	−0.348297
$695$	0	0
$696$	1.67008e7	1.30682
$697$	− 1.61365e7i	− 1.25813i
$698$	− 1.11798e7i	− 0.868549i
$699$	1.53115e7	1.18529
$700$	0	0
$701$	−1.85736e7	−1.42758	−0.713790	−	0.700360i	$-0.753023\pi$
$701$	−1.85736e7	−1.42758	−0.713790	+	0.700360i	$0.753023\pi$
$702$	− 7.26036e6i	− 0.556052i
$703$	− 415019.i	− 0.0316723i
$704$	1.44553e7	1.09925
$705$	0	0
$706$	8.59365e6	0.648883
$707$	2.24004e6i	0.168541i
$708$	2.36360e6i	0.177211i
$709$	1.71029e7	1.27778	0.638888	−	0.769300i	$-0.279395\pi$
$709$	1.71029e7	1.27778	0.638888	+	0.769300i	$0.279395\pi$
$710$	0	0
$711$	5.64654e6	0.418898
$712$	771727.i	0.0570511i
$713$	1.33965e7i	0.986890i
$714$	2.33157e6	0.171160
$715$	0	0
$716$	−1.20470e7	−0.878208
$717$	1.04984e7i	0.762651i
$718$	− 7.52204e6i	− 0.544534i
$719$	1.40945e7	1.01678	0.508390	−	0.861127i	$-0.330241\pi$
$719$	1.40945e7	1.01678	0.508390	+	0.861127i	$0.330241\pi$
$720$	0	0
$721$	−4.37463e6	−0.313403
$722$	7.19744e6i	0.513848i
$723$	− 1.90711e7i	− 1.35684i
$724$	−728256.	−0.0516343
$725$	0	0
$726$	−1.52231e7	−1.07192
$727$	2.24196e7i	1.57323i	0.617443	+	0.786616i	$0.288169\pi$
$727$	2.24196e7i	1.57323i	−0.617443	+	0.786616i	$0.711831\pi$
$728$	− 4.47726e6i	− 0.313101i
$729$	−1.59421e7	−1.11103
$730$	0	0
$731$	1.71276e7	1.18551
$732$	1.29892e7i	0.895993i
$733$	8.02464e6i	0.551653i	0.961208	+	0.275826i	$0.0889514\pi$
$733$	8.02464e6i	0.551653i	−0.961208	+	0.275826i	$0.911049\pi$
$734$	−1.09810e7	−0.752316
$735$	0	0
$736$	−1.96739e7	−1.33874
$737$	2.94435e6i	0.199674i
$738$	− 3.34643e6i	− 0.226173i
$739$	−1.15678e7	−0.779181	−0.389591	−	0.920988i	$-0.627384\pi$
$739$	−1.15678e7	−0.779181	−0.389591	+	0.920988i	$0.627384\pi$
$740$	0	0
$741$	−4.51655e6	−0.302177
$742$	− 4.49189e6i	− 0.299515i
$743$	− 3.79387e6i	− 0.252122i	−0.992023	−	0.126061i	$-0.959767\pi$
$743$	− 3.79387e6i	− 0.252122i	0.992023	−	0.126061i	$-0.0402334\pi$
$744$	9.70293e6	0.642644
$745$	0	0
$746$	1.11297e7	0.732214
$747$	− 4.11801e6i	− 0.270014i
$748$	1.33888e7i	0.874961i
$749$	−5.21017e6	−0.339349
$750$	0	0
$751$	−9.07884e6	−0.587395	−0.293698	−	0.955898i	$-0.594886\pi$
$751$	−9.07884e6	−0.587395	−0.293698	+	0.955898i	$0.594886\pi$
$752$	75337.8i	0.00485812i
$753$	2.22825e7i	1.43211i
$754$	−1.19722e7	−0.766912
$755$	0	0
$756$	−3.91863e6	−0.249362
$757$	− 2.33210e7i	− 1.47913i	−0.673085	−	0.739565i	$-0.735031\pi$
$757$	− 2.33210e7i	− 1.47913i	0.673085	−	0.739565i	$-0.264969\pi$
$758$	1.20888e6i	0.0764208i
$759$	3.21020e7	2.02269
$760$	0	0
$761$	1.44859e7	0.906745	0.453373	−	0.891321i	$-0.350221\pi$
$761$	1.44859e7	0.906745	0.453373	+	0.891321i	$0.350221\pi$
$762$	2.23249e6i	0.139284i
$763$	− 2.44101e6i	− 0.151795i
$764$	6.38200e6	0.395570
$765$	0	0
$766$	−1.30435e7	−0.803200
$767$	− 4.47045e6i	− 0.274387i
$768$	1.43985e7i	0.880876i
$769$	−1.59424e7	−0.972162	−0.486081	−	0.873914i	$-0.661574\pi$
$769$	−1.59424e7	−0.972162	−0.486081	+	0.873914i	$0.661574\pi$
$770$	0	0
$771$	−3.04036e6	−0.184200
$772$	5.19011e6i	0.313425i
$773$	3.36066e6i	0.202290i	0.994872	+	0.101145i	$0.0322507\pi$
$773$	3.36066e6i	0.202290i	−0.994872	+	0.101145i	$0.967749\pi$
$774$	3.55198e6	0.213117
$775$	0	0
$776$	−1.89925e7	−1.13221
$777$	− 417778.i	− 0.0248252i
$778$	7.24988e6i	0.429419i
$779$	−1.07707e7	−0.635916
$780$	0	0
$781$	−1.31326e7	−0.770411
$782$	1.19543e7i	0.699050i
$783$	2.76464e7i	1.61151i
$784$	42110.9	0.00244683
$785$	0	0
$786$	−1.83250e7	−1.05800
$787$	− 3.28918e6i	− 0.189300i	−0.995511	−	0.0946500i	$-0.969827\pi$
$787$	− 3.28918e6i	− 0.189300i	0.995511	−	0.0946500i	$-0.0301732\pi$
$788$	− 1.01145e7i	− 0.580268i
$789$	−4.07147e6	−0.232841
$790$	0	0
$791$	1.82088e6	0.103476
$792$	7.32586e6i	0.414998i
$793$	− 2.45675e7i	− 1.38732i
$794$	−8.05475e6	−0.453420
$795$	0	0
$796$	2.90402e6	0.162449
$797$	− 2.51939e6i	− 0.140492i	−0.997530	−	0.0702458i	$-0.977622\pi$
$797$	− 2.51939e6i	− 0.140492i	0.997530	−	0.0702458i	$-0.0223783\pi$
$798$	− 1.55626e6i	− 0.0865120i
$799$	−4.25816e6	−0.235969
$800$	0	0
$801$	−246917.	−0.0135978
$802$	− 1.17262e7i	− 0.643758i
$803$	7.00682e6i	0.383470i
$804$	−1.13042e6	−0.0616738
$805$	0	0
$806$	−6.95567e6	−0.377139
$807$	1.82614e6i	0.0987077i
$808$	8.31845e6i	0.448244i
$809$	−8.93808e6	−0.480146	−0.240073	−	0.970755i	$-0.577171\pi$
$809$	−8.93808e6	−0.480146	−0.240073	+	0.970755i	$0.577171\pi$
$810$	0	0
$811$	3.01341e7	1.60881	0.804406	−	0.594080i	$-0.202484\pi$
$811$	3.01341e7	1.60881	0.804406	+	0.594080i	$0.202484\pi$
$812$	6.46176e6i	0.343922i
$813$	− 2.62915e7i	− 1.39505i
$814$	−1.53158e6	−0.0810174
$815$	0	0
$816$	236342.	0.0124255
$817$	− 1.14323e7i	− 0.599207i
$818$	1.82184e7i	0.951980i
$819$	1.43252e6	0.0746261
$820$	0	0
$821$	−3.25440e7	−1.68505	−0.842524	−	0.538658i	$-0.818931\pi$
$821$	−3.25440e7	−1.68505	−0.842524	+	0.538658i	$0.818931\pi$
$822$	90938.1i	0.00469425i
$823$	499640.i	0.0257133i	0.999917	+	0.0128566i	$0.00409251\pi$
$823$	499640.i	0.0257133i	−0.999917	+	0.0128566i	$0.995907\pi$
$824$	−1.62453e7	−0.833509
$825$	0	0
$826$	1.54038e6	0.0785557
$827$	− 4.64084e6i	− 0.235957i	−0.993016	−	0.117978i	$-0.962359\pi$
$827$	− 4.64084e6i	− 0.235957i	0.993016	−	0.117978i	$-0.0376414\pi$
$828$	− 3.88328e6i	− 0.196844i
$829$	3.08794e7	1.56057	0.780283	−	0.625427i	$-0.215075\pi$
$829$	3.08794e7	1.56057	0.780283	+	0.625427i	$0.215075\pi$
$830$	0	0
$831$	2.41417e6	0.121273
$832$	− 1.04968e7i	− 0.525712i
$833$	2.38014e6i	0.118848i
$834$	9.68629e6	0.482217
$835$	0	0
$836$	8.93671e6	0.442244
$837$	1.60621e7i	0.792482i
$838$	− 1.71326e7i	− 0.842777i
$839$	−1.93485e7	−0.948948	−0.474474	−	0.880269i	$-0.657362\pi$
$839$	−1.93485e7	−0.948948	−0.474474	+	0.880269i	$0.657362\pi$
$840$	0	0
$841$	2.50772e7	1.22261
$842$	2.17082e7i	1.05522i
$843$	− 2.51656e7i	− 1.21966i
$844$	145372.	0.00702465
$845$	0	0
$846$	−883071.	−0.0424199
$847$	− 1.55403e7i	− 0.744303i
$848$	− 455325.i	− 0.0217436i
$849$	−3.26039e7	−1.55239
$850$	0	0
$851$	2.14201e6	0.101391
$852$	− 5.04197e6i	− 0.237959i
$853$	3.05998e7i	1.43994i	0.694003	+	0.719972i	$0.255845\pi$
$853$	3.05998e7i	1.43994i	−0.694003	+	0.719972i	$0.744155\pi$
$854$	8.46520e6	0.397185
$855$	0	0
$856$	−1.93481e7	−0.902515
$857$	− 1.33039e7i	− 0.618766i	−0.950937	−	0.309383i	$-0.899877\pi$
$857$	− 1.33039e7i	− 0.618766i	0.950937	−	0.309383i	$-0.100123\pi$
$858$	1.66678e7i	0.772967i
$859$	1.27708e7	0.590520	0.295260	−	0.955417i	$-0.404594\pi$
$859$	1.27708e7	0.590520	0.295260	+	0.955417i	$0.404594\pi$
$860$	0	0
$861$	−1.08423e7	−0.498440
$862$	− 1.25500e7i	− 0.575276i
$863$	− 1.37987e7i	− 0.630684i	−0.948978	−	0.315342i	$-0.897881\pi$
$863$	− 1.37987e7i	− 0.630684i	0.948978	−	0.315342i	$-0.102119\pi$
$864$	−2.35885e7	−1.07502
$865$	0	0
$866$	−9.98074e6	−0.452239
$867$	− 5.94241e6i	− 0.268482i
$868$	3.75418e6i	0.169128i
$869$	−6.70680e7	−3.01277
$870$	0	0
$871$	2.13806e6	0.0954934
$872$	− 9.06477e6i	− 0.403706i
$873$	− 6.07673e6i	− 0.269857i
$874$	7.97922e6	0.353331
$875$	0	0
$876$	−2.69012e6	−0.118443
$877$	6.68992e6i	0.293712i	0.989158	+	0.146856i	$0.0469154\pi$
$877$	6.68992e6i	0.293712i	−0.989158	+	0.146856i	$0.953085\pi$
$878$	1.63989e7i	0.717924i
$879$	−3.39728e7	−1.48306
$880$	0	0
$881$	−7.25051e6	−0.314723	−0.157362	−	0.987541i	$-0.550299\pi$
$881$	−7.25051e6	−0.314723	−0.157362	+	0.987541i	$0.550299\pi$
$882$	493602.i	0.0213651i
$883$	2.55944e7i	1.10470i	0.833613	+	0.552348i	$0.186268\pi$
$883$	2.55944e7i	1.10470i	−0.833613	+	0.552348i	$0.813732\pi$
$884$	9.72236e6	0.418447
$885$	0	0
$886$	2.17353e7	0.930210
$887$	− 2.33204e7i	− 0.995240i	−0.867395	−	0.497620i	$-0.834207\pi$
$887$	− 2.33204e7i	− 0.995240i	0.867395	−	0.497620i	$-0.165793\pi$
$888$	− 1.55143e6i	− 0.0660237i
$889$	−2.27900e6	−0.0967141
$890$	0	0
$891$	2.87064e7	1.21139
$892$	− 2.32726e6i	− 0.0979340i
$893$	2.84222e6i	0.119269i
$894$	−2.23794e7	−0.936493
$895$	0	0
$896$	−5.41620e6	−0.225385
$897$	− 2.33110e7i	− 0.967343i
$898$	1.29605e7i	0.536328i
$899$	2.64862e7	1.09300
$900$	0	0
$901$	2.57354e7	1.05613
$902$	3.97480e7i	1.62667i
$903$	− 1.15083e7i	− 0.469667i
$904$	6.76188e6	0.275199
$905$	0	0
$906$	5.86617e6	0.237429
$907$	3.71721e7i	1.50037i	0.661226	+	0.750187i	$0.270036\pi$
$907$	3.71721e7i	1.50037i	−0.661226	+	0.750187i	$0.729964\pi$
$908$	− 7.59453e6i	− 0.305694i
$909$	−2.66152e6	−0.106837
$910$	0	0
$911$	−2.89923e7	−1.15741	−0.578704	−	0.815537i	$-0.696441\pi$
$911$	−2.89923e7	−1.15741	−0.578704	+	0.815537i	$0.696441\pi$
$912$	− 157752.i	− 0.00628042i
$913$	4.89127e7i	1.94198i
$914$	−3.05911e6	−0.121124
$915$	0	0
$916$	−1.43089e7	−0.563468
$917$	− 1.87068e7i	− 0.734641i
$918$	1.43330e7i	0.561344i
$919$	1.25020e6	0.0488306	0.0244153	−	0.999702i	$-0.492228\pi$
$919$	1.25020e6	0.0488306	0.0244153	+	0.999702i	$0.492228\pi$
$920$	0	0
$921$	4.17582e7	1.62216
$922$	6.65872e6i	0.257966i
$923$	9.53627e6i	0.368447i
$924$	8.99613e6	0.346638
$925$	0	0
$926$	2.99799e7	1.14895
$927$	− 5.19776e6i	− 0.198663i
$928$	3.88970e7i	1.48268i
$929$	−6.87875e6	−0.261499	−0.130749	−	0.991415i	$-0.541738\pi$
$929$	−6.87875e6	−0.261499	−0.130749	+	0.991415i	$0.541738\pi$
$930$	0	0
$931$	1.58869e6	0.0600709
$932$	2.19997e7i	0.829616i
$933$	− 8.98840e6i	− 0.338048i
$934$	6.45272e6	0.242034
$935$	0	0
$936$	5.31971e6	0.198471
$937$	2.18600e7i	0.813396i	0.913563	+	0.406698i	$0.133320\pi$
$937$	2.18600e7i	0.813396i	−0.913563	+	0.406698i	$0.866680\pi$
$938$	736708.i	0.0273393i
$939$	4.47756e7	1.65721
$940$	0	0
$941$	2.26450e6	0.0833679	0.0416840	−	0.999131i	$-0.486728\pi$
$941$	2.26450e6	0.0833679	0.0416840	+	0.999131i	$0.486728\pi$
$942$	1.97106e7i	0.723725i
$943$	− 5.55901e7i	− 2.03572i
$944$	156142.	0.00570283
$945$	0	0
$946$	−4.21895e7	−1.53277
$947$	− 2.45846e7i	− 0.890815i	−0.895328	−	0.445408i	$-0.853059\pi$
$947$	− 2.45846e7i	− 0.890815i	0.895328	−	0.445408i	$-0.146941\pi$
$948$	− 2.57493e7i	− 0.930562i
$949$	5.08803e6	0.183394
$950$	0	0
$951$	8.69601e6	0.311795
$952$	8.83874e6i	0.316081i
$953$	− 4.59681e7i	− 1.63955i	−0.572687	−	0.819774i	$-0.694099\pi$
$953$	− 4.59681e7i	− 1.63955i	0.572687	−	0.819774i	$-0.305901\pi$
$954$	5.33708e6	0.189860
$955$	0	0
$956$	−1.50842e7	−0.533800
$957$	− 6.34686e7i	− 2.24016i
$958$	1.65520e7i	0.582687i
$959$	−92832.6	−0.00325952
$960$	0	0
$961$	−1.32411e7	−0.462503
$962$	1.11216e6i	0.0387463i
$963$	− 6.19051e6i	− 0.215110i
$964$	2.74015e7	0.949690
$965$	0	0
$966$	8.03226e6	0.276946
$967$	− 5.30202e7i	− 1.82337i	−0.410888	−	0.911686i	$-0.634781\pi$
$967$	− 5.30202e7i	− 1.82337i	0.410888	−	0.911686i	$-0.365219\pi$
$968$	− 5.77093e7i	− 1.97951i
$969$	8.91630e6	0.305053
$970$	0	0
$971$	−5.43523e7	−1.84999	−0.924996	−	0.379976i	$-0.875932\pi$
$971$	−5.43523e7	−1.84999	−0.924996	+	0.379976i	$0.875932\pi$
$972$	− 8.41202e6i	− 0.285585i
$973$	9.88809e6i	0.334835i
$974$	1.62309e7	0.548207
$975$	0	0
$976$	858083.	0.0288340
$977$	1.08929e7i	0.365097i	0.983197	+	0.182549i	$0.0584347\pi$
$977$	1.08929e7i	0.365097i	−0.983197	+	0.182549i	$0.941565\pi$
$978$	3.76922e6i	0.126010i
$979$	2.93282e6	0.0977976
$980$	0	0
$981$	2.90031e6	0.0962215
$982$	− 2.33796e7i	− 0.773674i
$983$	3.42136e7i	1.12932i	0.825325	+	0.564658i	$0.190992\pi$
$983$	3.42136e7i	1.12932i	−0.825325	+	0.564658i	$0.809008\pi$
$984$	−4.02632e7	−1.32562
$985$	0	0
$986$	2.36348e7	0.774211
$987$	2.86111e6i	0.0934850i
$988$	− 6.48943e6i	− 0.211502i
$989$	5.90047e7	1.91821
$990$	0	0
$991$	4.22117e6	0.136537	0.0682683	−	0.997667i	$-0.478253\pi$
$991$	4.22117e6	0.136537	0.0682683	+	0.997667i	$0.478253\pi$
$992$	2.25986e7i	0.729125i
$993$	1.54796e7i	0.498181i
$994$	−3.28591e6	−0.105485
$995$	0	0
$996$	−1.87790e7	−0.599824
$997$	4.30323e7i	1.37106i	0.728044	+	0.685530i	$0.240429\pi$
$997$	4.30323e7i	1.37106i	−0.728044	+	0.685530i	$0.759571\pi$
$998$	− 1.59830e7i	− 0.507964i
$999$	2.56822e6	0.0814177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.6.b.d.99.2		4
5.2	odd	4		175.6.a.d.1.2		2
5.3	odd	4		35.6.a.b.1.1	✓	2
5.4	even	2	inner	175.6.b.d.99.3		4
15.8	even	4		315.6.a.c.1.2		2
20.3	even	4		560.6.a.l.1.1		2
35.13	even	4		245.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.a.b.1.1	✓	2	5.3	odd	4
175.6.a.d.1.2		2	5.2	odd	4
175.6.b.d.99.2		4	1.1	even	1	trivial
175.6.b.d.99.3		4	5.4	even	2	inner
245.6.a.c.1.1		2	35.13	even	4
315.6.a.c.1.2		2	15.8	even	4
560.6.a.l.1.1		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.53113i q^{2} -13.5934i q^{3} +19.5311 q^{4} -48.0000 q^{6} -49.0000i q^{7} -181.963i q^{8} +58.2198 q^{9} +O(q^{10})\)	\(q-3.53113i q^{2} -13.5934i q^{3} +19.5311 q^{4} -48.0000 q^{6} -49.0000i q^{7} -181.963i q^{8} +58.2198 q^{9} -691.520 q^{11} -265.494i q^{12} +502.150i q^{13} -173.025 q^{14} -17.5389 q^{16} -991.313i q^{17} -205.582i q^{18} -661.677 q^{19} -666.076 q^{21} +2441.84i q^{22} -3415.08i q^{23} -2473.49 q^{24} +1773.16 q^{26} -4094.60i q^{27} -957.025i q^{28} -6751.92 q^{29} -3922.76 q^{31} -5760.89i q^{32} +9400.09i q^{33} -3500.46 q^{34} +1137.10 q^{36} +627.222i q^{37} +2336.47i q^{38} +6825.92 q^{39} +16277.9 q^{41} +2352.00i q^{42} +17277.7i q^{43} -13506.2 q^{44} -12059.1 q^{46} -4295.47i q^{47} +238.413i q^{48} -2401.00 q^{49} -13475.3 q^{51} +9807.55i q^{52} +25960.9i q^{53} -14458.6 q^{54} -8916.19 q^{56} +8994.43i q^{57} +23841.9i q^{58} -8902.63 q^{59} -48924.6 q^{61} +13851.8i q^{62} -2852.77i q^{63} -20903.7 q^{64} +33192.9 q^{66} -4257.80i q^{67} -19361.5i q^{68} -46422.5 q^{69} +18990.9 q^{71} -10593.9i q^{72} -10132.5i q^{73} +2214.80 q^{74} -12923.3 q^{76} +33884.5i q^{77} -24103.2i q^{78} +96986.5 q^{79} -41512.0 q^{81} -57479.2i q^{82} -70732.1i q^{83} -13009.2 q^{84} +61009.8 q^{86} +91781.4i q^{87} +125831. i q^{88} -4241.12 q^{89} +24605.3 q^{91} -66700.3i q^{92} +53323.6i q^{93} -15167.9 q^{94} -78309.9 q^{96} -104376. i q^{97} +8478.24i q^{98} -40260.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9}+O(q^{10}) \) 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9	\( 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9} - 1202 q^{11} + 98 q^{14} - 1086 q^{16} - 1260 q^{19} - 294 q^{21} - 9120 q^{24} + 2868 q^{26} - 11770 q^{29} - 792 q^{31} - 16356 q^{34} + 5274 q^{36} + 12066 q^{39} + 3548 q^{41} - 24936 q^{44} - 59072 q^{46} - 9604 q^{49} - 48822 q^{51} - 64800 q^{54} + 1470 q^{56} - 119200 q^{59} - 103692 q^{61} - 110978 q^{64} + 57696 q^{66} - 174564 q^{69} + 161488 q^{71} + 90804 q^{74} - 25120 q^{76} + 103590 q^{79} - 103356 q^{81} - 14112 q^{84} + 32808 q^{86} + 75300 q^{89} + 56546 q^{91} + 123664 q^{94} - 239712 q^{96} - 56844 q^{99}+O(q^{100}) \) 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9 - 1202 * q^11 + 98 * q^14 - 1086 * q^16 - 1260 * q^19 - 294 * q^21 - 9120 * q^24 + 2868 * q^26 - 11770 * q^29 - 792 * q^31 - 16356 * q^34 + 5274 * q^36 + 12066 * q^39 + 3548 * q^41 - 24936 * q^44 - 59072 * q^46 - 9604 * q^49 - 48822 * q^51 - 64800 * q^54 + 1470 * q^56 - 119200 * q^59 - 103692 * q^61 - 110978 * q^64 + 57696 * q^66 - 174564 * q^69 + 161488 * q^71 + 90804 * q^74 - 25120 * q^76 + 103590 * q^79 - 103356 * q^81 - 14112 * q^84 + 32808 * q^86 + 75300 * q^89 + 56546 * q^91 + 123664 * q^94 - 239712 * q^96 - 56844 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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