LMFDB - Embedded newform 175.8.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,8,Mod(1,175)]

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([0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("175.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	175.a (trivial)

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:	yes
Analytic conductor:	$54.6673794597$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 584x^{3} + 2550x^{2} + 46220x - 155664 $ x^5 - x^4 - 584x^3 + 2550x^2 + 46220*x - 155664
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.21232$ of defining polynomial
Character	$\chi$	$=$	175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.21232 q^{2} +60.7456 q^{3} -100.832 q^{4} -316.626 q^{6} -343.000 q^{7} +1192.74 q^{8} +1503.03 q^{9} +O(q^{10})$	\(q-5.21232 q^{2} +60.7456 q^{3} -100.832 q^{4} -316.626 q^{6} -343.000 q^{7} +1192.74 q^{8} +1503.03 q^{9} +7403.54 q^{11} -6125.09 q^{12} -3789.85 q^{13} +1787.83 q^{14} +6689.50 q^{16} -19226.5 q^{17} -7834.29 q^{18} -16627.4 q^{19} -20835.8 q^{21} -38589.6 q^{22} +31983.7 q^{23} +72454.0 q^{24} +19753.9 q^{26} -41548.0 q^{27} +34585.3 q^{28} +45395.3 q^{29} +92419.3 q^{31} -187539. q^{32} +449733. q^{33} +100215. q^{34} -151553. q^{36} +266269. q^{37} +86667.2 q^{38} -230217. q^{39} +627735. q^{41} +108603. q^{42} +557671. q^{43} -746512. q^{44} -166709. q^{46} -412354. q^{47} +406358. q^{48} +117649. q^{49} -1.16793e6 q^{51} +382137. q^{52} -1.34794e6 q^{53} +216561. q^{54} -409111. q^{56} -1.01004e6 q^{57} -236615. q^{58} +2.79079e6 q^{59} +1.92449e6 q^{61} -481719. q^{62} -515540. q^{63} +121257. q^{64} -2.34415e6 q^{66} +2.92866e6 q^{67} +1.93864e6 q^{68} +1.94287e6 q^{69} -2.95928e6 q^{71} +1.79273e6 q^{72} -3.95050e6 q^{73} -1.38788e6 q^{74} +1.67657e6 q^{76} -2.53941e6 q^{77} +1.19996e6 q^{78} +7.19828e6 q^{79} -5.81099e6 q^{81} -3.27196e6 q^{82} +7.88355e6 q^{83} +2.10091e6 q^{84} -2.90676e6 q^{86} +2.75756e6 q^{87} +8.83053e6 q^{88} +3.37393e6 q^{89} +1.29992e6 q^{91} -3.22497e6 q^{92} +5.61407e6 q^{93} +2.14932e6 q^{94} -1.13922e7 q^{96} +9.18493e6 q^{97} -613224. q^{98} +1.11278e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 11 q^{2} - 65 q^{3} + 553 q^{4} + 96 q^{6} - 1715 q^{7} + 1647 q^{8} - 12 q^{9}+O(q^{10}) $ 5 * q - 11 * q^2 - 65 * q^3 + 553 * q^4 + 96 * q^6 - 1715 * q^7 + 1647 * q^8 - 12 * q^9	\( 5 q - 11 q^{2} - 65 q^{3} + 553 q^{4} + 96 q^{6} - 1715 q^{7} + 1647 q^{8} - 12 q^{9} + 10027 q^{11} - 10024 q^{12} - 16141 q^{13} + 3773 q^{14} + 94169 q^{16} - 17427 q^{17} - 47547 q^{18} + 44394 q^{19} + 22295 q^{21} + 89168 q^{22} - 55698 q^{23} + 18324 q^{24} + 163750 q^{26} + 53389 q^{27} - 189679 q^{28} - 164219 q^{29} + 138440 q^{31} + 509615 q^{32} + 213271 q^{33} + 132338 q^{34} - 1408655 q^{36} - 181518 q^{37} + 1725140 q^{38} - 481621 q^{39} - 369676 q^{41} - 32928 q^{42} - 386 q^{43} - 37608 q^{44} - 2610592 q^{46} - 962985 q^{47} + 1271004 q^{48} + 588245 q^{49} - 1305827 q^{51} - 2349706 q^{52} - 396244 q^{53} + 2641788 q^{54} - 564921 q^{56} - 4322278 q^{57} + 4408538 q^{58} + 6284948 q^{59} + 1975828 q^{61} + 2246384 q^{62} + 4116 q^{63} + 13769617 q^{64} + 798284 q^{66} + 554376 q^{67} + 11618770 q^{68} + 9051046 q^{69} + 6526040 q^{71} + 3026835 q^{72} - 7479766 q^{73} - 3948102 q^{74} + 9203364 q^{76} - 3439261 q^{77} + 12300388 q^{78} + 4851305 q^{79} - 4256091 q^{81} + 29262330 q^{82} + 11813196 q^{83} + 3438232 q^{84} - 34717484 q^{86} + 263021 q^{87} + 54461828 q^{88} + 12879896 q^{89} + 5536363 q^{91} + 18484672 q^{92} + 17648536 q^{93} + 23744524 q^{94} - 39241860 q^{96} - 11704767 q^{97} - 1294139 q^{98} + 20972602 q^{99}+O(q^{100}) \) 5 * q - 11 * q^2 - 65 * q^3 + 553 * q^4 + 96 * q^6 - 1715 * q^7 + 1647 * q^8 - 12 * q^9 + 10027 * q^11 - 10024 * q^12 - 16141 * q^13 + 3773 * q^14 + 94169 * q^16 - 17427 * q^17 - 47547 * q^18 + 44394 * q^19 + 22295 * q^21 + 89168 * q^22 - 55698 * q^23 + 18324 * q^24 + 163750 * q^26 + 53389 * q^27 - 189679 * q^28 - 164219 * q^29 + 138440 * q^31 + 509615 * q^32 + 213271 * q^33 + 132338 * q^34 - 1408655 * q^36 - 181518 * q^37 + 1725140 * q^38 - 481621 * q^39 - 369676 * q^41 - 32928 * q^42 - 386 * q^43 - 37608 * q^44 - 2610592 * q^46 - 962985 * q^47 + 1271004 * q^48 + 588245 * q^49 - 1305827 * q^51 - 2349706 * q^52 - 396244 * q^53 + 2641788 * q^54 - 564921 * q^56 - 4322278 * q^57 + 4408538 * q^58 + 6284948 * q^59 + 1975828 * q^61 + 2246384 * q^62 + 4116 * q^63 + 13769617 * q^64 + 798284 * q^66 + 554376 * q^67 + 11618770 * q^68 + 9051046 * q^69 + 6526040 * q^71 + 3026835 * q^72 - 7479766 * q^73 - 3948102 * q^74 + 9203364 * q^76 - 3439261 * q^77 + 12300388 * q^78 + 4851305 * q^79 - 4256091 * q^81 + 29262330 * q^82 + 11813196 * q^83 + 3438232 * q^84 - 34717484 * q^86 + 263021 * q^87 + 54461828 * q^88 + 12879896 * q^89 + 5536363 * q^91 + 18484672 * q^92 + 17648536 * q^93 + 23744524 * q^94 - 39241860 * q^96 - 11704767 * q^97 - 1294139 * q^98 + 20972602 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−5.21232	−0.460708	−0.230354	−	0.973107i	$-0.573988\pi$
$2$	−5.21232	−0.460708	−0.230354	+	0.973107i	$0.573988\pi$
$3$	60.7456	1.29894	0.649472	−	0.760385i	$-0.274990\pi$
$3$	60.7456	1.29894	0.649472	+	0.760385i	$0.274990\pi$
$4$	−100.832	−0.787748
$5$	0	0
$5$	0	0
$6$	−316.626	−0.598435
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	1192.74	0.823630
$9$	1503.03	0.687258
$10$	0	0
$11$	7403.54	1.67712	0.838562	−	0.544807i	$-0.183397\pi$
$11$	7403.54	1.67712	0.838562	+	0.544807i	$0.183397\pi$
$12$	−6125.09	−1.02324
$13$	−3789.85	−0.478432	−0.239216	−	0.970966i	$-0.576890\pi$
$13$	−3789.85	−0.478432	−0.239216	+	0.970966i	$0.576890\pi$
$14$	1787.83	0.174131
$15$	0	0
$16$	6689.50	0.408295
$17$	−19226.5	−0.949137	−0.474568	−	0.880219i	$-0.657396\pi$
$17$	−19226.5	−0.949137	−0.474568	+	0.880219i	$0.657396\pi$
$18$	−7834.29	−0.316625
$19$	−16627.4	−0.556143	−0.278071	−	0.960560i	$-0.589695\pi$
$19$	−16627.4	−0.556143	−0.278071	+	0.960560i	$0.589695\pi$
$20$	0	0
$21$	−20835.8	−0.490955
$22$	−38589.6	−0.772665
$23$	31983.7	0.548127	0.274063	−	0.961712i	$-0.411632\pi$
$23$	31983.7	0.548127	0.274063	+	0.961712i	$0.411632\pi$
$24$	72454.0	1.06985
$25$	0	0
$26$	19753.9	0.220418
$27$	−41548.0	−0.406235
$28$	34585.3	0.297741
$29$	45395.3	0.345635	0.172817	−	0.984954i	$-0.444713\pi$
$29$	45395.3	0.345635	0.172817	+	0.984954i	$0.444713\pi$
$30$	0	0
$31$	92419.3	0.557182	0.278591	−	0.960410i	$-0.410133\pi$
$31$	92419.3	0.557182	0.278591	+	0.960410i	$0.410133\pi$
$32$	−187539.	−1.01173
$33$	449733.	2.17849
$34$	100215.	0.437275
$35$	0	0
$36$	−151553.	−0.541386
$37$	266269.	0.864201	0.432100	−	0.901826i	$-0.357773\pi$
$37$	266269.	0.864201	0.432100	+	0.901826i	$0.357773\pi$
$38$	86667.2	0.256220
$39$	−230217.	−0.621457
$40$	0	0
$41$	627735.	1.42244	0.711218	−	0.702971i	$-0.248144\pi$
$41$	627735.	1.42244	0.711218	+	0.702971i	$0.248144\pi$
$42$	108603.	0.226187
$43$	557671.	1.06964	0.534821	−	0.844966i	$-0.320379\pi$
$43$	557671.	1.06964	0.534821	+	0.844966i	$0.320379\pi$
$44$	−746512.	−1.32115
$45$	0	0
$46$	−166709.	−0.252526
$47$	−412354.	−0.579332	−0.289666	−	0.957128i	$-0.593544\pi$
$47$	−412354.	−0.579332	−0.289666	+	0.957128i	$0.593544\pi$
$48$	406358.	0.530352
$49$	117649.	0.142857
$50$	0	0
$51$	−1.16793e6	−1.23288
$52$	382137.	0.376884
$53$	−1.34794e6	−1.24367	−0.621835	−	0.783148i	$-0.713613\pi$
$53$	−1.34794e6	−1.24367	−0.621835	+	0.783148i	$0.713613\pi$
$54$	216561.	0.187156
$55$	0	0
$56$	−409111.	−0.311303
$57$	−1.01004e6	−0.722399
$58$	−236615.	−0.159237
$59$	2.79079e6	1.76907	0.884537	−	0.466471i	$-0.154475\pi$
$59$	2.79079e6	1.76907	0.884537	+	0.466471i	$0.154475\pi$
$60$	0	0
$61$	1.92449e6	1.08558	0.542789	−	0.839869i	$-0.317368\pi$
$61$	1.92449e6	1.08558	0.542789	+	0.839869i	$0.317368\pi$
$62$	−481719.	−0.256698
$63$	−515540.	−0.259759
$64$	121257.	0.0578199
$65$	0	0
$66$	−2.34415e6	−1.00365
$67$	2.92866e6	1.18962	0.594808	−	0.803868i	$-0.297228\pi$
$67$	2.92866e6	1.18962	0.594808	+	0.803868i	$0.297228\pi$
$68$	1.93864e6	0.747681
$69$	1.94287e6	0.711986
$70$	0	0
$71$	−2.95928e6	−0.981256	−0.490628	−	0.871369i	$-0.663233\pi$
$71$	−2.95928e6	−0.981256	−0.490628	+	0.871369i	$0.663233\pi$
$72$	1.79273e6	0.566046
$73$	−3.95050e6	−1.18856	−0.594281	−	0.804257i	$-0.702563\pi$
$73$	−3.95050e6	−1.18856	−0.594281	+	0.804257i	$0.702563\pi$
$74$	−1.38788e6	−0.398144
$75$	0	0
$76$	1.67657e6	0.438100
$77$	−2.53941e6	−0.633893
$78$	1.19996e6	0.286311
$79$	7.19828e6	1.64261	0.821304	−	0.570491i	$-0.193247\pi$
$79$	7.19828e6	1.64261	0.821304	+	0.570491i	$0.193247\pi$
$80$	0	0
$81$	−5.81099e6	−1.21493
$82$	−3.27196e6	−0.655328
$83$	7.88355e6	1.51338	0.756691	−	0.653773i	$-0.226815\pi$
$83$	7.88355e6	1.51338	0.756691	+	0.653773i	$0.226815\pi$
$84$	2.10091e6	0.386749
$85$	0	0
$86$	−2.90676e6	−0.492793
$87$	2.75756e6	0.448961
$88$	8.83053e6	1.38133
$89$	3.37393e6	0.507307	0.253654	−	0.967295i	$-0.418368\pi$
$89$	3.37393e6	0.507307	0.253654	+	0.967295i	$0.418368\pi$
$90$	0	0
$91$	1.29992e6	0.180830
$92$	−3.22497e6	−0.431786
$93$	5.61407e6	0.723748
$94$	2.14932e6	0.266903
$95$	0	0
$96$	−1.13922e7	−1.31419
$97$	9.18493e6	1.02182	0.510911	−	0.859634i	$-0.329308\pi$
$97$	9.18493e6	1.02182	0.510911	+	0.859634i	$0.329308\pi$
$98$	−613224.	−0.0658155
$99$	1.11278e7	1.15262
$100$	0	0
$101$	9.49820e6	0.917311	0.458655	−	0.888614i	$-0.348331\pi$
$101$	9.49820e6	0.917311	0.458655	+	0.888614i	$0.348331\pi$
$102$	6.08760e6	0.567996
$103$	1.40943e7	1.27090	0.635451	−	0.772141i	$-0.280814\pi$
$103$	1.40943e7	1.27090	0.635451	+	0.772141i	$0.280814\pi$
$104$	−4.52032e6	−0.394051
$105$	0	0
$106$	7.02590e6	0.572969
$107$	9.75316e6	0.769666	0.384833	−	0.922986i	$-0.374259\pi$
$107$	9.75316e6	0.769666	0.384833	+	0.922986i	$0.374259\pi$
$108$	4.18936e6	0.320011
$109$	8.22102e6	0.608041	0.304021	−	0.952665i	$-0.401671\pi$
$109$	8.22102e6	0.608041	0.304021	+	0.952665i	$0.401671\pi$
$110$	0	0
$111$	1.61747e7	1.12255
$112$	−2.29450e6	−0.154321
$113$	−1.63107e7	−1.06341	−0.531703	−	0.846931i	$-0.678448\pi$
$113$	−1.63107e7	−1.06341	−0.531703	+	0.846931i	$0.678448\pi$
$114$	5.26466e6	0.332815
$115$	0	0
$116$	−4.57728e6	−0.272273
$117$	−5.69627e6	−0.328806
$118$	−1.45465e7	−0.815027
$119$	6.59469e6	0.358740
$120$	0	0
$121$	3.53252e7	1.81274
$122$	−1.00310e7	−0.500135
$123$	3.81322e7	1.84767
$124$	−9.31880e6	−0.438919
$125$	0	0
$126$	2.68716e6	0.119673
$127$	2.17950e6	0.0944158	0.0472079	−	0.998885i	$-0.484968\pi$
$127$	2.17950e6	0.0944158	0.0472079	+	0.998885i	$0.484968\pi$
$128$	2.33730e7	0.985097
$129$	3.38761e7	1.38940
$130$	0	0
$131$	−3.74218e7	−1.45437	−0.727185	−	0.686442i	$-0.759172\pi$
$131$	−3.74218e7	−1.45437	−0.727185	+	0.686442i	$0.759172\pi$
$132$	−4.53473e7	−1.71610
$133$	5.70319e6	0.210202
$134$	−1.52651e7	−0.548066
$135$	0	0
$136$	−2.29323e7	−0.781738
$137$	4.62620e7	1.53710	0.768550	−	0.639790i	$-0.220978\pi$
$137$	4.62620e7	1.53710	0.768550	+	0.639790i	$0.220978\pi$
$138$	−1.01268e7	−0.328018
$139$	3.79070e7	1.19720	0.598601	−	0.801047i	$-0.295723\pi$
$139$	3.79070e7	1.19720	0.598601	+	0.801047i	$0.295723\pi$
$140$	0	0
$141$	−2.50487e7	−0.752520
$142$	1.54247e7	0.452073
$143$	−2.80583e7	−0.802390
$144$	1.00545e7	0.280604
$145$	0	0
$146$	2.05913e7	0.547580
$147$	7.14666e6	0.185564
$148$	−2.68484e7	−0.680772
$149$	−7.04498e7	−1.74473	−0.872365	−	0.488856i	$-0.837414\pi$
$149$	−7.04498e7	−1.74473	−0.872365	+	0.488856i	$0.837414\pi$
$150$	0	0
$151$	−6.29486e7	−1.48788	−0.743938	−	0.668249i	$-0.767044\pi$
$151$	−6.29486e7	−1.48788	−0.743938	+	0.668249i	$0.767044\pi$
$152$	−1.98322e7	−0.458056
$153$	−2.88981e7	−0.652302
$154$	1.32362e7	0.292040
$155$	0	0
$156$	2.32132e7	0.489552
$157$	−8.03559e7	−1.65718	−0.828588	−	0.559859i	$-0.810855\pi$
$157$	−8.03559e7	−1.65718	−0.828588	+	0.559859i	$0.810855\pi$
$158$	−3.75197e7	−0.756763
$159$	−8.18815e7	−1.61546
$160$	0	0
$161$	−1.09704e7	−0.207172
$162$	3.02888e7	0.559730
$163$	−4.93485e7	−0.892518	−0.446259	−	0.894904i	$-0.647244\pi$
$163$	−4.93485e7	−0.892518	−0.446259	+	0.894904i	$0.647244\pi$
$164$	−6.32956e7	−1.12052
$165$	0	0
$166$	−4.10916e7	−0.697227
$167$	2.78438e7	0.462616	0.231308	−	0.972881i	$-0.425699\pi$
$167$	2.78438e7	0.462616	0.231308	+	0.972881i	$0.425699\pi$
$168$	−2.48517e7	−0.404365
$169$	−4.83855e7	−0.771102
$170$	0	0
$171$	−2.49915e7	−0.382214
$172$	−5.62309e7	−0.842608
$173$	3.91017e7	0.574162	0.287081	−	0.957906i	$-0.407315\pi$
$173$	3.91017e7	0.574162	0.287081	+	0.957906i	$0.407315\pi$
$174$	−1.43733e7	−0.206840
$175$	0	0
$176$	4.95260e7	0.684761
$177$	1.69529e8	2.29793
$178$	−1.75860e7	−0.233721
$179$	−3.05418e7	−0.398024	−0.199012	−	0.979997i	$-0.563773\pi$
$179$	−3.05418e7	−0.398024	−0.199012	+	0.979997i	$0.563773\pi$
$180$	0	0
$181$	5.42309e7	0.679785	0.339892	−	0.940464i	$-0.389609\pi$
$181$	5.42309e7	0.679785	0.339892	+	0.940464i	$0.389609\pi$
$182$	−6.77560e6	−0.0833101
$183$	1.16904e8	1.41011
$184$	3.81483e7	0.451454
$185$	0	0
$186$	−2.92623e7	−0.333437
$187$	−1.42344e8	−1.59182
$188$	4.15784e7	0.456368
$189$	1.42510e7	0.153542
$190$	0	0
$191$	1.75614e8	1.82366	0.911828	−	0.410573i	$-0.134671\pi$
$191$	1.75614e8	1.82366	0.911828	+	0.410573i	$0.134671\pi$
$192$	7.36584e6	0.0751049
$193$	−1.10954e8	−1.11095	−0.555474	−	0.831534i	$-0.687463\pi$
$193$	−1.10954e8	−1.11095	−0.555474	+	0.831534i	$0.687463\pi$
$194$	−4.78748e7	−0.470762
$195$	0	0
$196$	−1.18628e7	−0.112535
$197$	−1.00536e8	−0.936895	−0.468448	−	0.883491i	$-0.655187\pi$
$197$	−1.00536e8	−0.936895	−0.468448	+	0.883491i	$0.655187\pi$
$198$	−5.80015e7	−0.531020
$199$	4.14944e7	0.373253	0.186627	−	0.982431i	$-0.440245\pi$
$199$	4.14944e7	0.373253	0.186627	+	0.982431i	$0.440245\pi$
$200$	0	0
$201$	1.77903e8	1.54524
$202$	−4.95076e7	−0.422613
$203$	−1.55706e7	−0.130638
$204$	1.17764e8	0.971196
$205$	0	0
$206$	−7.34638e7	−0.585515
$207$	4.80725e7	0.376704
$208$	−2.53522e7	−0.195341
$209$	−1.23102e8	−0.932720
$210$	0	0
$211$	2.53393e8	1.85698	0.928488	−	0.371362i	$-0.121109\pi$
$211$	2.53393e8	1.85698	0.928488	+	0.371362i	$0.121109\pi$
$212$	1.35915e8	0.979699
$213$	−1.79764e8	−1.27460
$214$	−5.08366e7	−0.354591
$215$	0	0
$216$	−4.95561e7	−0.334587
$217$	−3.16998e7	−0.210595
$218$	−4.28506e7	−0.280130
$219$	−2.39976e8	−1.54388
$220$	0	0
$221$	7.28656e7	0.454098
$222$	−8.43076e7	−0.517168
$223$	2.33810e8	1.41188	0.705938	−	0.708274i	$-0.250526\pi$
$223$	2.33810e8	1.41188	0.705938	+	0.708274i	$0.250526\pi$
$224$	6.43259e7	0.382400
$225$	0	0
$226$	8.50168e7	0.489920
$227$	−6.88221e7	−0.390515	−0.195257	−	0.980752i	$-0.562554\pi$
$227$	−6.88221e7	−0.390515	−0.195257	+	0.980752i	$0.562554\pi$
$228$	1.01844e8	0.569068
$229$	3.13920e7	0.172741	0.0863703	−	0.996263i	$-0.472473\pi$
$229$	3.13920e7	0.172741	0.0863703	+	0.996263i	$0.472473\pi$
$230$	0	0
$231$	−1.54258e8	−0.823392
$232$	5.41449e7	0.284675
$233$	−7.51549e7	−0.389235	−0.194617	−	0.980879i	$-0.562346\pi$
$233$	−7.51549e7	−0.389235	−0.194617	+	0.980879i	$0.562346\pi$
$234$	2.96908e7	0.151484
$235$	0	0
$236$	−2.81401e8	−1.39358
$237$	4.37264e8	2.13366
$238$	−3.43736e7	−0.165274
$239$	−1.92365e8	−0.911453	−0.455727	−	0.890120i	$-0.650621\pi$
$239$	−1.92365e8	−0.911453	−0.455727	+	0.890120i	$0.650621\pi$
$240$	0	0
$241$	1.32330e7	0.0608972	0.0304486	−	0.999536i	$-0.490306\pi$
$241$	1.32330e7	0.0608972	0.0304486	+	0.999536i	$0.490306\pi$
$242$	−1.84126e8	−0.835146
$243$	−2.62127e8	−1.17190
$244$	−1.94050e8	−0.855162
$245$	0	0
$246$	−1.98757e8	−0.851235
$247$	6.30154e7	0.266077
$248$	1.10233e8	0.458912
$249$	4.78891e8	1.96580
$250$	0	0
$251$	1.72781e8	0.689666	0.344833	−	0.938664i	$-0.387936\pi$
$251$	1.72781e8	0.689666	0.344833	+	0.938664i	$0.387936\pi$
$252$	5.19828e7	0.204625
$253$	2.36792e8	0.919276
$254$	−1.13603e7	−0.0434981
$255$	0	0
$256$	−1.37348e8	−0.511662
$257$	−2.03178e8	−0.746638	−0.373319	−	0.927703i	$-0.621780\pi$
$257$	−2.03178e8	−0.746638	−0.373319	+	0.927703i	$0.621780\pi$
$258$	−1.76573e8	−0.640110
$259$	−9.13303e7	−0.326637
$260$	0	0
$261$	6.82306e7	0.237540
$262$	1.95054e8	0.670040
$263$	−5.23865e6	−0.0177572	−0.00887859	−	0.999961i	$-0.502826\pi$
$263$	−5.23865e6	−0.0177572	−0.00887859	+	0.999961i	$0.502826\pi$
$264$	5.36416e8	1.79427
$265$	0	0
$266$	−2.97269e7	−0.0968419
$267$	2.04951e8	0.658964
$268$	−2.95301e8	−0.937117
$269$	−2.67408e8	−0.837609	−0.418805	−	0.908076i	$-0.637551\pi$
$269$	−2.67408e8	−0.837609	−0.418805	+	0.908076i	$0.637551\pi$
$270$	0	0
$271$	−5.64451e8	−1.72280	−0.861398	−	0.507931i	$-0.830410\pi$
$271$	−5.64451e8	−1.72280	−0.861398	+	0.507931i	$0.830410\pi$
$272$	−1.28616e8	−0.387528
$273$	7.89645e7	0.234889
$274$	−2.41132e8	−0.708155
$275$	0	0
$276$	−1.95903e8	−0.560866
$277$	−4.66139e8	−1.31776	−0.658880	−	0.752248i	$-0.728969\pi$
$277$	−4.66139e8	−1.31776	−0.658880	+	0.752248i	$0.728969\pi$
$278$	−1.97583e8	−0.551561
$279$	1.38909e8	0.382928
$280$	0	0
$281$	−3.15073e8	−0.847108	−0.423554	−	0.905871i	$-0.639218\pi$
$281$	−3.15073e8	−0.847108	−0.423554	+	0.905871i	$0.639218\pi$
$282$	1.30562e8	0.346692
$283$	8.18186e7	0.214585	0.107293	−	0.994227i	$-0.465782\pi$
$283$	8.18186e7	0.214585	0.107293	+	0.994227i	$0.465782\pi$
$284$	2.98390e8	0.772982
$285$	0	0
$286$	1.46249e8	0.369668
$287$	−2.15313e8	−0.537631
$288$	−2.81877e8	−0.695323
$289$	−4.06806e7	−0.0991390
$290$	0	0
$291$	5.57945e8	1.32729
$292$	3.98336e8	0.936287
$293$	1.99777e8	0.463989	0.231995	−	0.972717i	$-0.425475\pi$
$293$	1.99777e8	0.463989	0.231995	+	0.972717i	$0.425475\pi$
$294$	−3.72507e7	−0.0854907
$295$	0	0
$296$	3.17591e8	0.711782
$297$	−3.07602e8	−0.681306
$298$	3.67207e8	0.803811
$299$	−1.21213e8	−0.262242
$300$	0	0
$301$	−1.91281e8	−0.404286
$302$	3.28108e8	0.685477
$303$	5.76974e8	1.19154
$304$	−1.11229e8	−0.227070
$305$	0	0
$306$	1.50626e8	0.300521
$307$	4.82719e7	0.0952160	0.0476080	−	0.998866i	$-0.484840\pi$
$307$	4.82719e7	0.0952160	0.0476080	+	0.998866i	$0.484840\pi$
$308$	2.56054e8	0.499348
$309$	8.56165e8	1.65083
$310$	0	0
$311$	2.27544e8	0.428947	0.214473	−	0.976730i	$-0.431196\pi$
$311$	2.27544e8	0.428947	0.214473	+	0.976730i	$0.431196\pi$
$312$	−2.74590e8	−0.511851
$313$	−1.84509e8	−0.340104	−0.170052	−	0.985435i	$-0.554394\pi$
$313$	−1.84509e8	−0.340104	−0.170052	+	0.985435i	$0.554394\pi$
$314$	4.18840e8	0.763475
$315$	0	0
$316$	−7.25815e8	−1.29396
$317$	−1.33207e8	−0.234865	−0.117433	−	0.993081i	$-0.537466\pi$
$317$	−1.33207e8	−0.234865	−0.117433	+	0.993081i	$0.537466\pi$
$318$	4.26793e8	0.744255
$319$	3.36086e8	0.579672
$320$	0	0
$321$	5.92462e8	0.999754
$322$	5.71812e7	0.0954460
$323$	3.19686e8	0.527856
$324$	5.85933e8	0.957062
$325$	0	0
$326$	2.57220e8	0.411191
$327$	4.99391e8	0.789812
$328$	7.48727e8	1.17156
$329$	1.41437e8	0.218967
$330$	0	0
$331$	−2.48801e8	−0.377098	−0.188549	−	0.982064i	$-0.560378\pi$
$331$	−2.48801e8	−0.377098	−0.188549	+	0.982064i	$0.560378\pi$
$332$	−7.94912e8	−1.19216
$333$	4.00211e8	0.593929
$334$	−1.45131e8	−0.213131
$335$	0	0
$336$	−1.39381e8	−0.200454
$337$	1.76663e8	0.251444	0.125722	−	0.992065i	$-0.459875\pi$
$337$	1.76663e8	0.251444	0.125722	+	0.992065i	$0.459875\pi$
$338$	2.52201e8	0.355253
$339$	−9.90807e8	−1.38131
$340$	0	0
$341$	6.84230e8	0.934463
$342$	1.30264e8	0.176089
$343$	−4.03536e7	−0.0539949
$344$	6.65158e8	0.880989
$345$	0	0
$346$	−2.03811e8	−0.264521
$347$	4.82632e8	0.620101	0.310051	−	0.950720i	$-0.399654\pi$
$347$	4.82632e8	0.620101	0.310051	+	0.950720i	$0.399654\pi$
$348$	−2.78050e8	−0.353668
$349$	9.78940e7	0.123273	0.0616364	−	0.998099i	$-0.480368\pi$
$349$	9.78940e7	0.123273	0.0616364	+	0.998099i	$0.480368\pi$
$350$	0	0
$351$	1.57461e8	0.194356
$352$	−1.38845e9	−1.69680
$353$	−1.60663e8	−0.194404	−0.0972021	−	0.995265i	$-0.530989\pi$
$353$	−1.60663e8	−0.194404	−0.0972021	+	0.995265i	$0.530989\pi$
$354$	−8.83637e8	−1.05867
$355$	0	0
$356$	−3.40199e8	−0.399630
$357$	4.00599e8	0.465984
$358$	1.59194e8	0.183373
$359$	−1.27950e9	−1.45952	−0.729761	−	0.683702i	$-0.760369\pi$
$359$	−1.27950e9	−1.45952	−0.729761	+	0.683702i	$0.760369\pi$
$360$	0	0
$361$	−6.17402e8	−0.690705
$362$	−2.82669e8	−0.313182
$363$	2.14585e9	2.35465
$364$	−1.31073e8	−0.142449
$365$	0	0
$366$	−6.09342e8	−0.649647
$367$	1.35171e9	1.42743	0.713713	−	0.700439i	$-0.247012\pi$
$367$	1.35171e9	1.42743	0.713713	+	0.700439i	$0.247012\pi$
$368$	2.13955e8	0.223797
$369$	9.43507e8	0.977581
$370$	0	0
$371$	4.62344e8	0.470063
$372$	−5.66077e8	−0.570131
$373$	−1.65708e9	−1.65334	−0.826669	−	0.562689i	$-0.809767\pi$
$373$	−1.65708e9	−1.65334	−0.826669	+	0.562689i	$0.809767\pi$
$374$	7.41943e8	0.733365
$375$	0	0
$376$	−4.91833e8	−0.477155
$377$	−1.72041e8	−0.165363
$378$	−7.42806e7	−0.0707382
$379$	5.29980e8	0.500060	0.250030	−	0.968238i	$-0.419559\pi$
$379$	5.29980e8	0.500060	0.250030	+	0.968238i	$0.419559\pi$
$380$	0	0
$381$	1.32395e8	0.122641
$382$	−9.15357e8	−0.840173
$383$	1.00532e9	0.914338	0.457169	−	0.889380i	$-0.348864\pi$
$383$	1.00532e9	0.914338	0.457169	+	0.889380i	$0.348864\pi$
$384$	1.41981e9	1.27959
$385$	0	0
$386$	5.78329e8	0.511823
$387$	8.38197e8	0.735119
$388$	−9.26133e8	−0.804938
$389$	9.23418e7	0.0795380	0.0397690	−	0.999209i	$-0.487338\pi$
$389$	9.23418e7	0.0795380	0.0397690	+	0.999209i	$0.487338\pi$
$390$	0	0
$391$	−6.14934e8	−0.520247
$392$	1.40325e8	0.117661
$393$	−2.27321e9	−1.88915
$394$	5.24027e8	0.431635
$395$	0	0
$396$	−1.12203e9	−0.907971
$397$	1.43695e9	1.15259	0.576294	−	0.817242i	$-0.304498\pi$
$397$	1.43695e9	1.15259	0.576294	+	0.817242i	$0.304498\pi$
$398$	−2.16282e8	−0.171961
$399$	3.46444e8	0.273041
$400$	0	0
$401$	−1.72906e9	−1.33907	−0.669536	−	0.742780i	$-0.733507\pi$
$401$	−1.72906e9	−1.33907	−0.669536	+	0.742780i	$0.733507\pi$
$402$	−9.27288e8	−0.711907
$403$	−3.50256e8	−0.266574
$404$	−9.57720e8	−0.722610
$405$	0	0
$406$	8.11588e7	0.0601858
$407$	1.97133e9	1.44937
$408$	−1.39304e9	−1.01543
$409$	−5.81285e8	−0.420105	−0.210052	−	0.977690i	$-0.567363\pi$
$409$	−5.81285e8	−0.420105	−0.210052	+	0.977690i	$0.567363\pi$
$410$	0	0
$411$	2.81021e9	1.99661
$412$	−1.42115e9	−1.00115
$413$	−9.57243e8	−0.668647
$414$	−2.50569e8	−0.173551
$415$	0	0
$416$	7.10745e8	0.484047
$417$	2.30268e9	1.55510
$418$	6.41644e8	0.429712
$419$	2.05004e8	0.136149	0.0680743	−	0.997680i	$-0.478314\pi$
$419$	2.05004e8	0.136149	0.0680743	+	0.997680i	$0.478314\pi$
$420$	0	0
$421$	−1.32820e9	−0.867514	−0.433757	−	0.901030i	$-0.642812\pi$
$421$	−1.32820e9	−0.867514	−0.433757	+	0.901030i	$0.642812\pi$
$422$	−1.32077e9	−0.855524
$423$	−6.19782e8	−0.398151
$424$	−1.60775e9	−1.02432
$425$	0	0
$426$	9.36985e8	0.587218
$427$	−6.60100e8	−0.410310
$428$	−9.83428e8	−0.606303
$429$	−1.70442e9	−1.04226
$430$	0	0
$431$	1.71243e9	1.03025	0.515124	−	0.857116i	$-0.327746\pi$
$431$	1.71243e9	1.03025	0.515124	+	0.857116i	$0.327746\pi$
$432$	−2.77935e8	−0.165864
$433$	−1.26981e9	−0.751675	−0.375838	−	0.926686i	$-0.622645\pi$
$433$	−1.26981e9	−0.751675	−0.375838	+	0.926686i	$0.622645\pi$
$434$	1.65230e8	0.0970228
$435$	0	0
$436$	−8.28940e8	−0.478983
$437$	−5.31805e8	−0.304837
$438$	1.25083e9	0.711277
$439$	−7.13810e8	−0.402677	−0.201339	−	0.979522i	$-0.564529\pi$
$439$	−7.13810e8	−0.402677	−0.201339	+	0.979522i	$0.564529\pi$
$440$	0	0
$441$	1.76830e8	0.0981797
$442$	−3.79799e8	−0.209207
$443$	−1.33285e9	−0.728399	−0.364199	−	0.931321i	$-0.618657\pi$
$443$	−1.33285e9	−0.728399	−0.364199	+	0.931321i	$0.618657\pi$
$444$	−1.63092e9	−0.884286
$445$	0	0
$446$	−1.21869e9	−0.650463
$447$	−4.27952e9	−2.26631
$448$	−4.15912e7	−0.0218539
$449$	−8.55950e8	−0.446258	−0.223129	−	0.974789i	$-0.571627\pi$
$449$	−8.55950e8	−0.446258	−0.223129	+	0.974789i	$0.571627\pi$
$450$	0	0
$451$	4.64746e9	2.38560
$452$	1.64464e9	0.837696
$453$	−3.82385e9	−1.93267
$454$	3.58723e8	0.179913
$455$	0	0
$456$	−1.20472e9	−0.594990
$457$	3.18676e9	1.56186	0.780931	−	0.624618i	$-0.214745\pi$
$457$	3.18676e9	1.56186	0.780931	+	0.624618i	$0.214745\pi$
$458$	−1.63625e8	−0.0795830
$459$	7.98823e8	0.385572
$460$	0	0
$461$	3.02974e9	1.44030	0.720148	−	0.693821i	$-0.244074\pi$
$461$	3.02974e9	1.44030	0.720148	+	0.693821i	$0.244074\pi$
$462$	8.04044e8	0.379344
$463$	1.53211e9	0.717390	0.358695	−	0.933455i	$-0.383222\pi$
$463$	1.53211e9	0.717390	0.358695	+	0.933455i	$0.383222\pi$
$464$	3.03672e8	0.141121
$465$	0	0
$466$	3.91731e8	0.179324
$467$	3.47187e9	1.57745	0.788724	−	0.614748i	$-0.210742\pi$
$467$	3.47187e9	1.57745	0.788724	+	0.614748i	$0.210742\pi$
$468$	5.74365e8	0.259017
$469$	−1.00453e9	−0.449632
$470$	0	0
$471$	−4.88127e9	−2.15258
$472$	3.32870e9	1.45706
$473$	4.12874e9	1.79392
$474$	−2.27916e9	−0.982993
$475$	0	0
$476$	−6.64954e8	−0.282597
$477$	−2.02600e9	−0.854722
$478$	1.00267e9	0.419914
$479$	4.81799e8	0.200305	0.100152	−	0.994972i	$-0.468067\pi$
$479$	4.81799e8	0.200305	0.100152	+	0.994972i	$0.468067\pi$
$480$	0	0
$481$	−1.00912e9	−0.413462
$482$	−6.89744e7	−0.0280559
$483$	−6.66404e8	−0.269105
$484$	−3.56190e9	−1.42798
$485$	0	0
$486$	1.36629e9	0.539903
$487$	−2.95036e9	−1.15751	−0.578753	−	0.815503i	$-0.696461\pi$
$487$	−2.95036e9	−1.15751	−0.578753	+	0.815503i	$0.696461\pi$
$488$	2.29542e9	0.894115
$489$	−2.99770e9	−1.15933
$490$	0	0
$491$	−1.55876e8	−0.0594285	−0.0297143	−	0.999558i	$-0.509460\pi$
$491$	−1.55876e8	−0.0594285	−0.0297143	+	0.999558i	$0.509460\pi$
$492$	−3.84493e9	−1.45550
$493$	−8.72792e8	−0.328055
$494$	−3.28456e8	−0.122584
$495$	0	0
$496$	6.18239e8	0.227494
$497$	1.01503e9	0.370880
$498$	−2.49613e9	−0.905660
$499$	3.27549e9	1.18012	0.590058	−	0.807361i	$-0.299105\pi$
$499$	3.27549e9	1.18012	0.590058	+	0.807361i	$0.299105\pi$
$500$	0	0
$501$	1.69139e9	0.600913
$502$	−9.00591e8	−0.317735
$503$	2.93132e9	1.02701	0.513506	−	0.858086i	$-0.328346\pi$
$503$	2.93132e9	1.02701	0.513506	+	0.858086i	$0.328346\pi$
$504$	−6.14908e8	−0.213945
$505$	0	0
$506$	−1.23424e9	−0.423518
$507$	−2.93921e9	−1.00162
$508$	−2.19763e8	−0.0743759
$509$	−2.94603e9	−0.990205	−0.495103	−	0.868834i	$-0.664870\pi$
$509$	−2.94603e9	−0.990205	−0.495103	+	0.868834i	$0.664870\pi$
$510$	0	0
$511$	1.35502e9	0.449234
$512$	−2.27584e9	−0.749370
$513$	6.90835e8	0.225925
$514$	1.05903e9	0.343982
$515$	0	0
$516$	−3.41578e9	−1.09450
$517$	−3.05288e9	−0.971611
$518$	4.76042e8	0.150484
$519$	2.37526e9	0.745805
$520$	0	0
$521$	4.63933e9	1.43722	0.718610	−	0.695414i	$-0.244779\pi$
$521$	4.63933e9	1.43722	0.718610	+	0.695414i	$0.244779\pi$
$522$	−3.55639e8	−0.109437
$523$	−4.38133e9	−1.33921	−0.669607	−	0.742716i	$-0.733537\pi$
$523$	−4.38133e9	−1.33921	−0.669607	+	0.742716i	$0.733537\pi$
$524$	3.77330e9	1.14568
$525$	0	0
$526$	2.73055e7	0.00818088
$527$	−1.77690e9	−0.528842
$528$	3.00849e9	0.889466
$529$	−2.38187e9	−0.699557
$530$	0	0
$531$	4.19466e9	1.21581
$532$	−5.75063e8	−0.165586
$533$	−2.37902e9	−0.680540
$534$	−1.06827e9	−0.303590
$535$	0	0
$536$	3.49314e9	0.979803
$537$	−1.85528e9	−0.517011
$538$	1.39382e9	0.385894
$539$	8.71019e8	0.239589
$540$	0	0
$541$	−3.97296e9	−1.07876	−0.539379	−	0.842063i	$-0.681341\pi$
$541$	−3.97296e9	−1.07876	−0.539379	+	0.842063i	$0.681341\pi$
$542$	2.94210e9	0.793706
$543$	3.29429e9	0.883003
$544$	3.60572e9	0.960275
$545$	0	0
$546$	−4.11588e8	−0.108215
$547$	−2.57944e8	−0.0673859	−0.0336930	−	0.999432i	$-0.510727\pi$
$547$	−2.57944e8	−0.0673859	−0.0336930	+	0.999432i	$0.510727\pi$
$548$	−4.66468e9	−1.21085
$549$	2.89257e9	0.746072
$550$	0	0
$551$	−7.54804e8	−0.192222
$552$	2.31734e9	0.586413
$553$	−2.46901e9	−0.620847
$554$	2.42967e9	0.607103
$555$	0	0
$556$	−3.82223e9	−0.943094
$557$	3.66731e9	0.899197	0.449598	−	0.893231i	$-0.351567\pi$
$557$	3.66731e9	0.899197	0.449598	+	0.893231i	$0.351567\pi$
$558$	−7.24040e8	−0.176418
$559$	−2.11349e9	−0.511751
$560$	0	0
$561$	−8.64679e9	−2.06769
$562$	1.64226e9	0.390270
$563$	4.50414e9	1.06373	0.531866	−	0.846828i	$-0.321491\pi$
$563$	4.50414e9	1.06373	0.531866	+	0.846828i	$0.321491\pi$
$564$	2.52570e9	0.592796
$565$	0	0
$566$	−4.26464e8	−0.0988611
$567$	1.99317e9	0.459202
$568$	−3.52967e9	−0.808192
$569$	2.99694e9	0.682000	0.341000	−	0.940063i	$-0.389234\pi$
$569$	2.99694e9	0.682000	0.341000	+	0.940063i	$0.389234\pi$
$570$	0	0
$571$	8.03468e9	1.80610	0.903051	−	0.429534i	$-0.141322\pi$
$571$	8.03468e9	1.80610	0.903051	+	0.429534i	$0.141322\pi$
$572$	2.82917e9	0.632081
$573$	1.06678e10	2.36883
$574$	1.12228e9	0.247691
$575$	0	0
$576$	1.82253e8	0.0397372
$577$	1.52605e9	0.330714	0.165357	−	0.986234i	$-0.447122\pi$
$577$	1.52605e9	0.330714	0.165357	+	0.986234i	$0.447122\pi$
$578$	2.12040e8	0.0456742
$579$	−6.73999e9	−1.44306
$580$	0	0
$581$	−2.70406e9	−0.572004
$582$	−2.90819e9	−0.611493
$583$	−9.97953e9	−2.08579
$584$	−4.71193e9	−0.978936
$585$	0	0
$586$	−1.04130e9	−0.213764
$587$	4.10777e9	0.838247	0.419124	−	0.907929i	$-0.362337\pi$
$587$	4.10777e9	0.838247	0.419124	+	0.907929i	$0.362337\pi$
$588$	−7.20611e8	−0.146177
$589$	−1.53669e9	−0.309873
$590$	0	0
$591$	−6.10714e9	−1.21698
$592$	1.78121e9	0.352849
$593$	−5.07612e9	−0.999632	−0.499816	−	0.866132i	$-0.666599\pi$
$593$	−5.07612e9	−0.999632	−0.499816	+	0.866132i	$0.666599\pi$
$594$	1.60332e9	0.313883
$595$	0	0
$596$	7.10358e9	1.37441
$597$	2.52060e9	0.484836
$598$	6.31803e8	0.120817
$599$	2.86911e9	0.545448	0.272724	−	0.962092i	$-0.412075\pi$
$599$	2.86911e9	0.545448	0.272724	+	0.962092i	$0.412075\pi$
$600$	0	0
$601$	−1.37052e9	−0.257528	−0.128764	−	0.991675i	$-0.541101\pi$
$601$	−1.37052e9	−0.257528	−0.128764	+	0.991675i	$0.541101\pi$
$602$	9.97018e8	0.186258
$603$	4.40187e9	0.817573
$604$	6.34721e9	1.17207
$605$	0	0
$606$	−3.00737e9	−0.548951
$607$	−7.35729e9	−1.33523	−0.667617	−	0.744505i	$-0.732686\pi$
$607$	−7.35729e9	−1.33523	−0.667617	+	0.744505i	$0.732686\pi$
$608$	3.11828e9	0.562669
$609$	−9.45844e8	−0.169691
$610$	0	0
$611$	1.56276e9	0.277171
$612$	2.91384e9	0.513849
$613$	−3.55720e9	−0.623729	−0.311865	−	0.950127i	$-0.600954\pi$
$613$	−3.55720e9	−0.623729	−0.311865	+	0.950127i	$0.600954\pi$
$614$	−2.51608e8	−0.0438668
$615$	0	0
$616$	−3.02887e9	−0.522093
$617$	1.02547e10	1.75761	0.878806	−	0.477180i	$-0.158341\pi$
$617$	1.02547e10	1.75761	0.878806	+	0.477180i	$0.158341\pi$
$618$	−4.46260e9	−0.760552
$619$	1.10323e10	1.86961	0.934804	−	0.355164i	$-0.115575\pi$
$619$	1.10323e10	1.86961	0.934804	+	0.355164i	$0.115575\pi$
$620$	0	0
$621$	−1.32886e9	−0.222668
$622$	−1.18603e9	−0.197619
$623$	−1.15726e9	−0.191744
$624$	−1.54004e9	−0.253738
$625$	0	0
$626$	9.61719e8	0.156689
$627$	−7.47788e9	−1.21155
$628$	8.10242e9	1.30544
$629$	−5.11942e9	−0.820245
$630$	0	0
$631$	−9.48622e9	−1.50311	−0.751554	−	0.659671i	$-0.770696\pi$
$631$	−9.48622e9	−1.50311	−0.751554	+	0.659671i	$0.770696\pi$
$632$	8.58570e9	1.35290
$633$	1.53925e10	2.41211
$634$	6.94317e8	0.108204
$635$	0	0
$636$	8.25626e9	1.27257
$637$	−4.45872e8	−0.0683475
$638$	−1.75179e9	−0.267060
$639$	−4.44790e9	−0.674376
$640$	0	0
$641$	−6.00094e9	−0.899945	−0.449972	−	0.893042i	$-0.648566\pi$
$641$	−6.00094e9	−0.899945	−0.449972	+	0.893042i	$0.648566\pi$
$642$	−3.08810e9	−0.460595
$643$	1.05337e10	1.56258	0.781292	−	0.624165i	$-0.214561\pi$
$643$	1.05337e10	1.56258	0.781292	+	0.624165i	$0.214561\pi$
$644$	1.10616e9	0.163200
$645$	0	0
$646$	−1.66631e9	−0.243188
$647$	−4.76649e9	−0.691884	−0.345942	−	0.938256i	$-0.612441\pi$
$647$	−4.76649e9	−0.691884	−0.345942	+	0.938256i	$0.612441\pi$
$648$	−6.93103e9	−1.00066
$649$	2.06618e10	2.96695
$650$	0	0
$651$	−1.92563e9	−0.273551
$652$	4.97589e9	0.703079
$653$	6.45834e9	0.907663	0.453831	−	0.891088i	$-0.350057\pi$
$653$	6.45834e9	0.907663	0.453831	+	0.891088i	$0.350057\pi$
$654$	−2.60299e9	−0.363873
$655$	0	0
$656$	4.19923e9	0.580773
$657$	−5.93773e9	−0.816849
$658$	−7.37217e8	−0.100880
$659$	−3.10819e9	−0.423067	−0.211533	−	0.977371i	$-0.567846\pi$
$659$	−3.10819e9	−0.423067	−0.211533	+	0.977371i	$0.567846\pi$
$660$	0	0
$661$	−5.37479e9	−0.723863	−0.361931	−	0.932205i	$-0.617883\pi$
$661$	−5.37479e9	−0.723863	−0.361931	+	0.932205i	$0.617883\pi$
$662$	1.29683e9	0.173732
$663$	4.42627e9	0.589848
$664$	9.40305e9	1.24647
$665$	0	0
$666$	−2.08603e9	−0.273628
$667$	1.45191e9	0.189452
$668$	−2.80754e9	−0.364425
$669$	1.42029e10	1.83395
$670$	0	0
$671$	1.42480e10	1.82065
$672$	3.90752e9	0.496716
$673$	2.18575e9	0.276406	0.138203	−	0.990404i	$-0.455867\pi$
$673$	2.18575e9	0.276406	0.138203	+	0.990404i	$0.455867\pi$
$674$	−9.20825e8	−0.115842
$675$	0	0
$676$	4.87880e9	0.607434
$677$	−4.11027e9	−0.509108	−0.254554	−	0.967059i	$-0.581929\pi$
$677$	−4.11027e9	−0.509108	−0.254554	+	0.967059i	$0.581929\pi$
$678$	5.16440e9	0.636379
$679$	−3.15043e9	−0.386212
$680$	0	0
$681$	−4.18065e9	−0.507257
$682$	−3.56643e9	−0.430515
$683$	5.78846e8	0.0695169	0.0347585	−	0.999396i	$-0.488934\pi$
$683$	5.78846e8	0.0695169	0.0347585	+	0.999396i	$0.488934\pi$
$684$	2.51994e9	0.301088
$685$	0	0
$686$	2.10336e8	0.0248759
$687$	1.90693e9	0.224380
$688$	3.73054e9	0.436729
$689$	5.10850e9	0.595012
$690$	0	0
$691$	−1.15960e9	−0.133701	−0.0668506	−	0.997763i	$-0.521295\pi$
$691$	−1.15960e9	−0.133701	−0.0668506	+	0.997763i	$0.521295\pi$
$692$	−3.94269e9	−0.452295
$693$	−3.81682e9	−0.435648
$694$	−2.51563e9	−0.285686
$695$	0	0
$696$	3.28907e9	0.369777
$697$	−1.20691e10	−1.35009
$698$	−5.10255e8	−0.0567928
$699$	−4.56533e9	−0.505594
$700$	0	0
$701$	1.18229e10	1.29632	0.648159	−	0.761505i	$-0.275539\pi$
$701$	1.18229e10	1.29632	0.648159	+	0.761505i	$0.275539\pi$
$702$	−8.20736e8	−0.0895414
$703$	−4.42736e9	−0.480619
$704$	8.97732e8	0.0969711
$705$	0	0
$706$	8.37429e8	0.0895636
$707$	−3.25788e9	−0.346711
$708$	−1.70939e10	−1.81019
$709$	−1.23818e10	−1.30473	−0.652365	−	0.757905i	$-0.726223\pi$
$709$	−1.23818e10	−1.30473	−0.652365	+	0.757905i	$0.726223\pi$
$710$	0	0
$711$	1.08193e10	1.12890
$712$	4.02423e9	0.417833
$713$	2.95591e9	0.305406
$714$	−2.08805e9	−0.214682
$715$	0	0
$716$	3.07958e9	0.313542
$717$	−1.16854e10	−1.18393
$718$	6.66918e9	0.672414
$719$	−1.25293e10	−1.25712	−0.628559	−	0.777762i	$-0.716355\pi$
$719$	−1.25293e10	−1.25712	−0.628559	+	0.777762i	$0.716355\pi$
$720$	0	0
$721$	−4.83433e9	−0.480356
$722$	3.21809e9	0.318214
$723$	8.03845e8	0.0791022
$724$	−5.46819e9	−0.535499
$725$	0	0
$726$	−1.11849e10	−1.08481
$727$	−5.51212e9	−0.532045	−0.266022	−	0.963967i	$-0.585710\pi$
$727$	−5.51212e9	−0.532045	−0.266022	+	0.963967i	$0.585710\pi$
$728$	1.55047e9	0.148937
$729$	−3.21443e9	−0.307297
$730$	0	0
$731$	−1.07220e10	−1.01524
$732$	−1.17877e10	−1.11081
$733$	−1.21406e10	−1.13861	−0.569307	−	0.822125i	$-0.692789\pi$
$733$	−1.21406e10	−1.13861	−0.569307	+	0.822125i	$0.692789\pi$
$734$	−7.04556e9	−0.657627
$735$	0	0
$736$	−5.99819e9	−0.554559
$737$	2.16824e10	1.99513
$738$	−4.91786e9	−0.450380
$739$	−9.03077e9	−0.823132	−0.411566	−	0.911380i	$-0.635018\pi$
$739$	−9.03077e9	−0.823132	−0.411566	+	0.911380i	$0.635018\pi$
$740$	0	0
$741$	3.82791e9	0.345619
$742$	−2.40988e9	−0.216562
$743$	−1.08266e10	−0.968348	−0.484174	−	0.874972i	$-0.660880\pi$
$743$	−1.08266e10	−0.968348	−0.484174	+	0.874972i	$0.660880\pi$
$744$	6.69615e9	0.596101
$745$	0	0
$746$	8.63721e9	0.761706
$747$	1.18492e10	1.04008
$748$	1.43528e10	1.25395
$749$	−3.34533e9	−0.290906
$750$	0	0
$751$	1.06510e10	0.917598	0.458799	−	0.888540i	$-0.348280\pi$
$751$	1.06510e10	0.917598	0.458799	+	0.888540i	$0.348280\pi$
$752$	−2.75844e9	−0.236538
$753$	1.04957e10	0.895838
$754$	8.96734e8	0.0761840
$755$	0	0
$756$	−1.43695e9	−0.120953
$757$	3.90535e9	0.327208	0.163604	−	0.986526i	$-0.447688\pi$
$757$	3.90535e9	0.327208	0.163604	+	0.986526i	$0.447688\pi$
$758$	−2.76243e9	−0.230382
$759$	1.43841e10	1.19409
$760$	0	0
$761$	−7.33048e9	−0.602957	−0.301478	−	0.953473i	$-0.597480\pi$
$761$	−7.33048e9	−0.602957	−0.301478	+	0.953473i	$0.597480\pi$
$762$	−6.90087e8	−0.0565017
$763$	−2.81981e9	−0.229818
$764$	−1.77075e10	−1.43658
$765$	0	0
$766$	−5.24002e9	−0.421243
$767$	−1.05767e10	−0.846382
$768$	−8.34331e9	−0.664621
$769$	5.64716e9	0.447804	0.223902	−	0.974612i	$-0.428120\pi$
$769$	5.64716e9	0.447804	0.223902	+	0.974612i	$0.428120\pi$
$770$	0	0
$771$	−1.23422e10	−0.969842
$772$	1.11877e10	0.875146
$773$	5.49066e9	0.427559	0.213780	−	0.976882i	$-0.431423\pi$
$773$	5.49066e9	0.427559	0.213780	+	0.976882i	$0.431423\pi$
$774$	−4.36895e9	−0.338676
$775$	0	0
$776$	1.09553e10	0.841603
$777$	−5.54792e9	−0.424284
$778$	−4.81315e8	−0.0366438
$779$	−1.04376e10	−0.791078
$780$	0	0
$781$	−2.19092e10	−1.64569
$782$	3.20523e9	0.239682
$783$	−1.88608e9	−0.140409
$784$	7.87013e8	0.0583278
$785$	0	0
$786$	1.18487e10	0.870345
$787$	8.20098e9	0.599728	0.299864	−	0.953982i	$-0.403059\pi$
$787$	8.20098e9	0.599728	0.299864	+	0.953982i	$0.403059\pi$
$788$	1.01372e10	0.738037
$789$	−3.18225e8	−0.0230656
$790$	0	0
$791$	5.59459e9	0.401930
$792$	1.32726e10	0.949330
$793$	−7.29353e9	−0.519376
$794$	−7.48983e9	−0.531007
$795$	0	0
$796$	−4.18395e9	−0.294030
$797$	1.68355e10	1.17793	0.588967	−	0.808157i	$-0.299535\pi$
$797$	1.68355e10	1.17793	0.588967	+	0.808157i	$0.299535\pi$
$798$	−1.80578e9	−0.125792
$799$	7.92812e9	0.549866
$800$	0	0
$801$	5.07113e9	0.348651
$802$	9.01240e9	0.616922
$803$	−2.92477e10	−1.99337
$804$	−1.79383e10	−1.21726
$805$	0	0
$806$	1.82564e9	0.122813
$807$	−1.62439e10	−1.08801
$808$	1.13289e10	0.755525
$809$	−2.21360e10	−1.46987	−0.734937	−	0.678136i	$-0.762788\pi$
$809$	−2.21360e10	−1.46987	−0.734937	+	0.678136i	$0.762788\pi$
$810$	0	0
$811$	2.33276e10	1.53567	0.767835	−	0.640648i	$-0.221334\pi$
$811$	2.33276e10	1.53567	0.767835	+	0.640648i	$0.221334\pi$
$812$	1.57001e9	0.102910
$813$	−3.42879e10	−2.23782
$814$	−1.02752e10	−0.667737
$815$	0	0
$816$	−7.81284e9	−0.503377
$817$	−9.27260e9	−0.594873
$818$	3.02984e9	0.193546
$819$	1.95382e9	0.124277
$820$	0	0
$821$	1.93627e9	0.122114	0.0610570	−	0.998134i	$-0.480553\pi$
$821$	1.93627e9	0.122114	0.0610570	+	0.998134i	$0.480553\pi$
$822$	−1.46477e10	−0.919854
$823$	−3.03967e8	−0.0190076	−0.00950380	−	0.999955i	$-0.503025\pi$
$823$	−3.03967e8	−0.0190076	−0.00950380	+	0.999955i	$0.503025\pi$
$824$	1.68108e10	1.04675
$825$	0	0
$826$	4.98945e9	0.308051
$827$	−1.45016e10	−0.891551	−0.445776	−	0.895145i	$-0.647072\pi$
$827$	−1.45016e10	−0.891551	−0.445776	+	0.895145i	$0.647072\pi$
$828$	−4.84723e9	−0.296748
$829$	2.11779e9	0.129104	0.0645522	−	0.997914i	$-0.479438\pi$
$829$	2.11779e9	0.129104	0.0645522	+	0.997914i	$0.479438\pi$
$830$	0	0
$831$	−2.83159e10	−1.71170
$832$	−4.59547e8	−0.0276629
$833$	−2.26198e9	−0.135591
$834$	−1.20023e10	−0.716447
$835$	0	0
$836$	1.24125e10	0.734748
$837$	−3.83984e9	−0.226347
$838$	−1.06855e9	−0.0627248
$839$	6.30920e9	0.368814	0.184407	−	0.982850i	$-0.440964\pi$
$839$	6.30920e9	0.368814	0.184407	+	0.982850i	$0.440964\pi$
$840$	0	0
$841$	−1.51891e10	−0.880537
$842$	6.92301e9	0.399671
$843$	−1.91393e10	−1.10035
$844$	−2.55501e10	−1.46283
$845$	0	0
$846$	3.23050e9	0.183431
$847$	−1.21166e10	−0.685152
$848$	−9.01705e9	−0.507784
$849$	4.97012e9	0.278734
$850$	0	0
$851$	8.51626e9	0.473691
$852$	1.81259e10	1.00406
$853$	−1.88042e10	−1.03737	−0.518685	−	0.854965i	$-0.673578\pi$
$853$	−1.88042e10	−1.03737	−0.518685	+	0.854965i	$0.673578\pi$
$854$	3.44065e9	0.189033
$855$	0	0
$856$	1.16330e10	0.633920
$857$	6.55066e9	0.355510	0.177755	−	0.984075i	$-0.443116\pi$
$857$	6.55066e9	0.355510	0.177755	+	0.984075i	$0.443116\pi$
$858$	8.88399e9	0.480178
$859$	−9.36968e9	−0.504369	−0.252185	−	0.967679i	$-0.581149\pi$
$859$	−9.36968e9	−0.504369	−0.252185	+	0.967679i	$0.581149\pi$
$860$	0	0
$861$	−1.30793e10	−0.698352
$862$	−8.92571e9	−0.474643
$863$	−3.09680e10	−1.64012	−0.820060	−	0.572277i	$-0.806060\pi$
$863$	−3.09680e10	−1.64012	−0.820060	+	0.572277i	$0.806060\pi$
$864$	7.79187e9	0.411002
$865$	0	0
$866$	6.61864e9	0.346303
$867$	−2.47117e9	−0.128776
$868$	3.19635e9	0.165896
$869$	5.32927e10	2.75486
$870$	0	0
$871$	−1.10992e10	−0.569151
$872$	9.80557e9	0.500801
$873$	1.38053e10	0.702255
$874$	2.77194e9	0.140441
$875$	0	0
$876$	2.41972e10	1.21619
$877$	−1.81420e10	−0.908211	−0.454105	−	0.890948i	$-0.650041\pi$
$877$	−1.81420e10	−0.908211	−0.454105	+	0.890948i	$0.650041\pi$
$878$	3.72061e9	0.185517
$879$	1.21356e10	0.602697
$880$	0	0
$881$	2.30036e10	1.13339	0.566695	−	0.823928i	$-0.308222\pi$
$881$	2.30036e10	1.13339	0.566695	+	0.823928i	$0.308222\pi$
$882$	−9.21696e8	−0.0452322
$883$	3.55911e10	1.73972	0.869860	−	0.493299i	$-0.164209\pi$
$883$	3.55911e10	1.73972	0.869860	+	0.493299i	$0.164209\pi$
$884$	−7.34716e9	−0.357715
$885$	0	0
$886$	6.94725e9	0.335579
$887$	1.65876e10	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$887$	1.65876e10	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$888$	1.92923e10	0.924565
$889$	−7.47570e8	−0.0356858
$890$	0	0
$891$	−4.30219e10	−2.03760
$892$	−2.35755e10	−1.11220
$893$	6.85637e9	0.322191
$894$	2.23062e10	1.04411
$895$	0	0
$896$	−8.01693e9	−0.372332
$897$	−7.36319e9	−0.340637
$898$	4.46148e9	0.205595
$899$	4.19540e9	0.192581
$900$	0	0
$901$	2.59162e10	1.18041
$902$	−2.42241e10	−1.09907
$903$	−1.16195e10	−0.525146
$904$	−1.94545e10	−0.875854
$905$	0	0
$906$	1.99311e10	0.890396
$907$	1.96890e10	0.876189	0.438095	−	0.898929i	$-0.355653\pi$
$907$	1.96890e10	0.876189	0.438095	+	0.898929i	$0.355653\pi$
$908$	6.93946e9	0.307627
$909$	1.42761e10	0.630429
$910$	0	0
$911$	1.72050e10	0.753946	0.376973	−	0.926224i	$-0.376965\pi$
$911$	1.72050e10	0.753946	0.376973	+	0.926224i	$0.376965\pi$
$912$	−6.75667e9	−0.294952
$913$	5.83662e10	2.53813
$914$	−1.66104e10	−0.719563
$915$	0	0
$916$	−3.16531e9	−0.136076
$917$	1.28357e10	0.549700
$918$	−4.16372e9	−0.177636
$919$	−5.20001e9	−0.221004	−0.110502	−	0.993876i	$-0.535246\pi$
$919$	−5.20001e9	−0.221004	−0.110502	+	0.993876i	$0.535246\pi$
$920$	0	0
$921$	2.93231e9	0.123680
$922$	−1.57920e10	−0.663556
$923$	1.12152e10	0.469465
$924$	1.55541e10	0.648625
$925$	0	0
$926$	−7.98582e9	−0.330507
$927$	2.11841e10	0.873437
$928$	−8.51338e9	−0.349691
$929$	−4.68926e10	−1.91888	−0.959442	−	0.281905i	$-0.909034\pi$
$929$	−4.68926e10	−1.91888	−0.959442	+	0.281905i	$0.909034\pi$
$930$	0	0
$931$	−1.95620e9	−0.0794490
$932$	7.57800e9	0.306619
$933$	1.38223e10	0.557178
$934$	−1.80965e10	−0.726743
$935$	0	0
$936$	−6.79420e9	−0.270815
$937$	−1.78840e10	−0.710194	−0.355097	−	0.934829i	$-0.615552\pi$
$937$	−1.78840e10	−0.710194	−0.355097	+	0.934829i	$0.615552\pi$
$938$	5.23593e9	0.207149
$939$	−1.12081e10	−0.441776
$940$	0	0
$941$	4.81758e9	0.188480	0.0942400	−	0.995550i	$-0.469958\pi$
$941$	4.81758e9	0.188480	0.0942400	+	0.995550i	$0.469958\pi$
$942$	2.54427e10	0.991712
$943$	2.00773e10	0.779675
$944$	1.86690e10	0.722303
$945$	0	0
$946$	−2.15203e10	−0.826474
$947$	1.97403e10	0.755317	0.377659	−	0.925945i	$-0.376729\pi$
$947$	1.97403e10	0.755317	0.377659	+	0.925945i	$0.376729\pi$
$948$	−4.40901e10	−1.68078
$949$	1.49718e10	0.568647
$950$	0	0
$951$	−8.09174e9	−0.305077
$952$	7.86577e9	0.295469
$953$	2.22133e10	0.831358	0.415679	−	0.909511i	$-0.363544\pi$
$953$	2.22133e10	0.831358	0.415679	+	0.909511i	$0.363544\pi$
$954$	1.05602e10	0.393778
$955$	0	0
$956$	1.93965e10	0.717995
$957$	2.04157e10	0.752962
$958$	−2.51129e9	−0.0922822
$959$	−1.58679e10	−0.580969
$960$	0	0
$961$	−1.89713e10	−0.689549
$962$	5.25986e9	0.190485
$963$	1.46593e10	0.528959
$964$	−1.33430e9	−0.0479717
$965$	0	0
$966$	3.47351e9	0.123979
$967$	2.25674e10	0.802579	0.401290	−	0.915951i	$-0.368562\pi$
$967$	2.25674e10	0.802579	0.401290	+	0.915951i	$0.368562\pi$
$968$	4.21340e10	1.49303
$969$	1.94196e10	0.685656
$970$	0	0
$971$	1.95033e9	0.0683659	0.0341830	−	0.999416i	$-0.489117\pi$
$971$	1.95033e9	0.0683659	0.0341830	+	0.999416i	$0.489117\pi$
$972$	2.64307e10	0.923160
$973$	−1.30021e10	−0.452500
$974$	1.53782e10	0.533273
$975$	0	0
$976$	1.28739e10	0.443236
$977$	−7.74628e9	−0.265743	−0.132872	−	0.991133i	$-0.542420\pi$
$977$	−7.74628e9	−0.265743	−0.132872	+	0.991133i	$0.542420\pi$
$978$	1.56250e10	0.534114
$979$	2.49790e10	0.850817
$980$	0	0
$981$	1.23565e10	0.417881
$982$	8.12477e8	0.0273792
$983$	−4.84148e10	−1.62570	−0.812851	−	0.582472i	$-0.802085\pi$
$983$	−4.84148e10	−1.62570	−0.812851	+	0.582472i	$0.802085\pi$
$984$	4.54819e10	1.52179
$985$	0	0
$986$	4.54927e9	0.151138
$987$	8.59170e9	0.284426
$988$	−6.35395e9	−0.209601
$989$	1.78363e10	0.586299
$990$	0	0
$991$	−5.01522e10	−1.63694	−0.818468	−	0.574552i	$-0.805176\pi$
$991$	−5.01522e10	−1.63694	−0.818468	+	0.574552i	$0.805176\pi$
$992$	−1.73322e10	−0.563720
$993$	−1.51136e10	−0.489830
$994$	−5.29068e9	−0.170867
$995$	0	0
$996$	−4.82874e10	−1.54855
$997$	−4.44770e10	−1.42136	−0.710678	−	0.703517i	$-0.751612\pi$
$997$	−4.44770e10	−1.42136	−0.710678	+	0.703517i	$0.751612\pi$
$998$	−1.70729e10	−0.543689
$999$	−1.10629e10	−0.351068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.8.a.f.1.3		5
5.2	odd	4		175.8.b.f.99.5		10
5.3	odd	4		175.8.b.f.99.6		10
5.4	even	2		35.8.a.d.1.3	✓	5
15.14	odd	2		315.8.a.m.1.3		5
20.19	odd	2		560.8.a.s.1.5		5
35.34	odd	2		245.8.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.8.a.d.1.3	✓	5	5.4	even	2
175.8.a.f.1.3		5	1.1	even	1	trivial
175.8.b.f.99.5		10	5.2	odd	4
175.8.b.f.99.6		10	5.3	odd	4
245.8.a.f.1.3		5	35.34	odd	2
315.8.a.m.1.3		5	15.14	odd	2
560.8.a.s.1.5		5	20.19	odd	2

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\(q-5.21232 q^{2} +60.7456 q^{3} -100.832 q^{4} -316.626 q^{6} -343.000 q^{7} +1192.74 q^{8} +1503.03 q^{9} +O(q^{10})\)	\(q-5.21232 q^{2} +60.7456 q^{3} -100.832 q^{4} -316.626 q^{6} -343.000 q^{7} +1192.74 q^{8} +1503.03 q^{9} +7403.54 q^{11} -6125.09 q^{12} -3789.85 q^{13} +1787.83 q^{14} +6689.50 q^{16} -19226.5 q^{17} -7834.29 q^{18} -16627.4 q^{19} -20835.8 q^{21} -38589.6 q^{22} +31983.7 q^{23} +72454.0 q^{24} +19753.9 q^{26} -41548.0 q^{27} +34585.3 q^{28} +45395.3 q^{29} +92419.3 q^{31} -187539. q^{32} +449733. q^{33} +100215. q^{34} -151553. q^{36} +266269. q^{37} +86667.2 q^{38} -230217. q^{39} +627735. q^{41} +108603. q^{42} +557671. q^{43} -746512. q^{44} -166709. q^{46} -412354. q^{47} +406358. q^{48} +117649. q^{49} -1.16793e6 q^{51} +382137. q^{52} -1.34794e6 q^{53} +216561. q^{54} -409111. q^{56} -1.01004e6 q^{57} -236615. q^{58} +2.79079e6 q^{59} +1.92449e6 q^{61} -481719. q^{62} -515540. q^{63} +121257. q^{64} -2.34415e6 q^{66} +2.92866e6 q^{67} +1.93864e6 q^{68} +1.94287e6 q^{69} -2.95928e6 q^{71} +1.79273e6 q^{72} -3.95050e6 q^{73} -1.38788e6 q^{74} +1.67657e6 q^{76} -2.53941e6 q^{77} +1.19996e6 q^{78} +7.19828e6 q^{79} -5.81099e6 q^{81} -3.27196e6 q^{82} +7.88355e6 q^{83} +2.10091e6 q^{84} -2.90676e6 q^{86} +2.75756e6 q^{87} +8.83053e6 q^{88} +3.37393e6 q^{89} +1.29992e6 q^{91} -3.22497e6 q^{92} +5.61407e6 q^{93} +2.14932e6 q^{94} -1.13922e7 q^{96} +9.18493e6 q^{97} -613224. q^{98} +1.11278e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 11 q^{2} - 65 q^{3} + 553 q^{4} + 96 q^{6} - 1715 q^{7} + 1647 q^{8} - 12 q^{9}+O(q^{10}) \) 5 * q - 11 * q^2 - 65 * q^3 + 553 * q^4 + 96 * q^6 - 1715 * q^7 + 1647 * q^8 - 12 * q^9	\( 5 q - 11 q^{2} - 65 q^{3} + 553 q^{4} + 96 q^{6} - 1715 q^{7} + 1647 q^{8} - 12 q^{9} + 10027 q^{11} - 10024 q^{12} - 16141 q^{13} + 3773 q^{14} + 94169 q^{16} - 17427 q^{17} - 47547 q^{18} + 44394 q^{19} + 22295 q^{21} + 89168 q^{22} - 55698 q^{23} + 18324 q^{24} + 163750 q^{26} + 53389 q^{27} - 189679 q^{28} - 164219 q^{29} + 138440 q^{31} + 509615 q^{32} + 213271 q^{33} + 132338 q^{34} - 1408655 q^{36} - 181518 q^{37} + 1725140 q^{38} - 481621 q^{39} - 369676 q^{41} - 32928 q^{42} - 386 q^{43} - 37608 q^{44} - 2610592 q^{46} - 962985 q^{47} + 1271004 q^{48} + 588245 q^{49} - 1305827 q^{51} - 2349706 q^{52} - 396244 q^{53} + 2641788 q^{54} - 564921 q^{56} - 4322278 q^{57} + 4408538 q^{58} + 6284948 q^{59} + 1975828 q^{61} + 2246384 q^{62} + 4116 q^{63} + 13769617 q^{64} + 798284 q^{66} + 554376 q^{67} + 11618770 q^{68} + 9051046 q^{69} + 6526040 q^{71} + 3026835 q^{72} - 7479766 q^{73} - 3948102 q^{74} + 9203364 q^{76} - 3439261 q^{77} + 12300388 q^{78} + 4851305 q^{79} - 4256091 q^{81} + 29262330 q^{82} + 11813196 q^{83} + 3438232 q^{84} - 34717484 q^{86} + 263021 q^{87} + 54461828 q^{88} + 12879896 q^{89} + 5536363 q^{91} + 18484672 q^{92} + 17648536 q^{93} + 23744524 q^{94} - 39241860 q^{96} - 11704767 q^{97} - 1294139 q^{98} + 20972602 q^{99}+O(q^{100}) \) 5 * q - 11 * q^2 - 65 * q^3 + 553 * q^4 + 96 * q^6 - 1715 * q^7 + 1647 * q^8 - 12 * q^9 + 10027 * q^11 - 10024 * q^12 - 16141 * q^13 + 3773 * q^14 + 94169 * q^16 - 17427 * q^17 - 47547 * q^18 + 44394 * q^19 + 22295 * q^21 + 89168 * q^22 - 55698 * q^23 + 18324 * q^24 + 163750 * q^26 + 53389 * q^27 - 189679 * q^28 - 164219 * q^29 + 138440 * q^31 + 509615 * q^32 + 213271 * q^33 + 132338 * q^34 - 1408655 * q^36 - 181518 * q^37 + 1725140 * q^38 - 481621 * q^39 - 369676 * q^41 - 32928 * q^42 - 386 * q^43 - 37608 * q^44 - 2610592 * q^46 - 962985 * q^47 + 1271004 * q^48 + 588245 * q^49 - 1305827 * q^51 - 2349706 * q^52 - 396244 * q^53 + 2641788 * q^54 - 564921 * q^56 - 4322278 * q^57 + 4408538 * q^58 + 6284948 * q^59 + 1975828 * q^61 + 2246384 * q^62 + 4116 * q^63 + 13769617 * q^64 + 798284 * q^66 + 554376 * q^67 + 11618770 * q^68 + 9051046 * q^69 + 6526040 * q^71 + 3026835 * q^72 - 7479766 * q^73 - 3948102 * q^74 + 9203364 * q^76 - 3439261 * q^77 + 12300388 * q^78 + 4851305 * q^79 - 4256091 * q^81 + 29262330 * q^82 + 11813196 * q^83 + 3438232 * q^84 - 34717484 * q^86 + 263021 * q^87 + 54461828 * q^88 + 12879896 * q^89 + 5536363 * q^91 + 18484672 * q^92 + 17648536 * q^93 + 23744524 * q^94 - 39241860 * q^96 - 11704767 * q^97 - 1294139 * q^98 + 20972602 * q^99

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