LMFDB - Embedded newform 825.4.c.k.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.36142572544.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 53x^{4} + 632x^{2} + 484 $ x^6 + 53x^4 + 632x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-5.97123i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.k.199.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.97123i q^{2} +3.00000i q^{3} -16.7131 q^{4} +14.9137 q^{6} -5.48376i q^{7} +43.3148i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-4.97123i q^{2} +3.00000i q^{3} -16.7131 q^{4} +14.9137 q^{6} -5.48376i q^{7} +43.3148i q^{8} -9.00000 q^{9} +11.0000 q^{11} -50.1393i q^{12} +24.5738i q^{13} -27.2610 q^{14} +81.6231 q^{16} +59.3687i q^{17} +44.7411i q^{18} -5.89234 q^{19} +16.4513 q^{21} -54.6835i q^{22} -68.4570i q^{23} -129.945 q^{24} +122.162 q^{26} -27.0000i q^{27} +91.6506i q^{28} +265.022 q^{29} -196.554 q^{31} -59.2482i q^{32} +33.0000i q^{33} +295.135 q^{34} +150.418 q^{36} -166.575i q^{37} +29.2922i q^{38} -73.7214 q^{39} +424.101 q^{41} -81.7830i q^{42} -177.351i q^{43} -183.844 q^{44} -340.316 q^{46} -141.148i q^{47} +244.869i q^{48} +312.928 q^{49} -178.106 q^{51} -410.704i q^{52} -339.828i q^{53} -134.223 q^{54} +237.528 q^{56} -17.6770i q^{57} -1317.48i q^{58} -416.784 q^{59} +662.146 q^{61} +977.115i q^{62} +49.3538i q^{63} +358.448 q^{64} +164.051 q^{66} -313.713i q^{67} -992.235i q^{68} +205.371 q^{69} -153.693 q^{71} -389.834i q^{72} +153.866i q^{73} -828.084 q^{74} +98.4793 q^{76} -60.3213i q^{77} +366.486i q^{78} -403.000 q^{79} +81.0000 q^{81} -2108.30i q^{82} -652.313i q^{83} -274.952 q^{84} -881.650 q^{86} +795.065i q^{87} +476.463i q^{88} -1226.77 q^{89} +134.757 q^{91} +1144.13i q^{92} -589.662i q^{93} -701.677 q^{94} +177.745 q^{96} -959.746i q^{97} -1555.64i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 60 q^{4} - 12 q^{6} - 54 q^{9}+O(q^{10}) $ 6 * q - 60 * q^4 - 12 * q^6 - 54 * q^9	$ 6 q - 60 q^{4} - 12 q^{6} - 54 q^{9} + 66 q^{11} + 136 q^{14} + 356 q^{16} + 116 q^{19} + 60 q^{21} - 108 q^{24} - 240 q^{26} + 440 q^{29} + 496 q^{31} + 160 q^{34} + 540 q^{36} - 684 q^{39} + 312 q^{41} - 660 q^{44} - 2512 q^{46} - 558 q^{49} - 624 q^{51} + 108 q^{54} - 3288 q^{56} - 1096 q^{59} + 828 q^{61} + 116 q^{64} - 132 q^{66} - 720 q^{69} - 1824 q^{71} + 3224 q^{74} - 5504 q^{76} + 1084 q^{79} + 486 q^{81} - 4056 q^{84} - 3096 q^{86} - 1580 q^{89} - 1544 q^{91} + 848 q^{94} - 1548 q^{96} - 594 q^{99}+O(q^{100}) $ 6 * q - 60 * q^4 - 12 * q^6 - 54 * q^9 + 66 * q^11 + 136 * q^14 + 356 * q^16 + 116 * q^19 + 60 * q^21 - 108 * q^24 - 240 * q^26 + 440 * q^29 + 496 * q^31 + 160 * q^34 + 540 * q^36 - 684 * q^39 + 312 * q^41 - 660 * q^44 - 2512 * q^46 - 558 * q^49 - 624 * q^51 + 108 * q^54 - 3288 * q^56 - 1096 * q^59 + 828 * q^61 + 116 * q^64 - 132 * q^66 - 720 * q^69 - 1824 * q^71 + 3224 * q^74 - 5504 * q^76 + 1084 * q^79 + 486 * q^81 - 4056 * q^84 - 3096 * q^86 - 1580 * q^89 - 1544 * q^91 + 848 * q^94 - 1548 * q^96 - 594 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 4.97123i	− 1.75759i	−0.477195	−	0.878797i	$-0.658347\pi$
$2$	− 4.97123i	− 1.75759i	0.477195	−	0.878797i	$-0.341653\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−16.7131	−2.08914
$5$	0	0
$5$	0	0
$6$	14.9137	1.01475
$7$	− 5.48376i	− 0.296095i	−0.988980	−	0.148048i	$-0.952701\pi$
$7$	− 5.48376i	− 0.296095i	0.988980	−	0.148048i	$-0.0472989\pi$
$8$	43.3148i	1.91426i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	− 50.1393i	− 1.20616i
$13$	24.5738i	0.524272i	0.965031	+	0.262136i	$0.0844270\pi$
$13$	24.5738i	0.524272i	−0.965031	+	0.262136i	$0.915573\pi$
$14$	−27.2610	−0.520415
$15$	0	0
$16$	81.6231	1.27536
$17$	59.3687i	0.847001i	0.905896	+	0.423501i	$0.139199\pi$
$17$	59.3687i	0.847001i	−0.905896	+	0.423501i	$0.860801\pi$
$18$	44.7411i	0.585865i
$19$	−5.89234	−0.0711471	−0.0355736	−	0.999367i	$-0.511326\pi$
$19$	−5.89234	−0.0711471	−0.0355736	+	0.999367i	$0.511326\pi$
$20$	0	0
$21$	16.4513	0.170951
$22$	− 54.6835i	− 0.529935i
$23$	− 68.4570i	− 0.620621i	−0.950635	−	0.310310i	$-0.899567\pi$
$23$	− 68.4570i	− 0.620621i	0.950635	−	0.310310i	$-0.100433\pi$
$24$	−129.945	−1.10520
$25$	0	0
$26$	122.162	0.921458
$27$	− 27.0000i	− 0.192450i
$28$	91.6506i	0.618584i
$29$	265.022	1.69701	0.848505	−	0.529187i	$-0.177503\pi$
$29$	265.022	1.69701	0.848505	+	0.529187i	$0.177503\pi$
$30$	0	0
$31$	−196.554	−1.13878	−0.569389	−	0.822068i	$-0.692820\pi$
$31$	−196.554	−1.13878	−0.569389	+	0.822068i	$0.692820\pi$
$32$	− 59.2482i	− 0.327303i
$33$	33.0000i	0.174078i
$34$	295.135	1.48868
$35$	0	0
$36$	150.418	0.696379
$37$	− 166.575i	− 0.740131i	−0.929006	−	0.370065i	$-0.879335\pi$
$37$	− 166.575i	− 0.740131i	0.929006	−	0.370065i	$-0.120665\pi$
$38$	29.2922i	0.125048i
$39$	−73.7214	−0.302689
$40$	0	0
$41$	424.101	1.61545	0.807725	−	0.589559i	$-0.200699\pi$
$41$	424.101	1.61545	0.807725	+	0.589559i	$0.200699\pi$
$42$	− 81.7830i	− 0.300462i
$43$	− 177.351i	− 0.628970i	−0.949262	−	0.314485i	$-0.898168\pi$
$43$	− 177.351i	− 0.628970i	0.949262	−	0.314485i	$-0.101832\pi$
$44$	−183.844	−0.629899
$45$	0	0
$46$	−340.316	−1.09080
$47$	− 141.148i	− 0.438053i	−0.975719	−	0.219026i	$-0.929712\pi$
$47$	− 141.148i	− 0.438053i	0.975719	−	0.219026i	$-0.0702881\pi$
$48$	244.869i	0.736330i
$49$	312.928	0.912328
$50$	0	0
$51$	−178.106	−0.489016
$52$	− 410.704i	− 1.09528i
$53$	− 339.828i	− 0.880736i	−0.897817	−	0.440368i	$-0.854848\pi$
$53$	− 339.828i	− 0.880736i	0.897817	−	0.440368i	$-0.145152\pi$
$54$	−134.223	−0.338249
$55$	0	0
$56$	237.528	0.566804
$57$	− 17.6770i	− 0.0410768i
$58$	− 1317.48i	− 2.98266i
$59$	−416.784	−0.919672	−0.459836	−	0.888004i	$-0.652092\pi$
$59$	−416.784	−0.919672	−0.459836	+	0.888004i	$0.652092\pi$
$60$	0	0
$61$	662.146	1.38982	0.694911	−	0.719096i	$-0.255444\pi$
$61$	662.146	1.38982	0.694911	+	0.719096i	$0.255444\pi$
$62$	977.115i	2.00151i
$63$	49.3538i	0.0986984i
$64$	358.448	0.700094
$65$	0	0
$66$	164.051	0.305958
$67$	− 313.713i	− 0.572032i	−0.958225	−	0.286016i	$-0.907669\pi$
$67$	− 313.713i	− 0.572032i	0.958225	−	0.286016i	$-0.0923311\pi$
$68$	− 992.235i	− 1.76950i
$69$	205.371	0.358316
$70$	0	0
$71$	−153.693	−0.256902	−0.128451	−	0.991716i	$-0.541000\pi$
$71$	−153.693	−0.256902	−0.128451	+	0.991716i	$0.541000\pi$
$72$	− 389.834i	− 0.638088i
$73$	153.866i	0.246694i	0.992364	+	0.123347i	$0.0393628\pi$
$73$	153.866i	0.246694i	−0.992364	+	0.123347i	$0.960637\pi$
$74$	−828.084	−1.30085
$75$	0	0
$76$	98.4793	0.148636
$77$	− 60.3213i	− 0.0892760i
$78$	366.486i	0.532004i
$79$	−403.000	−0.573938	−0.286969	−	0.957940i	$-0.592648\pi$
$79$	−403.000	−0.573938	−0.286969	+	0.957940i	$0.592648\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 2108.30i	− 2.83931i
$83$	− 652.313i	− 0.862659i	−0.902195	−	0.431329i	$-0.858045\pi$
$83$	− 652.313i	− 0.862659i	0.902195	−	0.431329i	$-0.141955\pi$
$84$	−274.952	−0.357139
$85$	0	0
$86$	−881.650	−1.10547
$87$	795.065i	0.979769i
$88$	476.463i	0.577172i
$89$	−1226.77	−1.46109	−0.730545	−	0.682864i	$-0.760734\pi$
$89$	−1226.77	−1.46109	−0.730545	+	0.682864i	$0.760734\pi$
$90$	0	0
$91$	134.757	0.155234
$92$	1144.13i	1.29656i
$93$	− 589.662i	− 0.657474i
$94$	−701.677	−0.769919
$95$	0	0
$96$	177.745	0.188969
$97$	− 959.746i	− 1.00461i	−0.864690	−	0.502306i	$-0.832485\pi$
$97$	− 959.746i	− 1.00461i	0.864690	−	0.502306i	$-0.167515\pi$
$98$	− 1555.64i	− 1.60350i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−1258.52	−1.23988	−0.619939	−	0.784650i	$-0.712843\pi$
$101$	−1258.52	−1.23988	−0.619939	+	0.784650i	$0.712843\pi$
$102$	885.406i	0.859492i
$103$	− 493.175i	− 0.471786i	−0.971779	−	0.235893i	$-0.924198\pi$
$103$	− 493.175i	− 0.471786i	0.971779	−	0.235893i	$-0.0758015\pi$
$104$	−1064.41	−1.00360
$105$	0	0
$106$	−1689.36	−1.54798
$107$	1099.05i	0.992983i	0.868042	+	0.496492i	$0.165379\pi$
$107$	1099.05i	0.992983i	−0.868042	+	0.496492i	$0.834621\pi$
$108$	451.254i	0.402055i
$109$	−1275.04	−1.12043	−0.560216	−	0.828347i	$-0.689282\pi$
$109$	−1275.04	−1.12043	−0.560216	+	0.828347i	$0.689282\pi$
$110$	0	0
$111$	499.726	0.427315
$112$	− 447.601i	− 0.377628i
$113$	− 946.433i	− 0.787902i	−0.919131	−	0.393951i	$-0.871108\pi$
$113$	− 946.433i	− 0.787902i	0.919131	−	0.393951i	$-0.128892\pi$
$114$	−87.8765	−0.0721964
$115$	0	0
$116$	−4429.34	−3.54529
$117$	− 221.164i	− 0.174757i
$118$	2071.93i	1.61641i
$119$	325.563	0.250793
$120$	0	0
$121$	121.000	0.0909091
$122$	− 3291.68i	− 2.44274i
$123$	1272.30i	0.932681i
$124$	3285.03	2.37907
$125$	0	0
$126$	245.349	0.173472
$127$	− 2260.77i	− 1.57962i	−0.613354	−	0.789808i	$-0.710180\pi$
$127$	− 2260.77i	− 1.57962i	0.613354	−	0.789808i	$-0.289820\pi$
$128$	− 2255.91i	− 1.55778i
$129$	532.052	0.363136
$130$	0	0
$131$	−57.1277	−0.0381013	−0.0190507	−	0.999819i	$-0.506064\pi$
$131$	−57.1277	−0.0381013	−0.0190507	+	0.999819i	$0.506064\pi$
$132$	− 551.533i	− 0.363672i
$133$	32.3122i	0.0210663i
$134$	−1559.54	−1.00540
$135$	0	0
$136$	−2571.54	−1.62138
$137$	− 1919.27i	− 1.19689i	−0.801162	−	0.598447i	$-0.795785\pi$
$137$	− 1919.27i	− 1.19689i	0.801162	−	0.598447i	$-0.204215\pi$
$138$	− 1020.95i	− 0.629774i
$139$	1690.46	1.03153	0.515766	−	0.856729i	$-0.327507\pi$
$139$	1690.46	1.03153	0.515766	+	0.856729i	$0.327507\pi$
$140$	0	0
$141$	423.443	0.252910
$142$	764.044i	0.451529i
$143$	270.312i	0.158074i
$144$	−734.608	−0.425120
$145$	0	0
$146$	764.904	0.433588
$147$	938.785i	0.526733i
$148$	2783.99i	1.54624i
$149$	2077.88	1.14246	0.571231	−	0.820789i	$-0.306466\pi$
$149$	2077.88	1.14246	0.571231	+	0.820789i	$0.306466\pi$
$150$	0	0
$151$	2711.16	1.46113	0.730566	−	0.682842i	$-0.239256\pi$
$151$	2711.16	1.46113	0.730566	+	0.682842i	$0.239256\pi$
$152$	− 255.226i	− 0.136194i
$153$	− 534.318i	− 0.282334i
$154$	−299.871	−0.156911
$155$	0	0
$156$	1232.11	0.632359
$157$	− 2341.23i	− 1.19013i	−0.803677	−	0.595066i	$-0.797126\pi$
$157$	− 2341.23i	− 1.19013i	0.803677	−	0.595066i	$-0.202874\pi$
$158$	2003.41i	1.00875i
$159$	1019.48	0.508493
$160$	0	0
$161$	−375.402	−0.183763
$162$	− 402.669i	− 0.195288i
$163$	3523.01i	1.69291i	0.532464	+	0.846453i	$0.321266\pi$
$163$	3523.01i	1.69291i	−0.532464	+	0.846453i	$0.678734\pi$
$164$	−7088.05	−3.37490
$165$	0	0
$166$	−3242.80	−1.51620
$167$	− 4011.90i	− 1.85898i	−0.368842	−	0.929492i	$-0.620246\pi$
$167$	− 4011.90i	− 1.85898i	0.368842	−	0.929492i	$-0.379754\pi$
$168$	712.584i	0.327244i
$169$	1593.13	0.725138
$170$	0	0
$171$	53.0310	0.0237157
$172$	2964.08i	1.31400i
$173$	3325.30i	1.46137i	0.682713	+	0.730687i	$0.260800\pi$
$173$	3325.30i	1.46137i	−0.682713	+	0.730687i	$0.739200\pi$
$174$	3952.45	1.72204
$175$	0	0
$176$	897.854	0.384536
$177$	− 1250.35i	− 0.530973i
$178$	6098.54i	2.56800i
$179$	26.6350	0.0111217	0.00556087	−	0.999985i	$-0.498230\pi$
$179$	26.6350	0.0111217	0.00556087	+	0.999985i	$0.498230\pi$
$180$	0	0
$181$	1934.25	0.794320	0.397160	−	0.917749i	$-0.369996\pi$
$181$	1934.25	0.794320	0.397160	+	0.917749i	$0.369996\pi$
$182$	− 669.906i	− 0.272839i
$183$	1986.44i	0.802414i
$184$	2965.21	1.18803
$185$	0	0
$186$	−2931.34	−1.15557
$187$	653.055i	0.255380i
$188$	2359.01i	0.915153i
$189$	−148.061	−0.0569835
$190$	0	0
$191$	−3341.59	−1.26591	−0.632956	−	0.774188i	$-0.718158\pi$
$191$	−3341.59	−1.26591	−0.632956	+	0.774188i	$0.718158\pi$
$192$	1075.34i	0.404200i
$193$	− 1293.97i	− 0.482600i	−0.970451	−	0.241300i	$-0.922426\pi$
$193$	− 1293.97i	− 0.482600i	0.970451	−	0.241300i	$-0.0775737\pi$
$194$	−4771.12	−1.76570
$195$	0	0
$196$	−5230.01	−1.90598
$197$	2301.17i	0.832240i	0.909310	+	0.416120i	$0.136610\pi$
$197$	2301.17i	0.832240i	−0.909310	+	0.416120i	$0.863390\pi$
$198$	492.152i	0.176645i
$199$	3039.64	1.08279	0.541393	−	0.840770i	$-0.317897\pi$
$199$	3039.64	1.08279	0.541393	+	0.840770i	$0.317897\pi$
$200$	0	0
$201$	941.139	0.330263
$202$	6256.40i	2.17920i
$203$	− 1453.31i	− 0.502476i
$204$	2976.70	1.02162
$205$	0	0
$206$	−2451.69	−0.829209
$207$	616.113i	0.206874i
$208$	2005.79i	0.668636i
$209$	−64.8157	−0.0214517
$210$	0	0
$211$	2807.86	0.916119	0.458060	−	0.888921i	$-0.348545\pi$
$211$	2807.86	0.916119	0.458060	+	0.888921i	$0.348545\pi$
$212$	5679.58i	1.83998i
$213$	− 461.080i	− 0.148322i
$214$	5463.63	1.74526
$215$	0	0
$216$	1169.50	0.368400
$217$	1077.85i	0.337187i
$218$	6338.53i	1.96926i
$219$	−461.598	−0.142429
$220$	0	0
$221$	−1458.91	−0.444059
$222$	− 2484.25i	− 0.751046i
$223$	1443.09i	0.433348i	0.976244	+	0.216674i	$0.0695209\pi$
$223$	1443.09i	0.433348i	−0.976244	+	0.216674i	$0.930479\pi$
$224$	−324.903	−0.0969129
$225$	0	0
$226$	−4704.94	−1.38481
$227$	− 3658.71i	− 1.06977i	−0.844926	−	0.534884i	$-0.820355\pi$
$227$	− 3658.71i	− 1.06977i	0.844926	−	0.534884i	$-0.179645\pi$
$228$	295.438i	0.0858151i
$229$	1695.50	0.489265	0.244632	−	0.969616i	$-0.421333\pi$
$229$	1695.50	0.489265	0.244632	+	0.969616i	$0.421333\pi$
$230$	0	0
$231$	180.964	0.0515435
$232$	11479.4i	3.24853i
$233$	− 5782.24i	− 1.62578i	−0.582417	−	0.812890i	$-0.697893\pi$
$233$	− 5782.24i	− 1.62578i	0.582417	−	0.812890i	$-0.302107\pi$
$234$	−1099.46	−0.307153
$235$	0	0
$236$	6965.76	1.92132
$237$	− 1209.00i	− 0.331363i
$238$	− 1618.45i	− 0.440792i
$239$	5836.16	1.57954	0.789770	−	0.613404i	$-0.210200\pi$
$239$	5836.16	1.57954	0.789770	+	0.613404i	$0.210200\pi$
$240$	0	0
$241$	−982.308	−0.262556	−0.131278	−	0.991346i	$-0.541908\pi$
$241$	−982.308	−0.262556	−0.131278	+	0.991346i	$0.541908\pi$
$242$	− 601.519i	− 0.159781i
$243$	243.000i	0.0641500i
$244$	−11066.5	−2.90353
$245$	0	0
$246$	6324.91	1.63927
$247$	− 144.797i	− 0.0373005i
$248$	− 8513.70i	− 2.17992i
$249$	1956.94	0.498056
$250$	0	0
$251$	4320.23	1.08642	0.543209	−	0.839598i	$-0.317209\pi$
$251$	4320.23	1.08642	0.543209	+	0.839598i	$0.317209\pi$
$252$	− 824.856i	− 0.206195i
$253$	− 753.027i	− 0.187124i
$254$	−11238.8	−2.77633
$255$	0	0
$256$	−8347.07	−2.03786
$257$	6224.29i	1.51074i	0.655298	+	0.755371i	$0.272543\pi$
$257$	6224.29i	1.51074i	−0.655298	+	0.755371i	$0.727457\pi$
$258$	− 2644.95i	− 0.638246i
$259$	−913.459	−0.219149
$260$	0	0
$261$	−2385.20	−0.565670
$262$	283.995i	0.0669666i
$263$	− 2588.95i	− 0.607002i	−0.952831	−	0.303501i	$-0.901844\pi$
$263$	− 2588.95i	− 0.607002i	0.952831	−	0.303501i	$-0.0981555\pi$
$264$	−1429.39	−0.333230
$265$	0	0
$266$	160.631	0.0370260
$267$	− 3680.30i	− 0.843561i
$268$	5243.12i	1.19505i
$269$	5871.53	1.33083	0.665415	−	0.746473i	$-0.268254\pi$
$269$	5871.53	1.33083	0.665415	+	0.746473i	$0.268254\pi$
$270$	0	0
$271$	−3306.93	−0.741261	−0.370630	−	0.928780i	$-0.620858\pi$
$271$	−3306.93	−0.741261	−0.370630	+	0.928780i	$0.620858\pi$
$272$	4845.85i	1.08023i
$273$	404.270i	0.0896247i
$274$	−9541.14	−2.10366
$275$	0	0
$276$	−3432.39	−0.748571
$277$	2861.37i	0.620660i	0.950629	+	0.310330i	$0.100440\pi$
$277$	2861.37i	0.620660i	−0.950629	+	0.310330i	$0.899560\pi$
$278$	− 8403.66i	− 1.81302i
$279$	1768.99	0.379593
$280$	0	0
$281$	−2988.05	−0.634349	−0.317174	−	0.948367i	$-0.602734\pi$
$281$	−2988.05	−0.634349	−0.317174	+	0.948367i	$0.602734\pi$
$282$	− 2105.03i	− 0.444513i
$283$	1454.60i	0.305536i	0.988262	+	0.152768i	$0.0488188\pi$
$283$	1454.60i	0.305536i	−0.988262	+	0.152768i	$0.951181\pi$
$284$	2568.69	0.536704
$285$	0	0
$286$	1343.78	0.277830
$287$	− 2325.67i	− 0.478327i
$288$	533.234i	0.109101i
$289$	1388.36	0.282589
$290$	0	0
$291$	2879.24	0.580014
$292$	− 2571.58i	− 0.515378i
$293$	− 8586.52i	− 1.71205i	−0.516937	−	0.856023i	$-0.672928\pi$
$293$	− 8586.52i	− 1.71205i	0.516937	−	0.856023i	$-0.327072\pi$
$294$	4666.92	0.925782
$295$	0	0
$296$	7215.19	1.41681
$297$	− 297.000i	− 0.0580259i
$298$	− 10329.6i	− 2.00798i
$299$	1682.25	0.325374
$300$	0	0
$301$	−972.547	−0.186235
$302$	− 13477.8i	− 2.56808i
$303$	− 3775.57i	− 0.715844i
$304$	−480.951	−0.0907382
$305$	0	0
$306$	−2656.22	−0.496228
$307$	− 529.029i	− 0.0983495i	−0.998790	−	0.0491748i	$-0.984341\pi$
$307$	− 529.029i	− 0.0983495i	0.998790	−	0.0491748i	$-0.0156591\pi$
$308$	1008.16i	0.186510i
$309$	1479.53	0.272386
$310$	0	0
$311$	−5008.00	−0.913111	−0.456555	−	0.889695i	$-0.650917\pi$
$311$	−5008.00	−0.913111	−0.456555	+	0.889695i	$0.650917\pi$
$312$	− 3193.23i	− 0.579426i
$313$	7858.53i	1.41914i	0.704635	+	0.709570i	$0.251111\pi$
$313$	7858.53i	1.41914i	−0.704635	+	0.709570i	$0.748889\pi$
$314$	−11638.8	−2.09177
$315$	0	0
$316$	6735.39	1.19904
$317$	− 7747.04i	− 1.37261i	−0.727315	−	0.686304i	$-0.759232\pi$
$317$	− 7747.04i	− 1.37261i	0.727315	−	0.686304i	$-0.240768\pi$
$318$	− 5068.09i	− 0.893724i
$319$	2915.24	0.511668
$320$	0	0
$321$	−3297.15	−0.573299
$322$	1866.21i	0.322980i
$323$	− 349.820i	− 0.0602617i
$324$	−1353.76	−0.232126
$325$	0	0
$326$	17513.7	2.97544
$327$	− 3825.13i	− 0.646882i
$328$	18369.9i	3.09240i
$329$	−774.019	−0.129705
$330$	0	0
$331$	1355.08	0.225022	0.112511	−	0.993650i	$-0.464111\pi$
$331$	1355.08	0.225022	0.112511	+	0.993650i	$0.464111\pi$
$332$	10902.2i	1.80221i
$333$	1499.18i	0.246710i
$334$	−19944.1	−3.26734
$335$	0	0
$336$	1342.80	0.218024
$337$	− 2596.97i	− 0.419780i	−0.977725	−	0.209890i	$-0.932689\pi$
$337$	− 2596.97i	− 0.419780i	0.977725	−	0.209890i	$-0.0673106\pi$
$338$	− 7919.81i	− 1.27450i
$339$	2839.30	0.454896
$340$	0	0
$341$	−2162.09	−0.343355
$342$	− 263.629i	− 0.0416826i
$343$	− 3596.95i	− 0.566231i
$344$	7681.91	1.20401
$345$	0	0
$346$	16530.8	2.56850
$347$	− 266.820i	− 0.0412785i	−0.999787	−	0.0206392i	$-0.993430\pi$
$347$	− 266.820i	− 0.0412785i	0.999787	−	0.0206392i	$-0.00657014\pi$
$348$	− 13288.0i	− 2.04687i
$349$	10549.4	1.61804	0.809021	−	0.587780i	$-0.199998\pi$
$349$	10549.4	1.61804	0.809021	+	0.587780i	$0.199998\pi$
$350$	0	0
$351$	663.492	0.100896
$352$	− 651.730i	− 0.0986856i
$353$	7152.63i	1.07846i	0.842159	+	0.539229i	$0.181284\pi$
$353$	7152.63i	1.07846i	−0.842159	+	0.539229i	$0.818716\pi$
$354$	−6215.79	−0.933235
$355$	0	0
$356$	20503.1	3.05242
$357$	976.690i	0.144795i
$358$	− 132.409i	− 0.0195475i
$359$	−359.182	−0.0528047	−0.0264024	−	0.999651i	$-0.508405\pi$
$359$	−359.182	−0.0528047	−0.0264024	+	0.999651i	$0.508405\pi$
$360$	0	0
$361$	−6824.28	−0.994938
$362$	− 9615.62i	− 1.39609i
$363$	363.000i	0.0524864i
$364$	−2252.20	−0.324306
$365$	0	0
$366$	9875.04	1.41032
$367$	− 9042.47i	− 1.28614i	−0.765808	−	0.643070i	$-0.777661\pi$
$367$	− 9042.47i	− 1.28614i	0.765808	−	0.643070i	$-0.222339\pi$
$368$	− 5587.67i	− 0.791515i
$369$	−3816.91	−0.538483
$370$	0	0
$371$	−1863.54	−0.260781
$372$	9855.08i	1.37355i
$373$	− 11929.0i	− 1.65592i	−0.560787	−	0.827960i	$-0.689501\pi$
$373$	− 11929.0i	− 1.65592i	0.560787	−	0.827960i	$-0.310499\pi$
$374$	3246.49	0.448855
$375$	0	0
$376$	6113.78	0.838549
$377$	6512.59i	0.889696i
$378$	736.047i	0.100154i
$379$	−6556.08	−0.888557	−0.444279	−	0.895889i	$-0.646540\pi$
$379$	−6556.08	−0.888557	−0.444279	+	0.895889i	$0.646540\pi$
$380$	0	0
$381$	6782.32	0.911992
$382$	16611.8i	2.22496i
$383$	10703.1i	1.42795i	0.700172	+	0.713975i	$0.253107\pi$
$383$	10703.1i	1.42795i	−0.700172	+	0.713975i	$0.746893\pi$
$384$	6767.74	0.899388
$385$	0	0
$386$	−6432.60	−0.848214
$387$	1596.15i	0.209657i
$388$	16040.3i	2.09878i
$389$	6450.30	0.840728	0.420364	−	0.907355i	$-0.361902\pi$
$389$	6450.30	0.840728	0.420364	+	0.907355i	$0.361902\pi$
$390$	0	0
$391$	4064.20	0.525667
$392$	13554.4i	1.74644i
$393$	− 171.383i	− 0.0219978i
$394$	11439.6	1.46274
$395$	0	0
$396$	1654.60	0.209966
$397$	12526.0i	1.58353i	0.610828	+	0.791764i	$0.290837\pi$
$397$	12526.0i	1.58353i	−0.610828	+	0.791764i	$0.709163\pi$
$398$	− 15110.7i	− 1.90310i
$399$	−96.9365	−0.0121626
$400$	0	0
$401$	9119.52	1.13568	0.567839	−	0.823139i	$-0.307779\pi$
$401$	9119.52	1.13568	0.567839	+	0.823139i	$0.307779\pi$
$402$	− 4678.62i	− 0.580468i
$403$	− 4830.08i	− 0.597030i
$404$	21033.8	2.59028
$405$	0	0
$406$	−7224.76	−0.883150
$407$	− 1832.33i	− 0.223158i
$408$	− 7714.63i	− 0.936106i
$409$	596.921	0.0721658	0.0360829	−	0.999349i	$-0.488512\pi$
$409$	596.921	0.0721658	0.0360829	+	0.999349i	$0.488512\pi$
$410$	0	0
$411$	5757.82	0.691027
$412$	8242.49i	0.985627i
$413$	2285.54i	0.272310i
$414$	3062.84	0.363600
$415$	0	0
$416$	1455.95	0.171596
$417$	5071.38i	0.595555i
$418$	322.214i	0.0377033i
$419$	−11560.5	−1.34789	−0.673944	−	0.738782i	$-0.735401\pi$
$419$	−11560.5	−1.34789	−0.673944	+	0.738782i	$0.735401\pi$
$420$	0	0
$421$	15182.6	1.75761	0.878805	−	0.477182i	$-0.158342\pi$
$421$	15182.6	1.75761	0.878805	+	0.477182i	$0.158342\pi$
$422$	− 13958.5i	− 1.61017i
$423$	1270.33i	0.146018i
$424$	14719.6	1.68596
$425$	0	0
$426$	−2292.13	−0.260691
$427$	− 3631.05i	− 0.411519i
$428$	− 18368.5i	− 2.07448i
$429$	−810.935	−0.0912641
$430$	0	0
$431$	−14902.7	−1.66551	−0.832757	−	0.553639i	$-0.813239\pi$
$431$	−14902.7	−1.66551	−0.832757	+	0.553639i	$0.813239\pi$
$432$	− 2203.82i	− 0.245443i
$433$	− 2269.85i	− 0.251922i	−0.992035	−	0.125961i	$-0.959799\pi$
$433$	− 2269.85i	− 0.251922i	0.992035	−	0.125961i	$-0.0402014\pi$
$434$	5358.26	0.592637
$435$	0	0
$436$	21309.9	2.34074
$437$	403.372i	0.0441554i
$438$	2294.71i	0.250332i
$439$	13418.7	1.45886	0.729431	−	0.684054i	$-0.239785\pi$
$439$	13418.7	1.45886	0.729431	+	0.684054i	$0.239785\pi$
$440$	0	0
$441$	−2816.36	−0.304109
$442$	7252.59i	0.780476i
$443$	11507.5i	1.23417i	0.786896	+	0.617086i	$0.211687\pi$
$443$	11507.5i	1.23417i	−0.786896	+	0.617086i	$0.788313\pi$
$444$	−8351.98	−0.892719
$445$	0	0
$446$	7173.94	0.761650
$447$	6233.65i	0.659601i
$448$	− 1965.64i	− 0.207294i
$449$	16203.8	1.70313	0.851564	−	0.524250i	$-0.175654\pi$
$449$	16203.8	1.70313	0.851564	+	0.524250i	$0.175654\pi$
$450$	0	0
$451$	4665.11	0.487077
$452$	15817.8i	1.64604i
$453$	8133.48i	0.843585i
$454$	−18188.3	−1.88022
$455$	0	0
$456$	765.677	0.0786318
$457$	− 4912.79i	− 0.502868i	−0.967874	−	0.251434i	$-0.919098\pi$
$457$	− 4912.79i	− 0.502868i	0.967874	−	0.251434i	$-0.0809022\pi$
$458$	− 8428.70i	− 0.859929i
$459$	1602.95	0.163005
$460$	0	0
$461$	−14270.8	−1.44178	−0.720888	−	0.693051i	$-0.756266\pi$
$461$	−14270.8	−1.44178	−0.720888	+	0.693051i	$0.756266\pi$
$462$	− 899.613i	− 0.0905926i
$463$	6656.46i	0.668146i	0.942547	+	0.334073i	$0.108423\pi$
$463$	6656.46i	0.668146i	−0.942547	+	0.334073i	$0.891577\pi$
$464$	21631.9	2.16430
$465$	0	0
$466$	−28744.8	−2.85746
$467$	− 19347.7i	− 1.91714i	−0.284853	−	0.958571i	$-0.591945\pi$
$467$	− 19347.7i	− 1.91714i	0.284853	−	0.958571i	$-0.408055\pi$
$468$	3696.34i	0.365093i
$469$	−1720.33	−0.169376
$470$	0	0
$471$	7023.70	0.687123
$472$	− 18052.9i	− 1.76050i
$473$	− 1950.86i	− 0.189642i
$474$	−6010.22	−0.582402
$475$	0	0
$476$	−5441.18	−0.523941
$477$	3058.45i	0.293579i
$478$	− 29012.9i	− 2.77619i
$479$	−9826.69	−0.937355	−0.468678	−	0.883369i	$-0.655269\pi$
$479$	−9826.69	−0.937355	−0.468678	+	0.883369i	$0.655269\pi$
$480$	0	0
$481$	4093.39	0.388030
$482$	4883.28i	0.461467i
$483$	− 1126.21i	− 0.106095i
$484$	−2022.29	−0.189922
$485$	0	0
$486$	1208.01	0.112750
$487$	− 10278.1i	− 0.956353i	−0.878264	−	0.478176i	$-0.841298\pi$
$487$	− 10278.1i	− 0.956353i	0.878264	−	0.478176i	$-0.158702\pi$
$488$	28680.7i	2.66048i
$489$	−10569.0	−0.977400
$490$	0	0
$491$	11397.1	1.04755	0.523773	−	0.851858i	$-0.324524\pi$
$491$	11397.1	1.04755	0.523773	+	0.851858i	$0.324524\pi$
$492$	− 21264.1i	− 1.94850i
$493$	15734.0i	1.43737i
$494$	−719.819	−0.0655591
$495$	0	0
$496$	−16043.3	−1.45235
$497$	842.817i	0.0760674i
$498$	− 9728.39i	− 0.875381i
$499$	9179.17	0.823479	0.411740	−	0.911302i	$-0.364921\pi$
$499$	9179.17	0.823479	0.411740	+	0.911302i	$0.364921\pi$
$500$	0	0
$501$	12035.7	1.07329
$502$	− 21476.9i	− 1.90948i
$503$	6782.80i	0.601253i	0.953742	+	0.300626i	$0.0971957\pi$
$503$	6782.80i	0.601253i	−0.953742	+	0.300626i	$0.902804\pi$
$504$	−2137.75	−0.188935
$505$	0	0
$506$	−3743.47	−0.328889
$507$	4779.39i	0.418659i
$508$	37784.6i	3.30004i
$509$	−6814.24	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$509$	−6814.24	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$510$	0	0
$511$	843.765	0.0730449
$512$	23447.9i	2.02395i
$513$	159.093i	0.0136923i
$514$	30942.4	2.65527
$515$	0	0
$516$	−8892.23	−0.758641
$517$	− 1552.62i	− 0.132078i
$518$	4541.01i	0.385175i
$519$	−9975.89	−0.843724
$520$	0	0
$521$	206.712	0.0173824	0.00869118	−	0.999962i	$-0.497233\pi$
$521$	206.712	0.0173824	0.00869118	+	0.999962i	$0.497233\pi$
$522$	11857.4i	0.994219i
$523$	− 11375.9i	− 0.951118i	−0.879684	−	0.475559i	$-0.842246\pi$
$523$	− 11375.9i	− 0.951118i	0.879684	−	0.475559i	$-0.157754\pi$
$524$	954.781	0.0795989
$525$	0	0
$526$	−12870.3	−1.06686
$527$	− 11669.1i	− 0.964547i
$528$	2693.56i	0.222012i
$529$	7480.63	0.614830
$530$	0	0
$531$	3751.06	0.306557
$532$	− 540.036i	− 0.0440104i
$533$	10421.8i	0.846936i
$534$	−18295.6	−1.48264
$535$	0	0
$536$	13588.4	1.09502
$537$	79.9049i	0.00642114i
$538$	− 29188.7i	− 2.33906i
$539$	3442.21	0.275077
$540$	0	0
$541$	−10228.1	−0.812831	−0.406415	−	0.913688i	$-0.633221\pi$
$541$	−10228.1	−0.812831	−0.406415	+	0.913688i	$0.633221\pi$
$542$	16439.5i	1.30284i
$543$	5802.76i	0.458601i
$544$	3517.49	0.277226
$545$	0	0
$546$	2009.72	0.157524
$547$	15538.9i	1.21461i	0.794467	+	0.607307i	$0.207750\pi$
$547$	15538.9i	1.21461i	−0.794467	+	0.607307i	$0.792250\pi$
$548$	32077.0i	2.50048i
$549$	−5959.31	−0.463274
$550$	0	0
$551$	−1561.60	−0.120737
$552$	8895.62i	0.685911i
$553$	2209.96i	0.169940i
$554$	14224.5	1.09087
$555$	0	0
$556$	−28252.8	−2.15501
$557$	5191.17i	0.394896i	0.980313	+	0.197448i	$0.0632653\pi$
$557$	5191.17i	0.394896i	−0.980313	+	0.197448i	$0.936735\pi$
$558$	− 8794.03i	− 0.667170i
$559$	4358.17	0.329752
$560$	0	0
$561$	−1959.17	−0.147444
$562$	14854.3i	1.11493i
$563$	13722.9i	1.02727i	0.858009	+	0.513635i	$0.171701\pi$
$563$	13722.9i	1.02727i	−0.858009	+	0.513635i	$0.828299\pi$
$564$	−7077.04	−0.528364
$565$	0	0
$566$	7231.13	0.537009
$567$	− 444.184i	− 0.0328995i
$568$	− 6657.20i	− 0.491778i
$569$	−23209.0	−1.70997	−0.854986	−	0.518652i	$-0.826434\pi$
$569$	−23209.0	−1.70997	−0.854986	+	0.518652i	$0.826434\pi$
$570$	0	0
$571$	−13130.5	−0.962337	−0.481168	−	0.876628i	$-0.659787\pi$
$571$	−13130.5	−0.962337	−0.481168	+	0.876628i	$0.659787\pi$
$572$	− 4517.75i	− 0.330239i
$573$	− 10024.8i	− 0.730874i
$574$	−11561.4	−0.840705
$575$	0	0
$576$	−3226.03	−0.233365
$577$	12261.0i	0.884633i	0.896859	+	0.442317i	$0.145843\pi$
$577$	12261.0i	0.884633i	−0.896859	+	0.442317i	$0.854157\pi$
$578$	− 6901.86i	− 0.496677i
$579$	3881.90	0.278629
$580$	0	0
$581$	−3577.13	−0.255429
$582$	− 14313.3i	− 1.01943i
$583$	− 3738.11i	− 0.265552i
$584$	−6664.69	−0.472238
$585$	0	0
$586$	−42685.5	−3.00908
$587$	3553.42i	0.249856i	0.992166	+	0.124928i	$0.0398700\pi$
$587$	3553.42i	0.249856i	−0.992166	+	0.124928i	$0.960130\pi$
$588$	− 15690.0i	− 1.10042i
$589$	1158.16	0.0810208
$590$	0	0
$591$	−6903.50	−0.480494
$592$	− 13596.4i	− 0.943933i
$593$	1942.83i	0.134540i	0.997735	+	0.0672701i	$0.0214289\pi$
$593$	1942.83i	0.134540i	−0.997735	+	0.0672701i	$0.978571\pi$
$594$	−1476.45	−0.101986
$595$	0	0
$596$	−34727.9	−2.38676
$597$	9118.92i	0.625147i
$598$	− 8362.84i	− 0.571876i
$599$	−18585.4	−1.26774	−0.633871	−	0.773439i	$-0.718535\pi$
$599$	−18585.4	−1.26774	−0.633871	+	0.773439i	$0.718535\pi$
$600$	0	0
$601$	8010.10	0.543658	0.271829	−	0.962346i	$-0.412371\pi$
$601$	8010.10	0.543658	0.271829	+	0.962346i	$0.412371\pi$
$602$	4834.75i	0.327325i
$603$	2823.42i	0.190677i
$604$	−45311.9	−3.05251
$605$	0	0
$606$	−18769.2	−1.25816
$607$	1537.69i	0.102822i	0.998678	+	0.0514111i	$0.0163719\pi$
$607$	1537.69i	0.102822i	−0.998678	+	0.0514111i	$0.983628\pi$
$608$	349.110i	0.0232867i
$609$	4359.94	0.290105
$610$	0	0
$611$	3468.53	0.229659
$612$	8930.11i	0.589834i
$613$	2298.75i	0.151461i	0.997128	+	0.0757304i	$0.0241289\pi$
$613$	2298.75i	0.151461i	−0.997128	+	0.0757304i	$0.975871\pi$
$614$	−2629.93	−0.172859
$615$	0	0
$616$	2612.81	0.170898
$617$	− 6089.51i	− 0.397333i	−0.980067	−	0.198666i	$-0.936339\pi$
$617$	− 6089.51i	− 0.397333i	0.980067	−	0.198666i	$-0.0636610\pi$
$618$	− 7355.06i	− 0.478744i
$619$	−12255.4	−0.795775	−0.397887	−	0.917434i	$-0.630257\pi$
$619$	−12255.4	−0.795775	−0.397887	+	0.917434i	$0.630257\pi$
$620$	0	0
$621$	−1848.34	−0.119439
$622$	24895.9i	1.60488i
$623$	6727.29i	0.432622i
$624$	−6017.36	−0.386037
$625$	0	0
$626$	39066.6	2.49427
$627$	− 194.447i	− 0.0123851i
$628$	39129.3i	2.48635i
$629$	9889.36	0.626891
$630$	0	0
$631$	−25746.5	−1.62433	−0.812165	−	0.583428i	$-0.801711\pi$
$631$	−25746.5	−1.62433	−0.812165	+	0.583428i	$0.801711\pi$
$632$	− 17455.9i	− 1.09867i
$633$	8423.58i	0.528922i
$634$	−38512.3	−2.41249
$635$	0	0
$636$	−17038.8	−1.06231
$637$	7689.84i	0.478308i
$638$	− 14492.3i	− 0.899305i
$639$	1383.24	0.0856340
$640$	0	0
$641$	−598.022	−0.0368494	−0.0184247	−	0.999830i	$-0.505865\pi$
$641$	−598.022	−0.0368494	−0.0184247	+	0.999830i	$0.505865\pi$
$642$	16390.9i	1.00763i
$643$	14610.9i	0.896109i	0.894006	+	0.448054i	$0.147883\pi$
$643$	14610.9i	0.896109i	−0.894006	+	0.448054i	$0.852117\pi$
$644$	6274.13	0.383906
$645$	0	0
$646$	−1739.04	−0.105916
$647$	17252.3i	1.04831i	0.851622	+	0.524156i	$0.175619\pi$
$647$	17252.3i	1.04831i	−0.851622	+	0.524156i	$0.824381\pi$
$648$	3508.50i	0.212696i
$649$	−4584.63	−0.277292
$650$	0	0
$651$	−3233.56	−0.194675
$652$	− 58880.5i	− 3.53671i
$653$	− 17768.6i	− 1.06484i	−0.846481	−	0.532418i	$-0.821283\pi$
$653$	− 17768.6i	− 1.06484i	0.846481	−	0.532418i	$-0.178717\pi$
$654$	−19015.6	−1.13696
$655$	0	0
$656$	34616.4	2.06028
$657$	− 1384.80i	− 0.0822314i
$658$	3847.82i	0.227969i
$659$	17594.9	1.04006	0.520031	−	0.854147i	$-0.325920\pi$
$659$	17594.9	1.04006	0.520031	+	0.854147i	$0.325920\pi$
$660$	0	0
$661$	23355.9	1.37434	0.687170	−	0.726497i	$-0.258853\pi$
$661$	23355.9	1.37434	0.687170	+	0.726497i	$0.258853\pi$
$662$	− 6736.44i	− 0.395497i
$663$	− 4376.74i	− 0.256378i
$664$	28254.8	1.65136
$665$	0	0
$666$	7452.76	0.433617
$667$	− 18142.6i	− 1.05320i
$668$	67051.4i	3.88368i
$669$	−4329.28	−0.250194
$670$	0	0
$671$	7283.61	0.419047
$672$	− 974.708i	− 0.0559527i
$673$	− 2340.47i	− 0.134054i	−0.997751	−	0.0670271i	$-0.978649\pi$
$673$	− 2340.47i	− 0.134054i	0.997751	−	0.0670271i	$-0.0213514\pi$
$674$	−12910.1	−0.737803
$675$	0	0
$676$	−26626.1	−1.51491
$677$	16562.2i	0.940234i	0.882604	+	0.470117i	$0.155788\pi$
$677$	16562.2i	0.940234i	−0.882604	+	0.470117i	$0.844212\pi$
$678$	− 14114.8i	− 0.799522i
$679$	−5263.01	−0.297461
$680$	0	0
$681$	10976.1	0.617630
$682$	10748.3i	0.603478i
$683$	22303.4i	1.24951i	0.780821	+	0.624755i	$0.214801\pi$
$683$	22303.4i	1.24951i	−0.780821	+	0.624755i	$0.785199\pi$
$684$	−886.313	−0.0495454
$685$	0	0
$686$	−17881.3	−0.995204
$687$	5086.49i	0.282477i
$688$	− 14475.9i	− 0.802163i
$689$	8350.87	0.461745
$690$	0	0
$691$	6244.91	0.343802	0.171901	−	0.985114i	$-0.445009\pi$
$691$	6244.91	0.343802	0.171901	+	0.985114i	$0.445009\pi$
$692$	− 55576.0i	− 3.05301i
$693$	542.892i	0.0297587i
$694$	−1326.42	−0.0725508
$695$	0	0
$696$	−34438.1	−1.87554
$697$	25178.3i	1.36829i
$698$	− 52443.5i	− 2.84386i
$699$	17346.7	0.938645
$700$	0	0
$701$	−15277.7	−0.823154	−0.411577	−	0.911375i	$-0.635022\pi$
$701$	−15277.7	−0.823154	−0.411577	+	0.911375i	$0.635022\pi$
$702$	− 3298.37i	− 0.177335i
$703$	981.519i	0.0526582i
$704$	3942.93	0.211086
$705$	0	0
$706$	35557.3	1.89549
$707$	6901.43i	0.367122i
$708$	20897.3i	1.10928i
$709$	11732.1	0.621449	0.310725	−	0.950500i	$-0.399428\pi$
$709$	11732.1	0.621449	0.310725	+	0.950500i	$0.399428\pi$
$710$	0	0
$711$	3627.00	0.191313
$712$	− 53137.2i	− 2.79691i
$713$	13455.5i	0.706750i
$714$	4855.35	0.254491
$715$	0	0
$716$	−445.153	−0.0232349
$717$	17508.5i	0.911947i
$718$	1785.57i	0.0928093i
$719$	19006.2	0.985828	0.492914	−	0.870078i	$-0.335932\pi$
$719$	19006.2	0.985828	0.492914	+	0.870078i	$0.335932\pi$
$720$	0	0
$721$	−2704.45	−0.139694
$722$	33925.1i	1.74870i
$723$	− 2946.92i	− 0.151587i
$724$	−32327.4	−1.65945
$725$	0	0
$726$	1804.56	0.0922498
$727$	− 2650.77i	− 0.135229i	−0.997712	−	0.0676146i	$-0.978461\pi$
$727$	− 2650.77i	− 0.135229i	0.997712	−	0.0676146i	$-0.0215388\pi$
$728$	5836.96i	0.297160i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	10529.1	0.532738
$732$	− 33199.5i	− 1.67635i
$733$	− 8854.22i	− 0.446164i	−0.974800	−	0.223082i	$-0.928388\pi$
$733$	− 8854.22i	− 0.446164i	0.974800	−	0.223082i	$-0.0716117\pi$
$734$	−44952.2	−2.26051
$735$	0	0
$736$	−4055.96	−0.203131
$737$	− 3450.84i	− 0.172474i
$738$	18974.7i	0.946435i
$739$	−17174.8	−0.854921	−0.427460	−	0.904034i	$-0.640592\pi$
$739$	−17174.8	−0.854921	−0.427460	+	0.904034i	$0.640592\pi$
$740$	0	0
$741$	434.391	0.0215354
$742$	9264.06i	0.458348i
$743$	− 9800.30i	− 0.483901i	−0.970289	−	0.241950i	$-0.922213\pi$
$743$	− 9800.30i	− 0.483901i	0.970289	−	0.241950i	$-0.0777871\pi$
$744$	25541.1	1.25858
$745$	0	0
$746$	−59301.6	−2.91044
$747$	5870.82i	0.287553i
$748$	− 10914.6i	− 0.533525i
$749$	6026.92	0.294017
$750$	0	0
$751$	2781.25	0.135139	0.0675695	−	0.997715i	$-0.478476\pi$
$751$	2781.25	0.135139	0.0675695	+	0.997715i	$0.478476\pi$
$752$	− 11520.9i	− 0.558675i
$753$	12960.7i	0.627243i
$754$	32375.6	1.56372
$755$	0	0
$756$	2474.57	0.119046
$757$	− 3010.87i	− 0.144560i	−0.997384	−	0.0722800i	$-0.976972\pi$
$757$	− 3010.87i	− 0.144560i	0.997384	−	0.0722800i	$-0.0230275\pi$
$758$	32591.8i	1.56172i
$759$	2259.08	0.108036
$760$	0	0
$761$	−2709.54	−0.129068	−0.0645340	−	0.997916i	$-0.520556\pi$
$761$	−2709.54	−0.129068	−0.0645340	+	0.997916i	$0.520556\pi$
$762$	− 33716.5i	− 1.60291i
$763$	6992.03i	0.331754i
$764$	55848.4	2.64466
$765$	0	0
$766$	53207.7	2.50976
$767$	− 10242.0i	− 0.482159i
$768$	− 25041.2i	− 1.17656i
$769$	28121.1	1.31869	0.659345	−	0.751841i	$-0.270834\pi$
$769$	28121.1	1.31869	0.659345	+	0.751841i	$0.270834\pi$
$770$	0	0
$771$	−18672.9	−0.872227
$772$	21626.2i	1.00822i
$773$	4274.61i	0.198897i	0.995043	+	0.0994483i	$0.0317078\pi$
$773$	4274.61i	0.198897i	−0.995043	+	0.0994483i	$0.968292\pi$
$774$	7934.85	0.368491
$775$	0	0
$776$	41571.2	1.92309
$777$	− 2740.38i	− 0.126526i
$778$	− 32065.9i	− 1.47766i
$779$	−2498.95	−0.114935
$780$	0	0
$781$	−1690.63	−0.0774589
$782$	− 20204.1i	− 0.923909i
$783$	− 7155.59i	− 0.326590i
$784$	25542.2	1.16355
$785$	0	0
$786$	−851.985	−0.0386632
$787$	− 19107.9i	− 0.865469i	−0.901522	−	0.432734i	$-0.857549\pi$
$787$	− 19107.9i	− 0.865469i	0.901522	−	0.432734i	$-0.142451\pi$
$788$	− 38459.6i	− 1.73866i
$789$	7766.85	0.350453
$790$	0	0
$791$	−5190.01	−0.233294
$792$	− 4288.17i	− 0.192391i
$793$	16271.4i	0.728645i
$794$	62269.4	2.78320
$795$	0	0
$796$	−50801.9	−2.26209
$797$	− 27518.4i	− 1.22303i	−0.791234	−	0.611513i	$-0.790561\pi$
$797$	− 27518.4i	− 1.22303i	0.791234	−	0.611513i	$-0.209439\pi$
$798$	481.893i	0.0213770i
$799$	8379.74	0.371031
$800$	0	0
$801$	11040.9	0.487030
$802$	− 45335.2i	− 1.99606i
$803$	1692.53i	0.0743811i
$804$	−15729.4	−0.689965
$805$	0	0
$806$	−24011.4	−1.04934
$807$	17614.6i	0.768356i
$808$	− 54512.7i	− 2.37345i
$809$	−21986.6	−0.955511	−0.477755	−	0.878493i	$-0.658550\pi$
$809$	−21986.6	−0.955511	−0.477755	+	0.878493i	$0.658550\pi$
$810$	0	0
$811$	11835.1	0.512437	0.256218	−	0.966619i	$-0.417523\pi$
$811$	11835.1	0.512437	0.256218	+	0.966619i	$0.417523\pi$
$812$	24289.4i	1.04974i
$813$	− 9920.79i	− 0.427967i
$814$	−9108.93	−0.392221
$815$	0	0
$816$	−14537.6	−0.623672
$817$	1045.01i	0.0447494i
$818$	− 2967.43i	− 0.126838i
$819$	−1212.81	−0.0517448
$820$	0	0
$821$	−7890.58	−0.335424	−0.167712	−	0.985836i	$-0.553638\pi$
$821$	−7890.58	−0.335424	−0.167712	+	0.985836i	$0.553638\pi$
$822$	− 28623.4i	− 1.21455i
$823$	1000.74i	0.0423861i	0.999775	+	0.0211930i	$0.00674646\pi$
$823$	1000.74i	0.0423861i	−0.999775	+	0.0211930i	$0.993254\pi$
$824$	21361.8	0.903123
$825$	0	0
$826$	11362.0	0.478611
$827$	− 22764.2i	− 0.957181i	−0.878038	−	0.478590i	$-0.841148\pi$
$827$	− 22764.2i	− 0.957181i	0.878038	−	0.478590i	$-0.158852\pi$
$828$	− 10297.2i	− 0.432188i
$829$	−39835.8	−1.66894	−0.834471	−	0.551051i	$-0.814227\pi$
$829$	−39835.8	−1.66894	−0.834471	+	0.551051i	$0.814227\pi$
$830$	0	0
$831$	−8584.10	−0.358338
$832$	8808.43i	0.367040i
$833$	18578.1i	0.772742i
$834$	25211.0	1.04674
$835$	0	0
$836$	1083.27	0.0448155
$837$	5306.96i	0.219158i
$838$	57469.7i	2.36904i
$839$	−23330.9	−0.960037	−0.480019	−	0.877258i	$-0.659370\pi$
$839$	−23330.9	−0.960037	−0.480019	+	0.877258i	$0.659370\pi$
$840$	0	0
$841$	45847.5	1.87984
$842$	− 75476.0i	− 3.08916i
$843$	− 8964.14i	− 0.366241i
$844$	−46928.1	−1.91390
$845$	0	0
$846$	6315.09	0.256640
$847$	− 663.535i	− 0.0269177i
$848$	− 27737.8i	− 1.12326i
$849$	−4363.79	−0.176402
$850$	0	0
$851$	−11403.3	−0.459341
$852$	7706.08i	0.309866i
$853$	22937.3i	0.920701i	0.887737	+	0.460351i	$0.152276\pi$
$853$	22937.3i	0.920701i	−0.887737	+	0.460351i	$0.847724\pi$
$854$	−18050.8	−0.723284
$855$	0	0
$856$	−47605.2	−1.90083
$857$	38472.3i	1.53348i	0.641961	+	0.766738i	$0.278121\pi$
$857$	38472.3i	1.53348i	−0.641961	+	0.766738i	$0.721879\pi$
$858$	4031.34i	0.160405i
$859$	22970.3	0.912384	0.456192	−	0.889881i	$-0.349213\pi$
$859$	22970.3	0.912384	0.456192	+	0.889881i	$0.349213\pi$
$860$	0	0
$861$	6977.00	0.276162
$862$	74084.5i	2.92730i
$863$	− 33587.6i	− 1.32484i	−0.749133	−	0.662420i	$-0.769530\pi$
$863$	− 33587.6i	− 1.32484i	0.749133	−	0.662420i	$-0.230470\pi$
$864$	−1599.70	−0.0629895
$865$	0	0
$866$	−11284.0	−0.442777
$867$	4165.08i	0.163153i
$868$	− 18014.3i	− 0.704430i
$869$	−4433.00	−0.173049
$870$	0	0
$871$	7709.12	0.299901
$872$	− 55228.3i	− 2.14480i
$873$	8637.71i	0.334871i
$874$	2005.25	0.0776073
$875$	0	0
$876$	7714.74	0.297554
$877$	704.130i	0.0271115i	0.999908	+	0.0135558i	$0.00431506\pi$
$877$	704.130i	0.0271115i	−0.999908	+	0.0135558i	$0.995685\pi$
$878$	− 66707.5i	− 2.56409i
$879$	25759.5	0.988450
$880$	0	0
$881$	9746.33	0.372715	0.186358	−	0.982482i	$-0.440332\pi$
$881$	9746.33	0.372715	0.186358	+	0.982482i	$0.440332\pi$
$882$	14000.7i	0.534501i
$883$	− 8774.24i	− 0.334402i	−0.985923	−	0.167201i	$-0.946527\pi$
$883$	− 8774.24i	− 0.334402i	0.985923	−	0.167201i	$-0.0534728\pi$
$884$	24383.0	0.927701
$885$	0	0
$886$	57206.4	2.16917
$887$	13505.1i	0.511224i	0.966779	+	0.255612i	$0.0822770\pi$
$887$	13505.1i	0.511224i	−0.966779	+	0.255612i	$0.917723\pi$
$888$	21645.6i	0.817993i
$889$	−12397.5	−0.467717
$890$	0	0
$891$	891.000	0.0335013
$892$	− 24118.6i	− 0.905324i
$893$	831.689i	0.0311662i
$894$	30988.9	1.15931
$895$	0	0
$896$	−12370.9	−0.461252
$897$	5046.75i	0.187855i
$898$	− 80552.8i	− 2.99341i
$899$	−52091.1	−1.93252
$900$	0	0
$901$	20175.1	0.745984
$902$	− 23191.3i	− 0.856083i
$903$	− 2917.64i	− 0.107523i
$904$	40994.6	1.50825
$905$	0	0
$906$	40433.4	1.48268
$907$	2523.23i	0.0923732i	0.998933	+	0.0461866i	$0.0147069\pi$
$907$	2523.23i	0.0923732i	−0.998933	+	0.0461866i	$0.985293\pi$
$908$	61148.4i	2.23489i
$909$	11326.7	0.413293
$910$	0	0
$911$	−32285.3	−1.17416	−0.587080	−	0.809529i	$-0.699723\pi$
$911$	−32285.3	−1.17416	−0.587080	+	0.809529i	$0.699723\pi$
$912$	− 1442.85i	− 0.0523877i
$913$	− 7175.45i	− 0.260101i
$914$	−24422.6	−0.883838
$915$	0	0
$916$	−28337.0	−1.02214
$917$	313.274i	0.0112816i
$918$	− 7968.65i	− 0.286497i
$919$	40956.5	1.47011	0.735055	−	0.678008i	$-0.237156\pi$
$919$	40956.5	1.47011	0.735055	+	0.678008i	$0.237156\pi$
$920$	0	0
$921$	1587.09	0.0567821
$922$	70943.6i	2.53406i
$923$	− 3776.83i	− 0.134687i
$924$	−3024.47	−0.107682
$925$	0	0
$926$	33090.8	1.17433
$927$	4438.58i	0.157262i
$928$	− 15702.1i	− 0.555437i
$929$	6728.41	0.237623	0.118812	−	0.992917i	$-0.462092\pi$
$929$	6728.41	0.237623	0.118812	+	0.992917i	$0.462092\pi$
$930$	0	0
$931$	−1843.88	−0.0649095
$932$	96639.1i	3.39648i
$933$	− 15024.0i	− 0.527185i
$934$	−96181.9	−3.36956
$935$	0	0
$936$	9579.69	0.334532
$937$	36377.8i	1.26831i	0.773205	+	0.634157i	$0.218653\pi$
$937$	36377.8i	1.26831i	−0.773205	+	0.634157i	$0.781347\pi$
$938$	8552.13i	0.297694i
$939$	−23575.6	−0.819340
$940$	0	0
$941$	25907.0	0.897496	0.448748	−	0.893658i	$-0.351870\pi$
$941$	25907.0	0.897496	0.448748	+	0.893658i	$0.351870\pi$
$942$	− 34916.4i	− 1.20768i
$943$	− 29032.7i	− 1.00258i
$944$	−34019.2	−1.17291
$945$	0	0
$946$	−9698.15	−0.333313
$947$	− 37096.3i	− 1.27293i	−0.771304	−	0.636466i	$-0.780395\pi$
$947$	− 37096.3i	− 1.27293i	0.771304	−	0.636466i	$-0.219605\pi$
$948$	20206.2i	0.692263i
$949$	−3781.07	−0.129335
$950$	0	0
$951$	23241.1	0.792476
$952$	14101.7i	0.480084i
$953$	− 15323.7i	− 0.520865i	−0.965492	−	0.260432i	$-0.916135\pi$
$953$	− 15323.7i	− 0.520865i	0.965492	−	0.260432i	$-0.0838651\pi$
$954$	15204.3	0.515992
$955$	0	0
$956$	−97540.4	−3.29988
$957$	8745.72i	0.295412i
$958$	48850.7i	1.64749i
$959$	−10524.8	−0.354395
$960$	0	0
$961$	8842.46	0.296817
$962$	− 20349.2i	− 0.682000i
$963$	− 9891.45i	− 0.330994i
$964$	16417.4	0.548516
$965$	0	0
$966$	−5598.62	−0.186473
$967$	53388.0i	1.77543i	0.460391	+	0.887716i	$0.347709\pi$
$967$	53388.0i	1.77543i	−0.460391	+	0.887716i	$0.652291\pi$
$968$	5241.10i	0.174024i
$969$	1049.46	0.0347921
$970$	0	0
$971$	−36325.7	−1.20056	−0.600281	−	0.799789i	$-0.704945\pi$
$971$	−36325.7	−1.20056	−0.600281	+	0.799789i	$0.704945\pi$
$972$	− 4061.28i	− 0.134018i
$973$	− 9270.07i	− 0.305432i
$974$	−51094.6	−1.68088
$975$	0	0
$976$	54046.4	1.77252
$977$	− 48608.7i	− 1.59174i	−0.605468	−	0.795870i	$-0.707014\pi$
$977$	− 48608.7i	− 1.59174i	0.605468	−	0.795870i	$-0.292986\pi$
$978$	52541.1i	1.71787i
$979$	−13494.4	−0.440535
$980$	0	0
$981$	11475.4	0.373477
$982$	− 56657.7i	− 1.84116i
$983$	31762.6i	1.03059i	0.857013	+	0.515295i	$0.172318\pi$
$983$	31762.6i	1.03059i	−0.857013	+	0.515295i	$0.827682\pi$
$984$	−55109.6	−1.78540
$985$	0	0
$986$	78217.2	2.52631
$987$	− 2322.06i	− 0.0748854i
$988$	2420.01i	0.0779258i
$989$	−12140.9	−0.390352
$990$	0	0
$991$	−5073.82	−0.162639	−0.0813195	−	0.996688i	$-0.525913\pi$
$991$	−5073.82	−0.162639	−0.0813195	+	0.996688i	$0.525913\pi$
$992$	11645.5i	0.372726i
$993$	4065.25i	0.129916i
$994$	4189.83	0.133696
$995$	0	0
$996$	−32706.5	−1.04051
$997$	− 24607.0i	− 0.781657i	−0.920464	−	0.390828i	$-0.872189\pi$
$997$	− 24607.0i	− 0.781657i	0.920464	−	0.390828i	$-0.127811\pi$
$998$	− 45631.8i	− 1.44734i
$999$	−4497.54	−0.142438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.k.199.2		6
5.2	odd	4		165.4.a.e.1.3	✓	3
5.3	odd	4		825.4.a.r.1.1		3
5.4	even	2	inner	825.4.c.k.199.5		6
15.2	even	4		495.4.a.k.1.1		3
15.8	even	4		2475.4.a.t.1.3		3
55.32	even	4		1815.4.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.e.1.3	✓	3	5.2	odd	4
495.4.a.k.1.1		3	15.2	even	4
825.4.a.r.1.1		3	5.3	odd	4
825.4.c.k.199.2		6	1.1	even	1	trivial
825.4.c.k.199.5		6	5.4	even	2	inner
1815.4.a.r.1.1		3	55.32	even	4
2475.4.a.t.1.3		3	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.97123i q^{2} +3.00000i q^{3} -16.7131 q^{4} +14.9137 q^{6} -5.48376i q^{7} +43.3148i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-4.97123i q^{2} +3.00000i q^{3} -16.7131 q^{4} +14.9137 q^{6} -5.48376i q^{7} +43.3148i q^{8} -9.00000 q^{9} +11.0000 q^{11} -50.1393i q^{12} +24.5738i q^{13} -27.2610 q^{14} +81.6231 q^{16} +59.3687i q^{17} +44.7411i q^{18} -5.89234 q^{19} +16.4513 q^{21} -54.6835i q^{22} -68.4570i q^{23} -129.945 q^{24} +122.162 q^{26} -27.0000i q^{27} +91.6506i q^{28} +265.022 q^{29} -196.554 q^{31} -59.2482i q^{32} +33.0000i q^{33} +295.135 q^{34} +150.418 q^{36} -166.575i q^{37} +29.2922i q^{38} -73.7214 q^{39} +424.101 q^{41} -81.7830i q^{42} -177.351i q^{43} -183.844 q^{44} -340.316 q^{46} -141.148i q^{47} +244.869i q^{48} +312.928 q^{49} -178.106 q^{51} -410.704i q^{52} -339.828i q^{53} -134.223 q^{54} +237.528 q^{56} -17.6770i q^{57} -1317.48i q^{58} -416.784 q^{59} +662.146 q^{61} +977.115i q^{62} +49.3538i q^{63} +358.448 q^{64} +164.051 q^{66} -313.713i q^{67} -992.235i q^{68} +205.371 q^{69} -153.693 q^{71} -389.834i q^{72} +153.866i q^{73} -828.084 q^{74} +98.4793 q^{76} -60.3213i q^{77} +366.486i q^{78} -403.000 q^{79} +81.0000 q^{81} -2108.30i q^{82} -652.313i q^{83} -274.952 q^{84} -881.650 q^{86} +795.065i q^{87} +476.463i q^{88} -1226.77 q^{89} +134.757 q^{91} +1144.13i q^{92} -589.662i q^{93} -701.677 q^{94} +177.745 q^{96} -959.746i q^{97} -1555.64i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 60 q^{4} - 12 q^{6} - 54 q^{9}+O(q^{10}) \) 6 * q - 60 * q^4 - 12 * q^6 - 54 * q^9	\( 6 q - 60 q^{4} - 12 q^{6} - 54 q^{9} + 66 q^{11} + 136 q^{14} + 356 q^{16} + 116 q^{19} + 60 q^{21} - 108 q^{24} - 240 q^{26} + 440 q^{29} + 496 q^{31} + 160 q^{34} + 540 q^{36} - 684 q^{39} + 312 q^{41} - 660 q^{44} - 2512 q^{46} - 558 q^{49} - 624 q^{51} + 108 q^{54} - 3288 q^{56} - 1096 q^{59} + 828 q^{61} + 116 q^{64} - 132 q^{66} - 720 q^{69} - 1824 q^{71} + 3224 q^{74} - 5504 q^{76} + 1084 q^{79} + 486 q^{81} - 4056 q^{84} - 3096 q^{86} - 1580 q^{89} - 1544 q^{91} + 848 q^{94} - 1548 q^{96} - 594 q^{99}+O(q^{100}) \) 6 * q - 60 * q^4 - 12 * q^6 - 54 * q^9 + 66 * q^11 + 136 * q^14 + 356 * q^16 + 116 * q^19 + 60 * q^21 - 108 * q^24 - 240 * q^26 + 440 * q^29 + 496 * q^31 + 160 * q^34 + 540 * q^36 - 684 * q^39 + 312 * q^41 - 660 * q^44 - 2512 * q^46 - 558 * q^49 - 624 * q^51 + 108 * q^54 - 3288 * q^56 - 1096 * q^59 + 828 * q^61 + 116 * q^64 - 132 * q^66 - 720 * q^69 - 1824 * q^71 + 3224 * q^74 - 5504 * q^76 + 1084 * q^79 + 486 * q^81 - 4056 * q^84 - 3096 * q^86 - 1580 * q^89 - 1544 * q^91 + 848 * q^94 - 1548 * q^96 - 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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