LMFDB - Embedded newform 325.6.b.h.274.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 $ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.17
Root			$9.23858i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.23858i q^{2} -11.6375i q^{3} -35.8743 q^{4} +95.8765 q^{6} -195.071i q^{7} -31.9185i q^{8} +107.569 q^{9} +O(q^{10})$	\(q+8.23858i q^{2} -11.6375i q^{3} -35.8743 q^{4} +95.8765 q^{6} -195.071i q^{7} -31.9185i q^{8} +107.569 q^{9} +64.5579 q^{11} +417.487i q^{12} -169.000i q^{13} +1607.11 q^{14} -885.013 q^{16} +426.010i q^{17} +886.215i q^{18} +959.570 q^{19} -2270.14 q^{21} +531.866i q^{22} +499.768i q^{23} -371.452 q^{24} +1392.32 q^{26} -4079.74i q^{27} +6998.03i q^{28} -1288.17 q^{29} -6738.17 q^{31} -8312.65i q^{32} -751.292i q^{33} -3509.72 q^{34} -3858.95 q^{36} -6218.21i q^{37} +7905.50i q^{38} -1966.74 q^{39} -6498.90 q^{41} -18702.7i q^{42} +15640.4i q^{43} -2315.97 q^{44} -4117.38 q^{46} -6299.21i q^{47} +10299.3i q^{48} -21245.7 q^{49} +4957.69 q^{51} +6062.75i q^{52} -40397.3i q^{53} +33611.3 q^{54} -6226.38 q^{56} -11167.0i q^{57} -10612.7i q^{58} -25614.1 q^{59} +24197.9 q^{61} -55513.0i q^{62} -20983.6i q^{63} +40164.0 q^{64} +6189.58 q^{66} -39183.3i q^{67} -15282.8i q^{68} +5816.04 q^{69} -32681.2 q^{71} -3433.44i q^{72} -14507.0i q^{73} +51229.3 q^{74} -34423.9 q^{76} -12593.4i q^{77} -16203.1i q^{78} -79037.9 q^{79} -21338.7 q^{81} -53541.7i q^{82} -102366. i q^{83} +81439.5 q^{84} -128855. q^{86} +14991.1i q^{87} -2060.59i q^{88} +48103.8 q^{89} -32967.0 q^{91} -17928.8i q^{92} +78415.4i q^{93} +51896.6 q^{94} -96738.4 q^{96} +73361.2i q^{97} -175035. i q^{98} +6944.42 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.23858i	1.45639i	0.685370	+	0.728195i	$0.259640\pi$
$2$	8.23858i	1.45639i	−0.685370	+	0.728195i	$0.740360\pi$
$3$	− 11.6375i	− 0.746545i	−0.927722	−	0.373273i	$-0.878236\pi$
$3$	− 11.6375i	− 0.746545i	0.927722	−	0.373273i	$-0.121764\pi$
$4$	−35.8743	−1.12107
$5$	0	0
$5$	0	0
$6$	95.8765	1.08726
$7$	− 195.071i	− 1.50469i	−0.658768	−	0.752346i	$-0.728922\pi$
$7$	− 195.071i	− 1.50469i	0.658768	−	0.752346i	$-0.271078\pi$
$8$	− 31.9185i	− 0.176327i
$9$	107.569	0.442670
$10$	0	0
$11$	64.5579	0.160867	0.0804337	−	0.996760i	$-0.474369\pi$
$11$	64.5579	0.160867	0.0804337	+	0.996760i	$0.474369\pi$
$12$	417.487i	0.836930i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	1607.11	2.19142
$15$	0	0
$16$	−885.013	−0.864271
$17$	426.010i	0.357518i	0.983893	+	0.178759i	$0.0572082\pi$
$17$	426.010i	0.357518i	−0.983893	+	0.178759i	$0.942792\pi$
$18$	886.215i	0.644700i
$19$	959.570	0.609807	0.304904	−	0.952383i	$-0.401376\pi$
$19$	959.570	0.609807	0.304904	+	0.952383i	$0.401376\pi$
$20$	0	0
$21$	−2270.14	−1.12332
$22$	531.866i	0.234286i
$23$	499.768i	0.196992i	0.995137	+	0.0984960i	$0.0314032\pi$
$23$	499.768i	0.196992i	−0.995137	+	0.0984960i	$0.968597\pi$
$24$	−371.452	−0.131636
$25$	0	0
$26$	1392.32	0.403930
$27$	− 4079.74i	− 1.07702i
$28$	6998.03i	1.68687i
$29$	−1288.17	−0.284432	−0.142216	−	0.989836i	$-0.545423\pi$
$29$	−1288.17	−0.284432	−0.142216	+	0.989836i	$0.545423\pi$
$30$	0	0
$31$	−6738.17	−1.25932	−0.629662	−	0.776869i	$-0.716807\pi$
$31$	−6738.17	−1.25932	−0.629662	+	0.776869i	$0.716807\pi$
$32$	− 8312.65i	− 1.43504i
$33$	− 751.292i	− 0.120095i
$34$	−3509.72	−0.520685
$35$	0	0
$36$	−3858.95	−0.496264
$37$	− 6218.21i	− 0.746726i	−0.927685	−	0.373363i	$-0.878205\pi$
$37$	− 6218.21i	− 0.746726i	0.927685	−	0.373363i	$-0.121795\pi$
$38$	7905.50i	0.888117i
$39$	−1966.74	−0.207054
$40$	0	0
$41$	−6498.90	−0.603782	−0.301891	−	0.953342i	$-0.597618\pi$
$41$	−6498.90	−0.603782	−0.301891	+	0.953342i	$0.597618\pi$
$42$	− 18702.7i	− 1.63599i
$43$	15640.4i	1.28996i	0.764200	+	0.644980i	$0.223134\pi$
$43$	15640.4i	1.28996i	−0.764200	+	0.644980i	$0.776866\pi$
$44$	−2315.97	−0.180344
$45$	0	0
$46$	−4117.38	−0.286897
$47$	− 6299.21i	− 0.415951i	−0.978134	−	0.207975i	$-0.933313\pi$
$47$	− 6299.21i	− 0.415951i	0.978134	−	0.207975i	$-0.0666874\pi$
$48$	10299.3i	0.645217i
$49$	−21245.7	−1.26410
$50$	0	0
$51$	4957.69	0.266903
$52$	6062.75i	0.310929i
$53$	− 40397.3i	− 1.97543i	−0.156254	−	0.987717i	$-0.549942\pi$
$53$	− 40397.3i	− 1.97543i	0.156254	−	0.987717i	$-0.450058\pi$
$54$	33611.3	1.56856
$55$	0	0
$56$	−6226.38	−0.265317
$57$	− 11167.0i	− 0.455249i
$58$	− 10612.7i	− 0.414244i
$59$	−25614.1	−0.957965	−0.478982	−	0.877824i	$-0.658994\pi$
$59$	−25614.1	−0.957965	−0.478982	+	0.877824i	$0.658994\pi$
$60$	0	0
$61$	24197.9	0.832632	0.416316	−	0.909220i	$-0.363321\pi$
$61$	24197.9	0.832632	0.416316	+	0.909220i	$0.363321\pi$
$62$	− 55513.0i	− 1.83407i
$63$	− 20983.6i	− 0.666082i
$64$	40164.0	1.22571
$65$	0	0
$66$	6189.58	0.174905
$67$	− 39183.3i	− 1.06639i	−0.845994	−	0.533193i	$-0.820992\pi$
$67$	− 39183.3i	− 1.06639i	0.845994	−	0.533193i	$-0.179008\pi$
$68$	− 15282.8i	− 0.400803i
$69$	5816.04	0.147064
$70$	0	0
$71$	−32681.2	−0.769401	−0.384700	−	0.923041i	$-0.625695\pi$
$71$	−32681.2	−0.769401	−0.384700	+	0.923041i	$0.625695\pi$
$72$	− 3433.44i	− 0.0780545i
$73$	− 14507.0i	− 0.318617i	−0.987229	−	0.159309i	$-0.949073\pi$
$73$	− 14507.0i	− 0.318617i	0.987229	−	0.159309i	$-0.0509265\pi$
$74$	51229.3	1.08752
$75$	0	0
$76$	−34423.9	−0.683637
$77$	− 12593.4i	− 0.242056i
$78$	− 16203.1i	− 0.301552i
$79$	−79037.9	−1.42484	−0.712422	−	0.701751i	$-0.752402\pi$
$79$	−79037.9	−1.42484	−0.712422	+	0.701751i	$0.752402\pi$
$80$	0	0
$81$	−21338.7	−0.361373
$82$	− 53541.7i	− 0.879342i
$83$	− 102366.i	− 1.63102i	−0.578744	−	0.815509i	$-0.696457\pi$
$83$	− 102366.i	− 1.63102i	0.578744	−	0.815509i	$-0.303543\pi$
$84$	81439.5	1.25932
$85$	0	0
$86$	−128855.	−1.87868
$87$	14991.1i	0.212341i
$88$	− 2060.59i	− 0.0283652i
$89$	48103.8	0.643731	0.321866	−	0.946785i	$-0.395690\pi$
$89$	48103.8	0.643731	0.321866	+	0.946785i	$0.395690\pi$
$90$	0	0
$91$	−32967.0	−0.417327
$92$	− 17928.8i	− 0.220842i
$93$	78415.4i	0.940143i
$94$	51896.6	0.605786
$95$	0	0
$96$	−96738.4	−1.07132
$97$	73361.2i	0.791657i	0.918324	+	0.395829i	$0.129543\pi$
$97$	73361.2i	0.791657i	−0.918324	+	0.395829i	$0.870457\pi$
$98$	− 175035.i	− 1.84102i
$99$	6944.42	0.0712111
$100$	0	0
$101$	22470.8	0.219187	0.109594	−	0.993976i	$-0.465045\pi$
$101$	22470.8	0.219187	0.109594	+	0.993976i	$0.465045\pi$
$102$	40844.3i	0.388715i
$103$	116368.i	1.08078i	0.841413	+	0.540392i	$0.181724\pi$
$103$	116368.i	1.08078i	−0.841413	+	0.540392i	$0.818276\pi$
$104$	−5394.23	−0.0489042
$105$	0	0
$106$	332816.	2.87700
$107$	− 33242.7i	− 0.280697i	−0.990102	−	0.140348i	$-0.955178\pi$
$107$	− 33242.7i	− 0.280697i	0.990102	−	0.140348i	$-0.0448222\pi$
$108$	146358.i	1.20741i
$109$	87314.5	0.703915	0.351957	−	0.936016i	$-0.385516\pi$
$109$	87314.5	0.703915	0.351957	+	0.936016i	$0.385516\pi$
$110$	0	0
$111$	−72364.4	−0.557465
$112$	172640.i	1.30046i
$113$	− 211485.i	− 1.55806i	−0.626988	−	0.779029i	$-0.715713\pi$
$113$	− 211485.i	− 1.55806i	0.626988	−	0.779029i	$-0.284287\pi$
$114$	92000.1	0.663020
$115$	0	0
$116$	46212.2	0.318868
$117$	− 18179.1i	− 0.122775i
$118$	− 211024.i	− 1.39517i
$119$	83102.3	0.537954
$120$	0	0
$121$	−156883.	−0.974122
$122$	199356.i	1.21264i
$123$	75630.9i	0.450751i
$124$	241727.	1.41179
$125$	0	0
$126$	172875.	0.970075
$127$	− 74724.8i	− 0.411107i	−0.978646	−	0.205554i	$-0.934100\pi$
$127$	− 74724.8i	− 0.411107i	0.978646	−	0.205554i	$-0.0658995\pi$
$128$	64890.1i	0.350069i
$129$	182015.	0.963013
$130$	0	0
$131$	27550.2	0.140264	0.0701320	−	0.997538i	$-0.477658\pi$
$131$	27550.2	0.140264	0.0701320	+	0.997538i	$0.477658\pi$
$132$	26952.1i	0.134635i
$133$	− 187184.i	− 0.917572i
$134$	322815.	1.55307
$135$	0	0
$136$	13597.6	0.0630399
$137$	266022.i	1.21092i	0.795876	+	0.605460i	$0.207011\pi$
$137$	266022.i	1.21092i	−0.795876	+	0.605460i	$0.792989\pi$
$138$	47916.0i	0.214182i
$139$	263800.	1.15808	0.579039	−	0.815300i	$-0.303428\pi$
$139$	263800.	1.15808	0.579039	+	0.815300i	$0.303428\pi$
$140$	0	0
$141$	−73307.0	−0.310526
$142$	− 269247.i	− 1.12055i
$143$	− 10910.3i	− 0.0446166i
$144$	−95199.8	−0.382587
$145$	0	0
$146$	119517.	0.464031
$147$	247247.i	0.943708i
$148$	223074.i	0.837133i
$149$	−4746.74	−0.0175158	−0.00875790	−	0.999962i	$-0.502788\pi$
$149$	−4746.74	−0.0175158	−0.00875790	+	0.999962i	$0.502788\pi$
$150$	0	0
$151$	−94651.9	−0.337821	−0.168911	−	0.985631i	$-0.554025\pi$
$151$	−94651.9	−0.337821	−0.168911	+	0.985631i	$0.554025\pi$
$152$	− 30628.1i	− 0.107525i
$153$	45825.4i	0.158262i
$154$	103752.	0.352528
$155$	0	0
$156$	70555.2	0.232123
$157$	− 264887.i	− 0.857652i	−0.903387	−	0.428826i	$-0.858927\pi$
$157$	− 264887.i	− 0.857652i	0.903387	−	0.428826i	$-0.141073\pi$
$158$	− 651160.i	− 2.07513i
$159$	−470123.	−1.47475
$160$	0	0
$161$	97490.2	0.296412
$162$	− 175801.i	− 0.526300i
$163$	− 121957.i	− 0.359533i	−0.983709	−	0.179766i	$-0.942466\pi$
$163$	− 121957.i	− 0.359533i	0.983709	−	0.179766i	$-0.0575342\pi$
$164$	233143.	0.676883
$165$	0	0
$166$	843348.	2.37540
$167$	− 465583.i	− 1.29183i	−0.763408	−	0.645916i	$-0.776476\pi$
$167$	− 465583.i	− 1.29183i	0.763408	−	0.645916i	$-0.223524\pi$
$168$	72459.5i	0.198071i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	103220.	0.269943
$172$	− 561087.i	− 1.44614i
$173$	− 69307.4i	− 0.176062i	−0.996118	−	0.0880308i	$-0.971943\pi$
$173$	− 69307.4i	− 0.176062i	0.996118	−	0.0880308i	$-0.0280574\pi$
$174$	−123505.	−0.309252
$175$	0	0
$176$	−57134.6	−0.139033
$177$	298084.i	0.715164i
$178$	396307.i	0.937524i
$179$	−305640.	−0.712981	−0.356491	−	0.934299i	$-0.616027\pi$
$179$	−305640.	−0.712981	−0.356491	+	0.934299i	$0.616027\pi$
$180$	0	0
$181$	−284602.	−0.645715	−0.322858	−	0.946448i	$-0.604644\pi$
$181$	−284602.	−0.645715	−0.322858	+	0.946448i	$0.604644\pi$
$182$	− 271601.i	− 0.607790i
$183$	− 281603.i	− 0.621598i
$184$	15951.9	0.0347350
$185$	0	0
$186$	−646032.	−1.36921
$187$	27502.3i	0.0575129i
$188$	225980.i	0.466310i
$189$	−795839.	−1.62058
$190$	0	0
$191$	−536766.	−1.06464	−0.532319	−	0.846544i	$-0.678679\pi$
$191$	−536766.	−1.06464	−0.532319	+	0.846544i	$0.678679\pi$
$192$	− 467409.i	− 0.915048i
$193$	931130.i	1.79936i	0.436555	+	0.899678i	$0.356199\pi$
$193$	931130.i	1.79936i	−0.436555	+	0.899678i	$0.643801\pi$
$194$	−604393.	−1.15296
$195$	0	0
$196$	762175.	1.41715
$197$	210297.i	0.386071i	0.981192	+	0.193036i	$0.0618333\pi$
$197$	210297.i	0.386071i	−0.981192	+	0.193036i	$0.938167\pi$
$198$	57212.2i	0.103711i
$199$	−853286.	−1.52743	−0.763716	−	0.645552i	$-0.776627\pi$
$199$	−853286.	−1.52743	−0.763716	+	0.645552i	$0.776627\pi$
$200$	0	0
$201$	−455995.	−0.796105
$202$	185128.i	0.319222i
$203$	251285.i	0.427983i
$204$	−177854.	−0.299218
$205$	0	0
$206$	−958704.	−1.57404
$207$	53759.4i	0.0872025i
$208$	149567.i	0.239706i
$209$	61947.8	0.0980981
$210$	0	0
$211$	−913530.	−1.41259	−0.706296	−	0.707917i	$-0.749635\pi$
$211$	−913530.	−1.41259	−0.706296	+	0.707917i	$0.749635\pi$
$212$	1.44922e6i	2.21460i
$213$	380328.i	0.574393i
$214$	273873.	0.408804
$215$	0	0
$216$	−130219.	−0.189907
$217$	1.31442e6i	1.89490i
$218$	719348.i	1.02517i
$219$	−168825.	−0.237862
$220$	0	0
$221$	71995.7	0.0991576
$222$	− 596180.i	− 0.811886i
$223$	− 412081.i	− 0.554908i	−0.960739	−	0.277454i	$-0.910509\pi$
$223$	− 412081.i	− 0.554908i	0.960739	−	0.277454i	$-0.0894906\pi$
$224$	−1.62156e6	−2.15930
$225$	0	0
$226$	1.74234e6	2.26914
$227$	− 864103.i	− 1.11302i	−0.830843	−	0.556508i	$-0.812141\pi$
$227$	− 864103.i	− 1.11302i	0.830843	−	0.556508i	$-0.187859\pi$
$228$	400607.i	0.510366i
$229$	236005.	0.297395	0.148697	−	0.988883i	$-0.452492\pi$
$229$	236005.	0.297395	0.148697	+	0.988883i	$0.452492\pi$
$230$	0	0
$231$	−146555.	−0.180706
$232$	41116.5i	0.0501529i
$233$	− 1.24095e6i	− 1.49749i	−0.662859	−	0.748744i	$-0.730657\pi$
$233$	− 1.24095e6i	− 1.49749i	0.662859	−	0.748744i	$-0.269343\pi$
$234$	149770.	0.178808
$235$	0	0
$236$	918888.	1.07395
$237$	919803.i	1.06371i
$238$	684645.i	0.783471i
$239$	868458.	0.983454	0.491727	−	0.870749i	$-0.336366\pi$
$239$	868458.	0.983454	0.491727	+	0.870749i	$0.336366\pi$
$240$	0	0
$241$	−836767.	−0.928029	−0.464015	−	0.885827i	$-0.653592\pi$
$241$	−836767.	−0.928029	−0.464015	+	0.885827i	$0.653592\pi$
$242$	− 1.29250e6i	− 1.41870i
$243$	− 743048.i	− 0.807237i
$244$	−868082.	−0.933440
$245$	0	0
$246$	−623092.	−0.656469
$247$	− 162167.i	− 0.169130i
$248$	215072.i	0.222052i
$249$	−1.19128e6	−1.21763
$250$	0	0
$251$	−1.94946e6	−1.95313	−0.976565	−	0.215224i	$-0.930952\pi$
$251$	−1.94946e6	−1.95313	−0.976565	+	0.215224i	$0.930952\pi$
$252$	752770.i	0.746725i
$253$	32264.0i	0.0316896i
$254$	615626.	0.598733
$255$	0	0
$256$	750647.	0.715873
$257$	− 358481.i	− 0.338558i	−0.985568	−	0.169279i	$-0.945856\pi$
$257$	− 358481.i	− 0.338558i	0.985568	−	0.169279i	$-0.0541439\pi$
$258$	1.49954e6i	1.40252i
$259$	−1.21299e6	−1.12359
$260$	0	0
$261$	−138567.	−0.125909
$262$	226975.i	0.204279i
$263$	− 2.02422e6i	− 1.80454i	−0.431167	−	0.902272i	$-0.641898\pi$
$263$	− 2.02422e6i	− 1.80454i	0.431167	−	0.902272i	$-0.358102\pi$
$264$	−23980.2	−0.0211759
$265$	0	0
$266$	1.54213e6	1.33634
$267$	− 559808.i	− 0.480575i
$268$	1.40567e6i	1.19549i
$269$	1.72406e6	1.45268	0.726341	−	0.687334i	$-0.241219\pi$
$269$	1.72406e6	1.45268	0.726341	+	0.687334i	$0.241219\pi$
$270$	0	0
$271$	−618168.	−0.511309	−0.255654	−	0.966768i	$-0.582291\pi$
$271$	−618168.	−0.511309	−0.255654	+	0.966768i	$0.582291\pi$
$272$	− 377025.i	− 0.308992i
$273$	383653.i	0.311553i
$274$	−2.19164e6	−1.76357
$275$	0	0
$276$	−208646.	−0.164869
$277$	535823.i	0.419587i	0.977746	+	0.209793i	$0.0672791\pi$
$277$	535823.i	0.419587i	−0.977746	+	0.209793i	$0.932721\pi$
$278$	2.17334e6i	1.68661i
$279$	−724817.	−0.557465
$280$	0	0
$281$	1.19483e6	0.902695	0.451347	−	0.892348i	$-0.350944\pi$
$281$	1.19483e6	0.902695	0.451347	+	0.892348i	$0.350944\pi$
$282$	− 603946.i	− 0.452247i
$283$	2.59984e6i	1.92966i	0.262875	+	0.964830i	$0.415329\pi$
$283$	2.59984e6i	1.92966i	−0.262875	+	0.964830i	$0.584671\pi$
$284$	1.17242e6	0.862553
$285$	0	0
$286$	89885.3	0.0649791
$287$	1.26775e6i	0.908506i
$288$	− 894182.i	− 0.635250i
$289$	1.23837e6	0.872181
$290$	0	0
$291$	853741.	0.591008
$292$	520427.i	0.357193i
$293$	− 2.12976e6i	− 1.44931i	−0.689113	−	0.724654i	$-0.742000\pi$
$293$	− 2.12976e6i	− 1.44931i	0.689113	−	0.724654i	$-0.258000\pi$
$294$	−2.03696e6	−1.37441
$295$	0	0
$296$	−198476.	−0.131668
$297$	− 263380.i	− 0.173257i
$298$	− 39106.4i	− 0.0255098i
$299$	84460.8	0.0546358
$300$	0	0
$301$	3.05098e6	1.94099
$302$	− 779798.i	− 0.492000i
$303$	− 261504.i	− 0.163633i
$304$	−849232.	−0.527039
$305$	0	0
$306$	−377536.	−0.230492
$307$	1.52136e6i	0.921270i	0.887590	+	0.460635i	$0.152378\pi$
$307$	1.52136e6i	0.921270i	−0.887590	+	0.460635i	$0.847622\pi$
$308$	451778.i	0.271362i
$309$	1.35423e6	0.806854
$310$	0	0
$311$	1.78856e6	1.04858	0.524290	−	0.851540i	$-0.324331\pi$
$311$	1.78856e6	1.04858	0.524290	+	0.851540i	$0.324331\pi$
$312$	62775.3i	0.0365092i
$313$	− 451359.i	− 0.260412i	−0.991487	−	0.130206i	$-0.958436\pi$
$313$	− 451359.i	− 0.260412i	0.991487	−	0.130206i	$-0.0415639\pi$
$314$	2.18229e6	1.24908
$315$	0	0
$316$	2.83543e6	1.59735
$317$	1.39653e6i	0.780554i	0.920697	+	0.390277i	$0.127621\pi$
$317$	1.39653e6i	0.780554i	−0.920697	+	0.390277i	$0.872379\pi$
$318$	− 3.87315e6i	− 2.14781i
$319$	−83161.6	−0.0457558
$320$	0	0
$321$	−386862.	−0.209553
$322$	803182.i	0.431692i
$323$	408786.i	0.218017i
$324$	765512.	0.405125
$325$	0	0
$326$	1.00475e6	0.523620
$327$	− 1.01612e6i	− 0.525504i
$328$	207435.i	0.106463i
$329$	−1.22879e6	−0.625878
$330$	0	0
$331$	1.37630e6	0.690467	0.345233	−	0.938517i	$-0.387800\pi$
$331$	1.37630e6	0.690467	0.345233	+	0.938517i	$0.387800\pi$
$332$	3.67229e6i	1.82849i
$333$	− 668885.i	− 0.330553i
$334$	3.83575e6	1.88141
$335$	0	0
$336$	2.00910e6	0.970854
$337$	1.93389e6i	0.927590i	0.885943	+	0.463795i	$0.153513\pi$
$337$	1.93389e6i	0.927590i	−0.885943	+	0.463795i	$0.846487\pi$
$338$	− 235302.i	− 0.112030i
$339$	−2.46115e6	−1.16316
$340$	0	0
$341$	−435002.	−0.202584
$342$	850385.i	0.393143i
$343$	865866.i	0.397388i
$344$	499218.	0.227454
$345$	0	0
$346$	570995.	0.256414
$347$	4.24482e6i	1.89250i	0.323435	+	0.946250i	$0.395162\pi$
$347$	4.24482e6i	1.89250i	−0.323435	+	0.946250i	$0.604838\pi$
$348$	− 537794.i	− 0.238050i
$349$	3.06057e6	1.34505	0.672525	−	0.740074i	$-0.265210\pi$
$349$	3.06057e6	1.34505	0.672525	+	0.740074i	$0.265210\pi$
$350$	0	0
$351$	−689476.	−0.298711
$352$	− 536647.i	− 0.230851i
$353$	4.53104e6i	1.93536i	0.252186	+	0.967679i	$0.418851\pi$
$353$	4.53104e6i	1.93536i	−0.252186	+	0.967679i	$0.581149\pi$
$354$	−2.45579e6	−1.04156
$355$	0	0
$356$	−1.72569e6	−0.721668
$357$	− 967102.i	− 0.401607i
$358$	− 2.51804e6i	− 1.03838i
$359$	2.68308e6	1.09875	0.549373	−	0.835577i	$-0.314867\pi$
$359$	2.68308e6	1.09875	0.549373	+	0.835577i	$0.314867\pi$
$360$	0	0
$361$	−1.55532e6	−0.628135
$362$	− 2.34472e6i	− 0.940413i
$363$	1.82573e6i	0.727226i
$364$	1.18267e6	0.467853
$365$	0	0
$366$	2.32001e6	0.905288
$367$	659030.i	0.255411i	0.991812	+	0.127706i	$0.0407613\pi$
$367$	659030.i	0.255411i	−0.991812	+	0.127706i	$0.959239\pi$
$368$	− 442301.i	− 0.170254i
$369$	−699079.	−0.267276
$370$	0	0
$371$	−7.88034e6	−2.97242
$372$	− 2.81309e6i	− 1.05397i
$373$	1.73787e6i	0.646764i	0.946268	+	0.323382i	$0.104820\pi$
$373$	1.73787e6i	0.646764i	−0.946268	+	0.323382i	$0.895180\pi$
$374$	−226580.	−0.0837612
$375$	0	0
$376$	−201062.	−0.0733432
$377$	217701.i	0.0788872i
$378$	− 6.55659e6i	− 2.36020i
$379$	4.61878e6	1.65169	0.825847	−	0.563894i	$-0.190697\pi$
$379$	4.61878e6	1.65169	0.825847	+	0.563894i	$0.190697\pi$
$380$	0	0
$381$	−869609.	−0.306910
$382$	− 4.42219e6i	− 1.55053i
$383$	− 656507.i	− 0.228688i	−0.993441	−	0.114344i	$-0.963523\pi$
$383$	− 656507.i	− 0.228688i	0.993441	−	0.114344i	$-0.0364765\pi$
$384$	755158.	0.261342
$385$	0	0
$386$	−7.67119e6	−2.62056
$387$	1.68242e6i	0.571026i
$388$	− 2.63178e6i	− 0.887504i
$389$	1.50525e6	0.504352	0.252176	−	0.967681i	$-0.418854\pi$
$389$	1.50525e6	0.504352	0.252176	+	0.967681i	$0.418854\pi$
$390$	0	0
$391$	−212906.	−0.0704282
$392$	678132.i	0.222894i
$393$	− 320615.i	− 0.104714i
$394$	−1.73255e6	−0.562270
$395$	0	0
$396$	−249126.	−0.0798327
$397$	389426.i	0.124008i	0.998076	+	0.0620038i	$0.0197491\pi$
$397$	389426.i	0.124008i	−0.998076	+	0.0620038i	$0.980251\pi$
$398$	− 7.02987e6i	− 2.22454i
$399$	−2.17836e6	−0.685009
$400$	0	0
$401$	3.45560e6	1.07316	0.536578	−	0.843851i	$-0.319717\pi$
$401$	3.45560e6	1.07316	0.536578	+	0.843851i	$0.319717\pi$
$402$	− 3.75676e6i	− 1.15944i
$403$	1.13875e6i	0.349274i
$404$	−806125.	−0.245725
$405$	0	0
$406$	−2.07023e6	−0.623309
$407$	− 401435.i	− 0.120124i
$408$	− 158242.i	− 0.0470622i
$409$	5.22273e6	1.54379	0.771896	−	0.635748i	$-0.219308\pi$
$409$	5.22273e6	1.54379	0.771896	+	0.635748i	$0.219308\pi$
$410$	0	0
$411$	3.09582e6	0.904007
$412$	− 4.17460e6i	− 1.21164i
$413$	4.99658e6i	1.44144i
$414$	−442902.	−0.127001
$415$	0	0
$416$	−1.40484e6	−0.398009
$417$	− 3.06997e6i	− 0.864558i
$418$	510362.i	0.142869i
$419$	2.19501e6	0.610805	0.305402	−	0.952223i	$-0.401209\pi$
$419$	2.19501e6	0.610805	0.305402	+	0.952223i	$0.401209\pi$
$420$	0	0
$421$	−4.06919e6	−1.11893	−0.559465	−	0.828854i	$-0.688993\pi$
$421$	−4.06919e6	−1.11893	−0.559465	+	0.828854i	$0.688993\pi$
$422$	− 7.52619e6i	− 2.05728i
$423$	− 677599.i	− 0.184129i
$424$	−1.28942e6	−0.348322
$425$	0	0
$426$	−3.13336e6	−0.836540
$427$	− 4.72031e6i	− 1.25285i
$428$	1.19256e6i	0.314681i
$429$	−126968.	−0.0333083
$430$	0	0
$431$	−3.69314e6	−0.957641	−0.478821	−	0.877913i	$-0.658936\pi$
$431$	−3.69314e6	−0.957641	−0.478821	+	0.877913i	$0.658936\pi$
$432$	3.61062e6i	0.930836i
$433$	7.32688e6i	1.87802i	0.343896	+	0.939008i	$0.388253\pi$
$433$	7.32688e6i	1.87802i	−0.343896	+	0.939008i	$0.611747\pi$
$434$	−1.08290e7	−2.75971
$435$	0	0
$436$	−3.13234e6	−0.789139
$437$	479562.i	0.120127i
$438$	− 1.39088e6i	− 0.346420i
$439$	4.33121e6	1.07263	0.536313	−	0.844019i	$-0.319817\pi$
$439$	4.33121e6	1.07263	0.536313	+	0.844019i	$0.319817\pi$
$440$	0	0
$441$	−2.28538e6	−0.559579
$442$	593143.i	0.144412i
$443$	− 6.00225e6i	− 1.45313i	−0.687097	−	0.726566i	$-0.741115\pi$
$443$	− 6.00225e6i	− 1.45313i	0.687097	−	0.726566i	$-0.258885\pi$
$444$	2.59602e6	0.624958
$445$	0	0
$446$	3.39497e6	0.808162
$447$	55240.2i	0.0130763i
$448$	− 7.83484e6i	− 1.84432i
$449$	−1.63461e6	−0.382647	−0.191324	−	0.981527i	$-0.561278\pi$
$449$	−1.63461e6	−0.382647	−0.191324	+	0.981527i	$0.561278\pi$
$450$	0	0
$451$	−419556.	−0.0971288
$452$	7.58687e6i	1.74669i
$453$	1.10151e6i	0.252199i
$454$	7.11899e6	1.62098
$455$	0	0
$456$	−356434.	−0.0802725
$457$	− 8.14010e6i	− 1.82322i	−0.411056	−	0.911610i	$-0.634840\pi$
$457$	− 8.14010e6i	− 1.82322i	0.411056	−	0.911610i	$-0.365160\pi$
$458$	1.94435e6i	0.433123i
$459$	1.73801e6	0.385053
$460$	0	0
$461$	1.30948e6	0.286977	0.143489	−	0.989652i	$-0.454168\pi$
$461$	1.30948e6	0.286977	0.143489	+	0.989652i	$0.454168\pi$
$462$	− 1.20741e6i	− 0.263178i
$463$	1.54820e6i	0.335641i	0.985818	+	0.167820i	$0.0536728\pi$
$463$	1.54820e6i	0.335641i	−0.985818	+	0.167820i	$0.946327\pi$
$464$	1.14005e6	0.245826
$465$	0	0
$466$	1.02236e7	2.18093
$467$	6.32397e6i	1.34183i	0.741534	+	0.670915i	$0.234098\pi$
$467$	6.32397e6i	1.34183i	−0.741534	+	0.670915i	$0.765902\pi$
$468$	652163.i	0.137639i
$469$	−7.64353e6	−1.60458
$470$	0	0
$471$	−3.08262e6	−0.640276
$472$	817566.i	0.168915i
$473$	1.00971e6i	0.207512i
$474$	−7.57787e6	−1.54918
$475$	0	0
$476$	−2.98123e6	−0.603085
$477$	− 4.34549e6i	− 0.874465i
$478$	7.15486e6i	1.43229i
$479$	803117.	0.159934	0.0799669	−	0.996798i	$-0.474519\pi$
$479$	803117.	0.159934	0.0799669	+	0.996798i	$0.474519\pi$
$480$	0	0
$481$	−1.05088e6	−0.207104
$482$	− 6.89377e6i	− 1.35157i
$483$	− 1.13454e6i	− 0.221285i
$484$	5.62807e6	1.09206
$485$	0	0
$486$	6.12166e6	1.17565
$487$	− 6.13178e6i	− 1.17156i	−0.810471	−	0.585779i	$-0.800789\pi$
$487$	− 6.13178e6i	− 1.17156i	0.810471	−	0.585779i	$-0.199211\pi$
$488$	− 772362.i	− 0.146815i
$489$	−1.41928e6	−0.268407
$490$	0	0
$491$	−236451.	−0.0442626	−0.0221313	−	0.999755i	$-0.507045\pi$
$491$	−236451.	−0.0442626	−0.0221313	+	0.999755i	$0.507045\pi$
$492$	− 2.71320e6i	− 0.505324i
$493$	− 548774.i	− 0.101689i
$494$	1.33603e6	0.246319
$495$	0	0
$496$	5.96337e6	1.08840
$497$	6.37517e6i	1.15771i
$498$	− 9.81445e6i	− 1.77334i
$499$	1.81339e6	0.326018	0.163009	−	0.986625i	$-0.447880\pi$
$499$	1.81339e6	0.326018	0.163009	+	0.986625i	$0.447880\pi$
$500$	0	0
$501$	−5.41822e6	−0.964412
$502$	− 1.60608e7i	− 2.84452i
$503$	4.69515e6i	0.827427i	0.910407	+	0.413713i	$0.135768\pi$
$503$	4.69515e6i	0.827427i	−0.910407	+	0.413713i	$0.864232\pi$
$504$	−669765.	−0.117448
$505$	0	0
$506$	−265809.	−0.0461524
$507$	332378.i	0.0574266i
$508$	2.68070e6i	0.460881i
$509$	−7.10737e6	−1.21595	−0.607973	−	0.793958i	$-0.708017\pi$
$509$	−7.10737e6	−1.21595	−0.607973	+	0.793958i	$0.708017\pi$
$510$	0	0
$511$	−2.82989e6	−0.479421
$512$	8.26075e6i	1.39266i
$513$	− 3.91480e6i	− 0.656774i
$514$	2.95338e6	0.493073
$515$	0	0
$516$	−6.52965e6	−1.07961
$517$	− 406664.i	− 0.0669128i
$518$	− 9.99335e6i	− 1.63639i
$519$	−806564.	−0.131438
$520$	0	0
$521$	9.92332e6	1.60163	0.800816	−	0.598911i	$-0.204400\pi$
$521$	9.92332e6	1.60163	0.800816	+	0.598911i	$0.204400\pi$
$522$	− 1.14160e6i	− 0.183373i
$523$	1.66096e6i	0.265525i	0.991148	+	0.132763i	$0.0423848\pi$
$523$	1.66096e6i	0.265525i	−0.991148	+	0.132763i	$0.957615\pi$
$524$	−988343.	−0.157246
$525$	0	0
$526$	1.66767e7	2.62812
$527$	− 2.87053e6i	− 0.450231i
$528$	664904.i	0.103794i
$529$	6.18658e6	0.961194
$530$	0	0
$531$	−2.75528e6	−0.424062
$532$	6.71510e6i	1.02866i
$533$	1.09831e6i	0.167459i
$534$	4.61202e6	0.699904
$535$	0	0
$536$	−1.25067e6	−0.188032
$537$	3.55689e6i	0.532273i
$538$	1.42038e7i	2.11567i
$539$	−1.37158e6	−0.203352
$540$	0	0
$541$	8.08498e6	1.18764	0.593822	−	0.804597i	$-0.297619\pi$
$541$	8.08498e6	1.18764	0.593822	+	0.804597i	$0.297619\pi$
$542$	− 5.09283e6i	− 0.744665i
$543$	3.31205e6i	0.482056i
$544$	3.54127e6	0.513053
$545$	0	0
$546$	−3.16076e6	−0.453743
$547$	− 2.35716e6i	− 0.336838i	−0.985715	−	0.168419i	$-0.946134\pi$
$547$	− 2.35716e6i	− 0.336838i	0.985715	−	0.168419i	$-0.0538662\pi$
$548$	− 9.54333e6i	− 1.35753i
$549$	2.60294e6	0.368581
$550$	0	0
$551$	−1.23609e6	−0.173449
$552$	− 185640.i	− 0.0259312i
$553$	1.54180e7i	2.14395i
$554$	−4.41442e6	−0.611082
$555$	0	0
$556$	−9.46363e6	−1.29829
$557$	3.10870e6i	0.424562i	0.977209	+	0.212281i	$0.0680891\pi$
$557$	3.10870e6i	0.424562i	−0.977209	+	0.212281i	$0.931911\pi$
$558$	− 5.97146e6i	− 0.811886i
$559$	2.64322e6	0.357770
$560$	0	0
$561$	320058.	0.0429360
$562$	9.84372e6i	1.31468i
$563$	174928.i	0.0232589i	0.999932	+	0.0116295i	$0.00370186\pi$
$563$	174928.i	0.0232589i	−0.999932	+	0.0116295i	$0.996298\pi$
$564$	2.62984e6	0.348122
$565$	0	0
$566$	−2.14190e7	−2.81034
$567$	4.16257e6i	0.543756i
$568$	1.04314e6i	0.135666i
$569$	7.95974e6	1.03067	0.515333	−	0.856990i	$-0.327668\pi$
$569$	7.95974e6	1.03067	0.515333	+	0.856990i	$0.327668\pi$
$570$	0	0
$571$	3.22926e6	0.414488	0.207244	−	0.978289i	$-0.433551\pi$
$571$	3.22926e6	0.414488	0.207244	+	0.978289i	$0.433551\pi$
$572$	391399.i	0.0500183i
$573$	6.24661e6i	0.794800i
$574$	−1.04444e7	−1.32314
$575$	0	0
$576$	4.32040e6	0.542585
$577$	− 1.13083e7i	− 1.41403i	−0.707200	−	0.707014i	$-0.750042\pi$
$577$	− 1.13083e7i	− 1.41403i	0.707200	−	0.707014i	$-0.249958\pi$
$578$	1.02024e7i	1.27024i
$579$	1.08360e7	1.34330
$580$	0	0
$581$	−1.99686e7	−2.45418
$582$	7.03361e6i	0.860738i
$583$	− 2.60796e6i	− 0.317783i
$584$	−463041.	−0.0561807
$585$	0	0
$586$	1.75462e7	2.11076
$587$	9.47132e6i	1.13453i	0.823536	+	0.567264i	$0.191998\pi$
$587$	9.47132e6i	1.13453i	−0.823536	+	0.567264i	$0.808002\pi$
$588$	− 8.86980e6i	− 1.05796i
$589$	−6.46574e6	−0.767945
$590$	0	0
$591$	2.44733e6	0.288220
$592$	5.50320e6i	0.645373i
$593$	5.00001e6i	0.583894i	0.956434	+	0.291947i	$0.0943031\pi$
$593$	5.00001e6i	0.583894i	−0.956434	+	0.291947i	$0.905697\pi$
$594$	2.16988e6	0.252330
$595$	0	0
$596$	170286.	0.0196365
$597$	9.93011e6i	1.14030i
$598$	695837.i	0.0795710i
$599$	−9.49229e6	−1.08095	−0.540473	−	0.841361i	$-0.681755\pi$
$599$	−9.49229e6	−1.08095	−0.540473	+	0.841361i	$0.681755\pi$
$600$	0	0
$601$	7.36100e6	0.831286	0.415643	−	0.909528i	$-0.363557\pi$
$601$	7.36100e6	0.831286	0.415643	+	0.909528i	$0.363557\pi$
$602$	2.51358e7i	2.82684i
$603$	− 4.21490e6i	− 0.472057i
$604$	3.39557e6	0.378722
$605$	0	0
$606$	2.15442e6	0.238314
$607$	1.14445e7i	1.26074i	0.776295	+	0.630370i	$0.217097\pi$
$607$	1.14445e7i	1.26074i	−0.776295	+	0.630370i	$0.782903\pi$
$608$	− 7.97657e6i	− 0.875099i
$609$	2.92432e6	0.319508
$610$	0	0
$611$	−1.06457e6	−0.115364
$612$	− 1.64395e6i	− 0.177423i
$613$	− 1.01622e7i	− 1.09229i	−0.837691	−	0.546144i	$-0.816095\pi$
$613$	− 1.01622e7i	− 1.09229i	0.837691	−	0.546144i	$-0.183905\pi$
$614$	−1.25339e7	−1.34173
$615$	0	0
$616$	−401962.	−0.0426809
$617$	− 1.19269e7i	− 1.26129i	−0.776072	−	0.630644i	$-0.782791\pi$
$617$	− 1.19269e7i	− 1.26129i	0.776072	−	0.630644i	$-0.217209\pi$
$618$	1.11569e7i	1.17509i
$619$	−6.13819e6	−0.643893	−0.321946	−	0.946758i	$-0.604337\pi$
$619$	−6.13819e6	−0.643893	−0.321946	+	0.946758i	$0.604337\pi$
$620$	0	0
$621$	2.03892e6	0.212164
$622$	1.47352e7i	1.52714i
$623$	− 9.38366e6i	− 0.968617i
$624$	1.74059e6	0.178951
$625$	0	0
$626$	3.71856e6	0.379262
$627$	− 720917.i	− 0.0732347i
$628$	9.50262e6i	0.961489i
$629$	2.64902e6	0.266968
$630$	0	0
$631$	9.80829e6	0.980663	0.490332	−	0.871536i	$-0.336876\pi$
$631$	9.80829e6	0.980663	0.490332	+	0.871536i	$0.336876\pi$
$632$	2.52277e6i	0.251238i
$633$	1.06312e7i	1.05456i
$634$	−1.15055e7	−1.13679
$635$	0	0
$636$	1.68653e7	1.65330
$637$	3.59053e6i	0.350598i
$638$	− 685134.i	− 0.0666383i
$639$	−3.51548e6	−0.340591
$640$	0	0
$641$	1.57763e6	0.151657	0.0758283	−	0.997121i	$-0.475840\pi$
$641$	1.57763e6	0.151657	0.0758283	+	0.997121i	$0.475840\pi$
$642$	− 3.18719e6i	− 0.305190i
$643$	− 1.87484e7i	− 1.78829i	−0.447779	−	0.894144i	$-0.647785\pi$
$643$	− 1.87484e7i	− 1.78829i	0.447779	−	0.894144i	$-0.352215\pi$
$644$	−3.49739e6	−0.332299
$645$	0	0
$646$	−3.36782e6	−0.317518
$647$	1.32643e7i	1.24573i	0.782331	+	0.622863i	$0.214031\pi$
$647$	1.32643e7i	1.24573i	−0.782331	+	0.622863i	$0.785969\pi$
$648$	681101.i	0.0637198i
$649$	−1.65360e6	−0.154105
$650$	0	0
$651$	1.52966e7	1.41463
$652$	4.37513e6i	0.403062i
$653$	− 7.19177e6i	− 0.660013i	−0.943979	−	0.330007i	$-0.892949\pi$
$653$	− 7.19177e6i	− 0.660013i	0.943979	−	0.330007i	$-0.107051\pi$
$654$	8.37141e6	0.765339
$655$	0	0
$656$	5.75161e6	0.521831
$657$	− 1.56050e6i	− 0.141042i
$658$	− 1.01235e7i	− 0.911522i
$659$	2.13493e6	0.191501	0.0957503	−	0.995405i	$-0.469475\pi$
$659$	2.13493e6	0.191501	0.0957503	+	0.995405i	$0.469475\pi$
$660$	0	0
$661$	−3.54104e6	−0.315230	−0.157615	−	0.987501i	$-0.550380\pi$
$661$	−3.54104e6	−0.315230	−0.157615	+	0.987501i	$0.550380\pi$
$662$	1.13388e7i	1.00559i
$663$	− 837850.i	− 0.0740256i
$664$	−3.26736e6	−0.287592
$665$	0	0
$666$	5.51067e6	0.481414
$667$	− 643786.i	− 0.0560308i
$668$	1.67025e7i	1.44824i
$669$	−4.79559e6	−0.414264
$670$	0	0
$671$	1.56217e6	0.133943
$672$	1.88709e7i	1.61201i
$673$	1.59930e7i	1.36110i	0.732700	+	0.680551i	$0.238260\pi$
$673$	1.59930e7i	1.36110i	−0.732700	+	0.680551i	$0.761740\pi$
$674$	−1.59325e7	−1.35093
$675$	0	0
$676$	1.02461e6	0.0862362
$677$	− 6.51782e6i	− 0.546551i	−0.961936	−	0.273275i	$-0.911893\pi$
$677$	− 6.51782e6i	− 0.546551i	0.961936	−	0.273275i	$-0.0881070\pi$
$678$	− 2.02764e7i	− 1.69401i
$679$	1.43107e7	1.19120
$680$	0	0
$681$	−1.00560e7	−0.830916
$682$	− 3.58380e6i	− 0.295041i
$683$	2.30581e7i	1.89135i	0.325118	+	0.945673i	$0.394596\pi$
$683$	2.30581e7i	1.89135i	−0.325118	+	0.945673i	$0.605404\pi$
$684$	−3.70293e6	−0.302626
$685$	0	0
$686$	−7.13351e6	−0.578752
$687$	− 2.74651e6i	− 0.222019i
$688$	− 1.38419e7i	− 1.11487i
$689$	−6.82714e6	−0.547887
$690$	0	0
$691$	1.77699e7	1.41576	0.707880	−	0.706333i	$-0.249652\pi$
$691$	1.77699e7	1.41576	0.707880	+	0.706333i	$0.249652\pi$
$692$	2.48635e6i	0.197377i
$693$	− 1.35466e6i	− 0.107151i
$694$	−3.49713e7	−2.75622
$695$	0	0
$696$	478493.	0.0374414
$697$	− 2.76860e6i	− 0.215863i
$698$	2.52148e7i	1.95892i
$699$	−1.44415e7	−1.11794
$700$	0	0
$701$	−7.07397e6	−0.543711	−0.271856	−	0.962338i	$-0.587637\pi$
$701$	−7.07397e6	−0.543711	−0.271856	+	0.962338i	$0.587637\pi$
$702$	− 5.68031e6i	− 0.435040i
$703$	− 5.96681e6i	− 0.455359i
$704$	2.59291e6	0.197177
$705$	0	0
$706$	−3.73294e7	−2.81863
$707$	− 4.38341e6i	− 0.329810i
$708$	− 1.06936e7i	− 0.801750i
$709$	8.17421e6	0.610703	0.305352	−	0.952240i	$-0.401226\pi$
$709$	8.17421e6	0.610703	0.305352	+	0.952240i	$0.401226\pi$
$710$	0	0
$711$	−8.50201e6	−0.630736
$712$	− 1.53540e6i	− 0.113507i
$713$	− 3.36752e6i	− 0.248077i
$714$	7.96755e6	0.584897
$715$	0	0
$716$	1.09646e7	0.799303
$717$	− 1.01067e7i	− 0.734193i
$718$	2.21048e7i	1.60020i
$719$	5.47393e6	0.394891	0.197446	−	0.980314i	$-0.436735\pi$
$719$	5.47393e6	0.394891	0.197446	+	0.980314i	$0.436735\pi$
$720$	0	0
$721$	2.26999e7	1.62625
$722$	− 1.28137e7i	− 0.914810i
$723$	9.73786e6i	0.692816i
$724$	1.02099e7	0.723893
$725$	0	0
$726$	−1.50414e7	−1.05912
$727$	− 1.55536e7i	− 1.09143i	−0.837972	−	0.545714i	$-0.816259\pi$
$727$	− 1.55536e7i	− 1.09143i	0.837972	−	0.545714i	$-0.183741\pi$
$728$	1.05226e6i	0.0735858i
$729$	−1.38325e7	−0.964012
$730$	0	0
$731$	−6.66296e6	−0.461183
$732$	1.01023e7i	0.696855i
$733$	− 1.49050e7i	− 1.02464i	−0.858795	−	0.512320i	$-0.828786\pi$
$733$	− 1.49050e7i	− 1.02464i	0.858795	−	0.512320i	$-0.171214\pi$
$734$	−5.42948e6	−0.371979
$735$	0	0
$736$	4.15439e6	0.282692
$737$	− 2.52959e6i	− 0.171547i
$738$	− 5.75942e6i	− 0.389258i
$739$	−1.31574e6	−0.0886256	−0.0443128	−	0.999018i	$-0.514110\pi$
$739$	−1.31574e6	−0.0886256	−0.0443128	+	0.999018i	$0.514110\pi$
$740$	0	0
$741$	−1.88722e6	−0.126263
$742$	− 6.49229e7i	− 4.32900i
$743$	− 1.76450e7i	− 1.17260i	−0.810094	−	0.586300i	$-0.800584\pi$
$743$	− 1.76450e7i	− 1.17260i	0.810094	−	0.586300i	$-0.199416\pi$
$744$	2.50290e6	0.165772
$745$	0	0
$746$	−1.43176e7	−0.941941
$747$	− 1.10113e7i	− 0.722003i
$748$	− 986626.i	− 0.0644761i
$749$	−6.48469e6	−0.422362
$750$	0	0
$751$	−1.08240e7	−0.700308	−0.350154	−	0.936692i	$-0.613871\pi$
$751$	−1.08240e7	−0.700308	−0.350154	+	0.936692i	$0.613871\pi$
$752$	5.57489e6i	0.359494i
$753$	2.26869e7i	1.45810i
$754$	−1.79355e6	−0.114891
$755$	0	0
$756$	2.85502e7	1.81679
$757$	− 2.74661e7i	− 1.74204i	−0.491251	−	0.871018i	$-0.663460\pi$
$757$	− 2.74661e7i	− 1.74204i	0.491251	−	0.871018i	$-0.336540\pi$
$758$	3.80522e7i	2.40551i
$759$	375472.	0.0236577
$760$	0	0
$761$	3.08096e6	0.192852	0.0964261	−	0.995340i	$-0.469259\pi$
$761$	3.08096e6	0.192852	0.0964261	+	0.995340i	$0.469259\pi$
$762$	− 7.16435e6i	− 0.446981i
$763$	− 1.70325e7i	− 1.05918i
$764$	1.92561e7	1.19353
$765$	0	0
$766$	5.40869e6	0.333058
$767$	4.32879e6i	0.265692i
$768$	− 8.73565e6i	− 0.534432i
$769$	−3.97374e6	−0.242317	−0.121159	−	0.992633i	$-0.538661\pi$
$769$	−3.97374e6	−0.242317	−0.121159	+	0.992633i	$0.538661\pi$
$770$	0	0
$771$	−4.17182e6	−0.252749
$772$	− 3.34036e7i	− 2.01721i
$773$	− 1.29080e6i	− 0.0776978i	−0.999245	−	0.0388489i	$-0.987631\pi$
$773$	− 1.29080e6i	− 0.0776978i	0.999245	−	0.0388489i	$-0.0123691\pi$
$774$	−1.38607e7	−0.831637
$775$	0	0
$776$	2.34158e6	0.139590
$777$	1.41162e7i	0.838813i
$778$	1.24011e7i	0.734533i
$779$	−6.23615e6	−0.368191
$780$	0	0
$781$	−2.10983e6	−0.123771
$782$	− 1.75405e6i	− 0.102571i
$783$	5.25540e6i	0.306339i
$784$	1.88027e7	1.09252
$785$	0	0
$786$	2.64142e6	0.152504
$787$	2.52309e7i	1.45210i	0.687642	+	0.726050i	$0.258646\pi$
$787$	2.52309e7i	1.45210i	−0.687642	+	0.726050i	$0.741354\pi$
$788$	− 7.54425e6i	− 0.432813i
$789$	−2.35568e7	−1.34717
$790$	0	0
$791$	−4.12546e7	−2.34440
$792$	− 221656.i	− 0.0125564i
$793$	− 4.08944e6i	− 0.230931i
$794$	−3.20832e6	−0.180603
$795$	0	0
$796$	3.06110e7	1.71236
$797$	2.89490e7i	1.61431i	0.590339	+	0.807155i	$0.298994\pi$
$797$	2.89490e7i	1.61431i	−0.590339	+	0.807155i	$0.701006\pi$
$798$	− 1.79466e7i	− 0.997641i
$799$	2.68353e6	0.148710
$800$	0	0
$801$	5.17447e6	0.284960
$802$	2.84693e7i	1.56293i
$803$	− 936540.i	− 0.0512551i
$804$	1.63585e7	0.892490
$805$	0	0
$806$	−9.38169e6	−0.508679
$807$	− 2.00637e7i	− 1.08449i
$808$	− 717236.i	− 0.0386486i
$809$	1.89823e7	1.01971	0.509855	−	0.860260i	$-0.329699\pi$
$809$	1.89823e7	1.01971	0.509855	+	0.860260i	$0.329699\pi$
$810$	0	0
$811$	−2.83661e7	−1.51442	−0.757212	−	0.653169i	$-0.773439\pi$
$811$	−2.83661e7	−1.51442	−0.757212	+	0.653169i	$0.773439\pi$
$812$	− 9.01466e6i	− 0.479799i
$813$	7.19392e6i	0.381715i
$814$	3.30725e6	0.174947
$815$	0	0
$816$	−4.38762e6	−0.230677
$817$	1.50080e7i	0.786627i
$818$	4.30279e7i	2.24836i
$819$	−3.54622e6	−0.184738
$820$	0	0
$821$	3.27149e7	1.69390	0.846951	−	0.531671i	$-0.178436\pi$
$821$	3.27149e7	1.69390	0.846951	+	0.531671i	$0.178436\pi$
$822$	2.55052e7i	1.31659i
$823$	1.40256e7i	0.721809i	0.932603	+	0.360904i	$0.117532\pi$
$823$	1.40256e7i	0.721809i	−0.932603	+	0.360904i	$0.882468\pi$
$824$	3.71428e6	0.190571
$825$	0	0
$826$	−4.11647e7	−2.09930
$827$	− 2.33094e7i	− 1.18513i	−0.805522	−	0.592566i	$-0.798115\pi$
$827$	− 2.33094e7i	− 1.18513i	0.805522	−	0.592566i	$-0.201885\pi$
$828$	− 1.92858e6i	− 0.0977602i
$829$	−2.47785e7	−1.25224	−0.626122	−	0.779725i	$-0.715359\pi$
$829$	−2.47785e7	−1.25224	−0.626122	+	0.779725i	$0.715359\pi$
$830$	0	0
$831$	6.23563e6	0.313241
$832$	− 6.78772e6i	− 0.339951i
$833$	− 9.05089e6i	− 0.451938i
$834$	2.52922e7	1.25913
$835$	0	0
$836$	−2.22233e6	−0.109975
$837$	2.74900e7i	1.35632i
$838$	1.80838e7i	0.889569i
$839$	2.58285e7	1.26676	0.633380	−	0.773841i	$-0.281667\pi$
$839$	2.58285e7	1.26676	0.633380	+	0.773841i	$0.281667\pi$
$840$	0	0
$841$	−1.88518e7	−0.919098
$842$	− 3.35244e7i	− 1.62960i
$843$	− 1.39048e7i	− 0.673903i
$844$	3.27722e7	1.58361
$845$	0	0
$846$	5.58245e6	0.268163
$847$	3.06034e7i	1.46575i
$848$	3.57521e7i	1.70731i
$849$	3.02556e7	1.44058
$850$	0	0
$851$	3.10766e6	0.147099
$852$	− 1.36440e7i	− 0.643935i
$853$	− 1.49617e7i	− 0.704060i	−0.935989	−	0.352030i	$-0.885492\pi$
$853$	− 1.49617e7i	− 0.704060i	0.935989	−	0.352030i	$-0.114508\pi$
$854$	3.88887e7	1.82465
$855$	0	0
$856$	−1.06106e6	−0.0494943
$857$	7.36028e6i	0.342328i	0.985243	+	0.171164i	$0.0547528\pi$
$857$	7.36028e6i	0.342328i	−0.985243	+	0.171164i	$0.945247\pi$
$858$	− 1.04604e6i	− 0.0485099i
$859$	−2.71276e7	−1.25438	−0.627190	−	0.778866i	$-0.715795\pi$
$859$	−2.71276e7	−1.25438	−0.627190	+	0.778866i	$0.715795\pi$
$860$	0	0
$861$	1.47534e7	0.678241
$862$	− 3.04263e7i	− 1.39470i
$863$	− 3.14458e7i	− 1.43726i	−0.695393	−	0.718630i	$-0.744770\pi$
$863$	− 3.14458e7i	− 1.43726i	0.695393	−	0.718630i	$-0.255230\pi$
$864$	−3.39135e7	−1.54557
$865$	0	0
$866$	−6.03631e7	−2.73512
$867$	− 1.44115e7i	− 0.651123i
$868$	− 4.71539e7i	− 2.12431i
$869$	−5.10252e6	−0.229211
$870$	0	0
$871$	−6.62198e6	−0.295762
$872$	− 2.78695e6i	− 0.124119i
$873$	7.89138e6i	0.350443i
$874$	−3.95091e6	−0.174952
$875$	0	0
$876$	6.05646e6	0.266661
$877$	3.94061e7i	1.73007i	0.501708	+	0.865037i	$0.332705\pi$
$877$	3.94061e7i	1.73007i	−0.501708	+	0.865037i	$0.667295\pi$
$878$	3.56831e7i	1.56216i
$879$	−2.47850e7	−1.08197
$880$	0	0
$881$	−2.67060e7	−1.15923	−0.579614	−	0.814891i	$-0.696797\pi$
$881$	−2.67060e7	−1.15923	−0.579614	+	0.814891i	$0.696797\pi$
$882$	− 1.88283e7i	− 0.814965i
$883$	− 4.33871e7i	− 1.87266i	−0.351120	−	0.936330i	$-0.614199\pi$
$883$	− 4.33871e7i	− 1.87266i	0.351120	−	0.936330i	$-0.385801\pi$
$884$	−2.58279e6	−0.111163
$885$	0	0
$886$	4.94501e7	2.11633
$887$	− 3.25024e7i	− 1.38710i	−0.720410	−	0.693549i	$-0.756046\pi$
$887$	− 3.25024e7i	− 1.38710i	0.720410	−	0.693549i	$-0.243954\pi$
$888$	2.30976e6i	0.0982959i
$889$	−1.45766e7	−0.618590
$890$	0	0
$891$	−1.37758e6	−0.0581332
$892$	1.47831e7i	0.622091i
$893$	− 6.04453e6i	− 0.253650i
$894$	−455101.	−0.0190443
$895$	0	0
$896$	1.26582e7	0.526746
$897$	− 982911.i	− 0.0407881i
$898$	− 1.34669e7i	− 0.557283i
$899$	8.67991e6	0.358192
$900$	0	0
$901$	1.72097e7	0.706253
$902$	− 3.45654e6i	− 0.141457i
$903$	− 3.55058e7i	− 1.44904i
$904$	−6.75029e6	−0.274727
$905$	0	0
$906$	−9.07489e6	−0.367300
$907$	− 3.43395e7i	− 1.38604i	−0.720918	−	0.693020i	$-0.756280\pi$
$907$	− 3.43395e7i	− 1.38604i	0.720918	−	0.693020i	$-0.243720\pi$
$908$	3.09991e7i	1.24777i
$909$	2.41716e6	0.0970277
$910$	0	0
$911$	3.18788e7	1.27264	0.636320	−	0.771425i	$-0.280456\pi$
$911$	3.18788e7	1.27264	0.636320	+	0.771425i	$0.280456\pi$
$912$	9.88293e6i	0.393458i
$913$	− 6.60851e6i	− 0.262378i
$914$	6.70629e7	2.65532
$915$	0	0
$916$	−8.46652e6	−0.333401
$917$	− 5.37425e6i	− 0.211054i
$918$	1.43188e7i	0.560788i
$919$	2.12110e7	0.828460	0.414230	−	0.910172i	$-0.364051\pi$
$919$	2.12110e7	0.828460	0.414230	+	0.910172i	$0.364051\pi$
$920$	0	0
$921$	1.77049e7	0.687770
$922$	1.07883e7i	0.417951i
$923$	5.52313e6i	0.213393i
$924$	5.25757e6	0.202584
$925$	0	0
$926$	−1.27550e7	−0.488823
$927$	1.25175e7i	0.478431i
$928$	1.07081e7i	0.408172i
$929$	−1.66040e6	−0.0631210	−0.0315605	−	0.999502i	$-0.510048\pi$
$929$	−1.66040e6	−0.0631210	−0.0315605	+	0.999502i	$0.510048\pi$
$930$	0	0
$931$	−2.03867e7	−0.770857
$932$	4.45181e7i	1.67879i
$933$	− 2.08143e7i	− 0.782813i
$934$	−5.21005e7	−1.95423
$935$	0	0
$936$	−580251.	−0.0216484
$937$	− 6.93039e6i	− 0.257875i	−0.991653	−	0.128937i	$-0.958843\pi$
$937$	− 6.93039e6i	− 0.257875i	0.991653	−	0.128937i	$-0.0411566\pi$
$938$	− 6.29719e7i	− 2.33690i
$939$	−5.25269e6	−0.194410
$940$	0	0
$941$	−1.29119e7	−0.475352	−0.237676	−	0.971344i	$-0.576386\pi$
$941$	−1.29119e7	−0.475352	−0.237676	+	0.971344i	$0.576386\pi$
$942$	− 2.53964e7i	− 0.932492i
$943$	− 3.24794e6i	− 0.118940i
$944$	2.26688e7	0.827941
$945$	0	0
$946$	−8.31858e6	−0.302219
$947$	− 2.15320e7i	− 0.780207i	−0.920771	−	0.390104i	$-0.872439\pi$
$947$	− 2.15320e7i	− 0.780207i	0.920771	−	0.390104i	$-0.127561\pi$
$948$	− 3.29973e7i	− 1.19250i
$949$	−2.45168e6	−0.0883686
$950$	0	0
$951$	1.62521e7	0.582719
$952$	− 2.65250e6i	− 0.0948557i
$953$	− 3.37555e7i	− 1.20396i	−0.798511	−	0.601980i	$-0.794379\pi$
$953$	− 3.37555e7i	− 1.20396i	0.798511	−	0.601980i	$-0.205621\pi$
$954$	3.58007e7	1.27356
$955$	0	0
$956$	−3.11553e7	−1.10252
$957$	967793.i	0.0341588i
$958$	6.61655e6i	0.232926i
$959$	5.18931e7	1.82206
$960$	0	0
$961$	1.67737e7	0.585897
$962$	− 8.65774e6i	− 0.301625i
$963$	− 3.57588e6i	− 0.124256i
$964$	3.00184e7	1.04039
$965$	0	0
$966$	9.34702e6	0.322278
$967$	− 2.32723e6i	− 0.0800336i	−0.999199	−	0.0400168i	$-0.987259\pi$
$967$	− 2.32723e6i	− 0.0800336i	0.999199	−	0.0400168i	$-0.0127411\pi$
$968$	5.00748e6i	0.171764i
$969$	4.75725e6	0.162760
$970$	0	0
$971$	2.32618e7	0.791762	0.395881	−	0.918302i	$-0.370439\pi$
$971$	2.32618e7	0.791762	0.395881	+	0.918302i	$0.370439\pi$
$972$	2.66563e7i	0.904970i
$973$	− 5.14597e7i	− 1.74255i
$974$	5.05172e7	1.70625
$975$	0	0
$976$	−2.14155e7	−0.719619
$977$	1.85849e7i	0.622908i	0.950261	+	0.311454i	$0.100816\pi$
$977$	1.85849e7i	0.622908i	−0.950261	+	0.311454i	$0.899184\pi$
$978$	− 1.16928e7i	− 0.390906i
$979$	3.10548e6	0.103555
$980$	0	0
$981$	9.39232e6	0.311602
$982$	− 1.94802e6i	− 0.0644636i
$983$	− 1.42083e6i	− 0.0468986i	−0.999725	−	0.0234493i	$-0.992535\pi$
$983$	− 1.42083e6i	− 0.0468986i	0.999725	−	0.0234493i	$-0.00746482\pi$
$984$	2.41403e6	0.0794794
$985$	0	0
$986$	4.52112e6	0.148100
$987$	1.43001e7i	0.467246i
$988$	5.81763e6i	0.189607i
$989$	−7.81656e6	−0.254112
$990$	0	0
$991$	−2.86425e6	−0.0926462	−0.0463231	−	0.998927i	$-0.514750\pi$
$991$	−2.86425e6	−0.0926462	−0.0463231	+	0.998927i	$0.514750\pi$
$992$	5.60120e7i	1.80718i
$993$	− 1.60167e7i	− 0.515465i
$994$	−5.25223e7	−1.68608
$995$	0	0
$996$	4.27363e7	1.36505
$997$	4.95795e7i	1.57966i	0.613325	+	0.789831i	$0.289832\pi$
$997$	4.95795e7i	1.57966i	−0.613325	+	0.789831i	$0.710168\pi$
$998$	1.49398e7i	0.474809i
$999$	−2.53687e7	−0.804238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.17		18
5.2	odd	4		325.6.a.i.1.1	yes	9
5.3	odd	4		325.6.a.h.1.9	✓	9
5.4	even	2	inner	325.6.b.h.274.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.9	✓	9	5.3	odd	4
325.6.a.i.1.1	yes	9	5.2	odd	4
325.6.b.h.274.2		18	5.4	even	2	inner
325.6.b.h.274.17		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.23858i q^{2} -11.6375i q^{3} -35.8743 q^{4} +95.8765 q^{6} -195.071i q^{7} -31.9185i q^{8} +107.569 q^{9} +O(q^{10})\)	\(q+8.23858i q^{2} -11.6375i q^{3} -35.8743 q^{4} +95.8765 q^{6} -195.071i q^{7} -31.9185i q^{8} +107.569 q^{9} +64.5579 q^{11} +417.487i q^{12} -169.000i q^{13} +1607.11 q^{14} -885.013 q^{16} +426.010i q^{17} +886.215i q^{18} +959.570 q^{19} -2270.14 q^{21} +531.866i q^{22} +499.768i q^{23} -371.452 q^{24} +1392.32 q^{26} -4079.74i q^{27} +6998.03i q^{28} -1288.17 q^{29} -6738.17 q^{31} -8312.65i q^{32} -751.292i q^{33} -3509.72 q^{34} -3858.95 q^{36} -6218.21i q^{37} +7905.50i q^{38} -1966.74 q^{39} -6498.90 q^{41} -18702.7i q^{42} +15640.4i q^{43} -2315.97 q^{44} -4117.38 q^{46} -6299.21i q^{47} +10299.3i q^{48} -21245.7 q^{49} +4957.69 q^{51} +6062.75i q^{52} -40397.3i q^{53} +33611.3 q^{54} -6226.38 q^{56} -11167.0i q^{57} -10612.7i q^{58} -25614.1 q^{59} +24197.9 q^{61} -55513.0i q^{62} -20983.6i q^{63} +40164.0 q^{64} +6189.58 q^{66} -39183.3i q^{67} -15282.8i q^{68} +5816.04 q^{69} -32681.2 q^{71} -3433.44i q^{72} -14507.0i q^{73} +51229.3 q^{74} -34423.9 q^{76} -12593.4i q^{77} -16203.1i q^{78} -79037.9 q^{79} -21338.7 q^{81} -53541.7i q^{82} -102366. i q^{83} +81439.5 q^{84} -128855. q^{86} +14991.1i q^{87} -2060.59i q^{88} +48103.8 q^{89} -32967.0 q^{91} -17928.8i q^{92} +78415.4i q^{93} +51896.6 q^{94} -96738.4 q^{96} +73361.2i q^{97} -175035. i q^{98} +6944.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

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