Properties

 Label 1200.4.f.b.49.1 Level $1200$ Weight $4$ Character 1200.49 Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (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: $$70.8022920069$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 49.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.49 Dual form 1200.4.f.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} -52.0000 q^{11} +22.0000i q^{13} +14.0000i q^{17} -20.0000 q^{19} -72.0000 q^{21} +168.000i q^{23} +27.0000i q^{27} -230.000 q^{29} +288.000 q^{31} +156.000i q^{33} +34.0000i q^{37} +66.0000 q^{39} +122.000 q^{41} +188.000i q^{43} +256.000i q^{47} -233.000 q^{49} +42.0000 q^{51} -338.000i q^{53} +60.0000i q^{57} +100.000 q^{59} +742.000 q^{61} +216.000i q^{63} -84.0000i q^{67} +504.000 q^{69} +328.000 q^{71} -38.0000i q^{73} +1248.00i q^{77} -240.000 q^{79} +81.0000 q^{81} -1212.00i q^{83} +690.000i q^{87} -330.000 q^{89} +528.000 q^{91} -864.000i q^{93} -866.000i q^{97} +468.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 104 * q^11 - 40 * q^19 - 144 * q^21 - 460 * q^29 + 576 * q^31 + 132 * q^39 + 244 * q^41 - 466 * q^49 + 84 * q^51 + 200 * q^59 + 1484 * q^61 + 1008 * q^69 + 656 * q^71 - 480 * q^79 + 162 * q^81 - 660 * q^89 + 1056 * q^91 + 936 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 3.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 24.0000i − 1.29588i −0.761692 0.647939i $$-0.775631\pi$$
0.761692 0.647939i $$-0.224369\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −52.0000 −1.42533 −0.712663 0.701506i $$-0.752511\pi$$
−0.712663 + 0.701506i $$0.752511\pi$$
$$12$$ 0 0
$$13$$ 22.0000i 0.469362i 0.972072 + 0.234681i $$0.0754045\pi$$
−0.972072 + 0.234681i $$0.924595\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 14.0000i 0.199735i 0.995001 + 0.0998676i $$0.0318419\pi$$
−0.995001 + 0.0998676i $$0.968158\pi$$
$$18$$ 0 0
$$19$$ −20.0000 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ 0 0
$$21$$ −72.0000 −0.748176
$$22$$ 0 0
$$23$$ 168.000i 1.52306i 0.648129 + 0.761531i $$0.275552\pi$$
−0.648129 + 0.761531i $$0.724448\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 27.0000i 0.192450i
$$28$$ 0 0
$$29$$ −230.000 −1.47276 −0.736378 0.676570i $$-0.763465\pi$$
−0.736378 + 0.676570i $$0.763465\pi$$
$$30$$ 0 0
$$31$$ 288.000 1.66859 0.834296 0.551317i $$-0.185875\pi$$
0.834296 + 0.551317i $$0.185875\pi$$
$$32$$ 0 0
$$33$$ 156.000i 0.822913i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 34.0000i 0.151069i 0.997143 + 0.0755347i $$0.0240664\pi$$
−0.997143 + 0.0755347i $$0.975934\pi$$
$$38$$ 0 0
$$39$$ 66.0000 0.270986
$$40$$ 0 0
$$41$$ 122.000 0.464712 0.232356 0.972631i $$-0.425357\pi$$
0.232356 + 0.972631i $$0.425357\pi$$
$$42$$ 0 0
$$43$$ 188.000i 0.666738i 0.942796 + 0.333369i $$0.108185\pi$$
−0.942796 + 0.333369i $$0.891815\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 256.000i 0.794499i 0.917711 + 0.397249i $$0.130035\pi$$
−0.917711 + 0.397249i $$0.869965\pi$$
$$48$$ 0 0
$$49$$ −233.000 −0.679300
$$50$$ 0 0
$$51$$ 42.0000 0.115317
$$52$$ 0 0
$$53$$ − 338.000i − 0.875998i −0.898976 0.437999i $$-0.855687\pi$$
0.898976 0.437999i $$-0.144313\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 60.0000i 0.139424i
$$58$$ 0 0
$$59$$ 100.000 0.220659 0.110330 0.993895i $$-0.464809\pi$$
0.110330 + 0.993895i $$0.464809\pi$$
$$60$$ 0 0
$$61$$ 742.000 1.55743 0.778716 0.627376i $$-0.215871\pi$$
0.778716 + 0.627376i $$0.215871\pi$$
$$62$$ 0 0
$$63$$ 216.000i 0.431959i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 84.0000i − 0.153168i −0.997063 0.0765838i $$-0.975599\pi$$
0.997063 0.0765838i $$-0.0244013\pi$$
$$68$$ 0 0
$$69$$ 504.000 0.879340
$$70$$ 0 0
$$71$$ 328.000 0.548260 0.274130 0.961693i $$-0.411610\pi$$
0.274130 + 0.961693i $$0.411610\pi$$
$$72$$ 0 0
$$73$$ − 38.0000i − 0.0609255i −0.999536 0.0304628i $$-0.990302\pi$$
0.999536 0.0304628i $$-0.00969810\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1248.00i 1.84705i
$$78$$ 0 0
$$79$$ −240.000 −0.341799 −0.170899 0.985288i $$-0.554667\pi$$
−0.170899 + 0.985288i $$0.554667\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ − 1212.00i − 1.60282i −0.598114 0.801411i $$-0.704083\pi$$
0.598114 0.801411i $$-0.295917\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 690.000i 0.850296i
$$88$$ 0 0
$$89$$ −330.000 −0.393033 −0.196516 0.980501i $$-0.562963\pi$$
−0.196516 + 0.980501i $$0.562963\pi$$
$$90$$ 0 0
$$91$$ 528.000 0.608236
$$92$$ 0 0
$$93$$ − 864.000i − 0.963362i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 866.000i − 0.906484i −0.891387 0.453242i $$-0.850267\pi$$
0.891387 0.453242i $$-0.149733\pi$$
$$98$$ 0 0
$$99$$ 468.000 0.475109
$$100$$ 0 0
$$101$$ −1218.00 −1.19996 −0.599978 0.800017i $$-0.704824\pi$$
−0.599978 + 0.800017i $$0.704824\pi$$
$$102$$ 0 0
$$103$$ 88.0000i 0.0841835i 0.999114 + 0.0420917i $$0.0134022\pi$$
−0.999114 + 0.0420917i $$0.986598\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 36.0000i 0.0325257i 0.999868 + 0.0162629i $$0.00517686\pi$$
−0.999868 + 0.0162629i $$0.994823\pi$$
$$108$$ 0 0
$$109$$ 970.000 0.852378 0.426189 0.904634i $$-0.359856\pi$$
0.426189 + 0.904634i $$0.359856\pi$$
$$110$$ 0 0
$$111$$ 102.000 0.0872199
$$112$$ 0 0
$$113$$ 1042.00i 0.867461i 0.901043 + 0.433731i $$0.142803\pi$$
−0.901043 + 0.433731i $$0.857197\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 198.000i − 0.156454i
$$118$$ 0 0
$$119$$ 336.000 0.258833
$$120$$ 0 0
$$121$$ 1373.00 1.03156
$$122$$ 0 0
$$123$$ − 366.000i − 0.268302i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1936.00i 1.35269i 0.736583 + 0.676347i $$0.236438\pi$$
−0.736583 + 0.676347i $$0.763562\pi$$
$$128$$ 0 0
$$129$$ 564.000 0.384941
$$130$$ 0 0
$$131$$ −732.000 −0.488207 −0.244104 0.969749i $$-0.578494\pi$$
−0.244104 + 0.969749i $$0.578494\pi$$
$$132$$ 0 0
$$133$$ 480.000i 0.312942i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2214.00i 1.38069i 0.723479 + 0.690346i $$0.242542\pi$$
−0.723479 + 0.690346i $$0.757458\pi$$
$$138$$ 0 0
$$139$$ 20.0000 0.0122042 0.00610208 0.999981i $$-0.498058\pi$$
0.00610208 + 0.999981i $$0.498058\pi$$
$$140$$ 0 0
$$141$$ 768.000 0.458704
$$142$$ 0 0
$$143$$ − 1144.00i − 0.668994i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 699.000i 0.392194i
$$148$$ 0 0
$$149$$ 1330.00 0.731261 0.365630 0.930760i $$-0.380853\pi$$
0.365630 + 0.930760i $$0.380853\pi$$
$$150$$ 0 0
$$151$$ 1208.00 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ − 126.000i − 0.0665784i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3514.00i 1.78629i 0.449768 + 0.893146i $$0.351507\pi$$
−0.449768 + 0.893146i $$0.648493\pi$$
$$158$$ 0 0
$$159$$ −1014.00 −0.505757
$$160$$ 0 0
$$161$$ 4032.00 1.97370
$$162$$ 0 0
$$163$$ 2068.00i 0.993732i 0.867827 + 0.496866i $$0.165516\pi$$
−0.867827 + 0.496866i $$0.834484\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 24.0000i − 0.0111208i −0.999985 0.00556041i $$-0.998230\pi$$
0.999985 0.00556041i $$-0.00176994\pi$$
$$168$$ 0 0
$$169$$ 1713.00 0.779700
$$170$$ 0 0
$$171$$ 180.000 0.0804967
$$172$$ 0 0
$$173$$ − 618.000i − 0.271593i −0.990737 0.135797i $$-0.956641\pi$$
0.990737 0.135797i $$-0.0433594\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 300.000i − 0.127398i
$$178$$ 0 0
$$179$$ 3340.00 1.39466 0.697328 0.716752i $$-0.254372\pi$$
0.697328 + 0.716752i $$0.254372\pi$$
$$180$$ 0 0
$$181$$ −178.000 −0.0730974 −0.0365487 0.999332i $$-0.511636\pi$$
−0.0365487 + 0.999332i $$0.511636\pi$$
$$182$$ 0 0
$$183$$ − 2226.00i − 0.899184i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 728.000i − 0.284688i
$$188$$ 0 0
$$189$$ 648.000 0.249392
$$190$$ 0 0
$$191$$ 1888.00 0.715240 0.357620 0.933867i $$-0.383588\pi$$
0.357620 + 0.933867i $$0.383588\pi$$
$$192$$ 0 0
$$193$$ 1922.00i 0.716832i 0.933562 + 0.358416i $$0.116683\pi$$
−0.933562 + 0.358416i $$0.883317\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 2526.00i − 0.913554i −0.889581 0.456777i $$-0.849004\pi$$
0.889581 0.456777i $$-0.150996\pi$$
$$198$$ 0 0
$$199$$ −1160.00 −0.413217 −0.206609 0.978424i $$-0.566243\pi$$
−0.206609 + 0.978424i $$0.566243\pi$$
$$200$$ 0 0
$$201$$ −252.000 −0.0884314
$$202$$ 0 0
$$203$$ 5520.00i 1.90851i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 1512.00i − 0.507687i
$$208$$ 0 0
$$209$$ 1040.00 0.344202
$$210$$ 0 0
$$211$$ 4468.00 1.45777 0.728886 0.684635i $$-0.240039\pi$$
0.728886 + 0.684635i $$0.240039\pi$$
$$212$$ 0 0
$$213$$ − 984.000i − 0.316538i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 6912.00i − 2.16229i
$$218$$ 0 0
$$219$$ −114.000 −0.0351754
$$220$$ 0 0
$$221$$ −308.000 −0.0937481
$$222$$ 0 0
$$223$$ − 6032.00i − 1.81136i −0.423965 0.905678i $$-0.639362\pi$$
0.423965 0.905678i $$-0.360638\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2636.00i 0.770738i 0.922763 + 0.385369i $$0.125926\pi$$
−0.922763 + 0.385369i $$0.874074\pi$$
$$228$$ 0 0
$$229$$ −4830.00 −1.39378 −0.696889 0.717179i $$-0.745433\pi$$
−0.696889 + 0.717179i $$0.745433\pi$$
$$230$$ 0 0
$$231$$ 3744.00 1.06639
$$232$$ 0 0
$$233$$ 2682.00i 0.754093i 0.926194 + 0.377046i $$0.123060\pi$$
−0.926194 + 0.377046i $$0.876940\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 720.000i 0.197338i
$$238$$ 0 0
$$239$$ 2320.00 0.627901 0.313950 0.949439i $$-0.398347\pi$$
0.313950 + 0.949439i $$0.398347\pi$$
$$240$$ 0 0
$$241$$ 2002.00 0.535104 0.267552 0.963543i $$-0.413785\pi$$
0.267552 + 0.963543i $$0.413785\pi$$
$$242$$ 0 0
$$243$$ − 243.000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 440.000i − 0.113346i
$$248$$ 0 0
$$249$$ −3636.00 −0.925390
$$250$$ 0 0
$$251$$ −132.000 −0.0331943 −0.0165971 0.999862i $$-0.505283\pi$$
−0.0165971 + 0.999862i $$0.505283\pi$$
$$252$$ 0 0
$$253$$ − 8736.00i − 2.17086i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 7614.00i 1.84805i 0.382335 + 0.924024i $$0.375120\pi$$
−0.382335 + 0.924024i $$0.624880\pi$$
$$258$$ 0 0
$$259$$ 816.000 0.195767
$$260$$ 0 0
$$261$$ 2070.00 0.490919
$$262$$ 0 0
$$263$$ 4888.00i 1.14603i 0.819543 + 0.573017i $$0.194227\pi$$
−0.819543 + 0.573017i $$0.805773\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 990.000i 0.226918i
$$268$$ 0 0
$$269$$ −1270.00 −0.287856 −0.143928 0.989588i $$-0.545973\pi$$
−0.143928 + 0.989588i $$0.545973\pi$$
$$270$$ 0 0
$$271$$ −1072.00 −0.240293 −0.120146 0.992756i $$-0.538336\pi$$
−0.120146 + 0.992756i $$0.538336\pi$$
$$272$$ 0 0
$$273$$ − 1584.00i − 0.351165i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 5394.00i 1.17001i 0.811028 + 0.585007i $$0.198908\pi$$
−0.811028 + 0.585007i $$0.801092\pi$$
$$278$$ 0 0
$$279$$ −2592.00 −0.556197
$$280$$ 0 0
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ 0 0
$$283$$ − 2772.00i − 0.582255i −0.956684 0.291128i $$-0.905970\pi$$
0.956684 0.291128i $$-0.0940305\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 2928.00i − 0.602210i
$$288$$ 0 0
$$289$$ 4717.00 0.960106
$$290$$ 0 0
$$291$$ −2598.00 −0.523359
$$292$$ 0 0
$$293$$ 4542.00i 0.905619i 0.891607 + 0.452810i $$0.149578\pi$$
−0.891607 + 0.452810i $$0.850422\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 1404.00i − 0.274304i
$$298$$ 0 0
$$299$$ −3696.00 −0.714867
$$300$$ 0 0
$$301$$ 4512.00 0.864011
$$302$$ 0 0
$$303$$ 3654.00i 0.692795i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5116.00i 0.951093i 0.879691 + 0.475546i $$0.157750\pi$$
−0.879691 + 0.475546i $$0.842250\pi$$
$$308$$ 0 0
$$309$$ 264.000 0.0486034
$$310$$ 0 0
$$311$$ 2808.00 0.511984 0.255992 0.966679i $$-0.417598\pi$$
0.255992 + 0.966679i $$0.417598\pi$$
$$312$$ 0 0
$$313$$ − 7318.00i − 1.32153i −0.750594 0.660763i $$-0.770233\pi$$
0.750594 0.660763i $$-0.229767\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2246.00i − 0.397943i −0.980005 0.198971i $$-0.936240\pi$$
0.980005 0.198971i $$-0.0637601\pi$$
$$318$$ 0 0
$$319$$ 11960.0 2.09916
$$320$$ 0 0
$$321$$ 108.000 0.0187787
$$322$$ 0 0
$$323$$ − 280.000i − 0.0482341i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2910.00i − 0.492120i
$$328$$ 0 0
$$329$$ 6144.00 1.02957
$$330$$ 0 0
$$331$$ −1332.00 −0.221188 −0.110594 0.993866i $$-0.535275\pi$$
−0.110594 + 0.993866i $$0.535275\pi$$
$$332$$ 0 0
$$333$$ − 306.000i − 0.0503564i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 11534.0i 1.86438i 0.361966 + 0.932191i $$0.382106\pi$$
−0.361966 + 0.932191i $$0.617894\pi$$
$$338$$ 0 0
$$339$$ 3126.00 0.500829
$$340$$ 0 0
$$341$$ −14976.0 −2.37829
$$342$$ 0 0
$$343$$ − 2640.00i − 0.415588i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 11956.0i 1.84966i 0.380382 + 0.924830i $$0.375793\pi$$
−0.380382 + 0.924830i $$0.624207\pi$$
$$348$$ 0 0
$$349$$ −4870.00 −0.746949 −0.373474 0.927640i $$-0.621834\pi$$
−0.373474 + 0.927640i $$0.621834\pi$$
$$350$$ 0 0
$$351$$ −594.000 −0.0903287
$$352$$ 0 0
$$353$$ 10722.0i 1.61664i 0.588742 + 0.808321i $$0.299623\pi$$
−0.588742 + 0.808321i $$0.700377\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 1008.00i − 0.149437i
$$358$$ 0 0
$$359$$ 120.000 0.0176417 0.00882083 0.999961i $$-0.497192\pi$$
0.00882083 + 0.999961i $$0.497192\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 0 0
$$363$$ − 4119.00i − 0.595569i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3936.00i 0.559830i 0.960025 + 0.279915i $$0.0903063\pi$$
−0.960025 + 0.279915i $$0.909694\pi$$
$$368$$ 0 0
$$369$$ −1098.00 −0.154904
$$370$$ 0 0
$$371$$ −8112.00 −1.13519
$$372$$ 0 0
$$373$$ 3022.00i 0.419499i 0.977755 + 0.209750i $$0.0672649\pi$$
−0.977755 + 0.209750i $$0.932735\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 5060.00i − 0.691255i
$$378$$ 0 0
$$379$$ −13340.0 −1.80799 −0.903997 0.427539i $$-0.859381\pi$$
−0.903997 + 0.427539i $$0.859381\pi$$
$$380$$ 0 0
$$381$$ 5808.00 0.780979
$$382$$ 0 0
$$383$$ 1008.00i 0.134481i 0.997737 + 0.0672407i $$0.0214195\pi$$
−0.997737 + 0.0672407i $$0.978580\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 1692.00i − 0.222246i
$$388$$ 0 0
$$389$$ −9630.00 −1.25517 −0.627584 0.778549i $$-0.715956\pi$$
−0.627584 + 0.778549i $$0.715956\pi$$
$$390$$ 0 0
$$391$$ −2352.00 −0.304209
$$392$$ 0 0
$$393$$ 2196.00i 0.281867i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7126.00i − 0.900866i −0.892810 0.450433i $$-0.851270\pi$$
0.892810 0.450433i $$-0.148730\pi$$
$$398$$ 0 0
$$399$$ 1440.00 0.180677
$$400$$ 0 0
$$401$$ −8718.00 −1.08568 −0.542838 0.839837i $$-0.682650\pi$$
−0.542838 + 0.839837i $$0.682650\pi$$
$$402$$ 0 0
$$403$$ 6336.00i 0.783173i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 1768.00i − 0.215323i
$$408$$ 0 0
$$409$$ 10870.0 1.31415 0.657074 0.753826i $$-0.271794\pi$$
0.657074 + 0.753826i $$0.271794\pi$$
$$410$$ 0 0
$$411$$ 6642.00 0.797143
$$412$$ 0 0
$$413$$ − 2400.00i − 0.285947i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 60.0000i − 0.00704607i
$$418$$ 0 0
$$419$$ −9700.00 −1.13097 −0.565484 0.824759i $$-0.691311\pi$$
−0.565484 + 0.824759i $$0.691311\pi$$
$$420$$ 0 0
$$421$$ 862.000 0.0997893 0.0498947 0.998754i $$-0.484111\pi$$
0.0498947 + 0.998754i $$0.484111\pi$$
$$422$$ 0 0
$$423$$ − 2304.00i − 0.264833i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 17808.0i − 2.01824i
$$428$$ 0 0
$$429$$ −3432.00 −0.386244
$$430$$ 0 0
$$431$$ −15792.0 −1.76490 −0.882452 0.470402i $$-0.844109\pi$$
−0.882452 + 0.470402i $$0.844109\pi$$
$$432$$ 0 0
$$433$$ 11602.0i 1.28766i 0.765169 + 0.643830i $$0.222655\pi$$
−0.765169 + 0.643830i $$0.777345\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3360.00i − 0.367805i
$$438$$ 0 0
$$439$$ −440.000 −0.0478361 −0.0239181 0.999714i $$-0.507614\pi$$
−0.0239181 + 0.999714i $$0.507614\pi$$
$$440$$ 0 0
$$441$$ 2097.00 0.226433
$$442$$ 0 0
$$443$$ 10188.0i 1.09266i 0.837571 + 0.546328i $$0.183975\pi$$
−0.837571 + 0.546328i $$0.816025\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 3990.00i − 0.422194i
$$448$$ 0 0
$$449$$ 13310.0 1.39897 0.699485 0.714647i $$-0.253413\pi$$
0.699485 + 0.714647i $$0.253413\pi$$
$$450$$ 0 0
$$451$$ −6344.00 −0.662367
$$452$$ 0 0
$$453$$ − 3624.00i − 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 3226.00i − 0.330210i −0.986276 0.165105i $$-0.947204\pi$$
0.986276 0.165105i $$-0.0527963\pi$$
$$458$$ 0 0
$$459$$ −378.000 −0.0384391
$$460$$ 0 0
$$461$$ 6582.00 0.664977 0.332488 0.943107i $$-0.392112\pi$$
0.332488 + 0.943107i $$0.392112\pi$$
$$462$$ 0 0
$$463$$ − 15072.0i − 1.51286i −0.654073 0.756431i $$-0.726941\pi$$
0.654073 0.756431i $$-0.273059\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 476.000i 0.0471663i 0.999722 + 0.0235831i $$0.00750744\pi$$
−0.999722 + 0.0235831i $$0.992493\pi$$
$$468$$ 0 0
$$469$$ −2016.00 −0.198487
$$470$$ 0 0
$$471$$ 10542.0 1.03132
$$472$$ 0 0
$$473$$ − 9776.00i − 0.950319i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 3042.00i 0.291999i
$$478$$ 0 0
$$479$$ −19680.0 −1.87725 −0.938624 0.344941i $$-0.887899\pi$$
−0.938624 + 0.344941i $$0.887899\pi$$
$$480$$ 0 0
$$481$$ −748.000 −0.0709062
$$482$$ 0 0
$$483$$ − 12096.0i − 1.13952i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 5944.00i − 0.553077i −0.961003 0.276538i $$-0.910813\pi$$
0.961003 0.276538i $$-0.0891873\pi$$
$$488$$ 0 0
$$489$$ 6204.00 0.573731
$$490$$ 0 0
$$491$$ −10772.0 −0.990089 −0.495044 0.868868i $$-0.664848\pi$$
−0.495044 + 0.868868i $$0.664848\pi$$
$$492$$ 0 0
$$493$$ − 3220.00i − 0.294161i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 7872.00i − 0.710478i
$$498$$ 0 0
$$499$$ 8140.00 0.730253 0.365127 0.930958i $$-0.381026\pi$$
0.365127 + 0.930958i $$0.381026\pi$$
$$500$$ 0 0
$$501$$ −72.0000 −0.00642060
$$502$$ 0 0
$$503$$ 13768.0i 1.22045i 0.792229 + 0.610223i $$0.208920\pi$$
−0.792229 + 0.610223i $$0.791080\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 5139.00i − 0.450160i
$$508$$ 0 0
$$509$$ −22150.0 −1.92884 −0.964422 0.264368i $$-0.914837\pi$$
−0.964422 + 0.264368i $$0.914837\pi$$
$$510$$ 0 0
$$511$$ −912.000 −0.0789521
$$512$$ 0 0
$$513$$ − 540.000i − 0.0464748i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 13312.0i − 1.13242i
$$518$$ 0 0
$$519$$ −1854.00 −0.156805
$$520$$ 0 0
$$521$$ 1562.00 0.131348 0.0656741 0.997841i $$-0.479080\pi$$
0.0656741 + 0.997841i $$0.479080\pi$$
$$522$$ 0 0
$$523$$ 668.000i 0.0558501i 0.999610 + 0.0279250i $$0.00888997\pi$$
−0.999610 + 0.0279250i $$0.991110\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4032.00i 0.333276i
$$528$$ 0 0
$$529$$ −16057.0 −1.31972
$$530$$ 0 0
$$531$$ −900.000 −0.0735531
$$532$$ 0 0
$$533$$ 2684.00i 0.218118i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 10020.0i − 0.805205i
$$538$$ 0 0
$$539$$ 12116.0 0.968225
$$540$$ 0 0
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ 0 0
$$543$$ 534.000i 0.0422028i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 10484.0i − 0.819494i −0.912199 0.409747i $$-0.865617\pi$$
0.912199 0.409747i $$-0.134383\pi$$
$$548$$ 0 0
$$549$$ −6678.00 −0.519144
$$550$$ 0 0
$$551$$ 4600.00 0.355656
$$552$$ 0 0
$$553$$ 5760.00i 0.442930i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 3606.00i − 0.274311i −0.990550 0.137155i $$-0.956204\pi$$
0.990550 0.137155i $$-0.0437960\pi$$
$$558$$ 0 0
$$559$$ −4136.00 −0.312941
$$560$$ 0 0
$$561$$ −2184.00 −0.164365
$$562$$ 0 0
$$563$$ − 12252.0i − 0.917159i −0.888654 0.458579i $$-0.848359\pi$$
0.888654 0.458579i $$-0.151641\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 1944.00i − 0.143986i
$$568$$ 0 0
$$569$$ 14550.0 1.07200 0.536000 0.844218i $$-0.319935\pi$$
0.536000 + 0.844218i $$0.319935\pi$$
$$570$$ 0 0
$$571$$ 25468.0 1.86655 0.933277 0.359157i $$-0.116936\pi$$
0.933277 + 0.359157i $$0.116936\pi$$
$$572$$ 0 0
$$573$$ − 5664.00i − 0.412944i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 12866.0i − 0.928282i −0.885761 0.464141i $$-0.846363\pi$$
0.885761 0.464141i $$-0.153637\pi$$
$$578$$ 0 0
$$579$$ 5766.00 0.413863
$$580$$ 0 0
$$581$$ −29088.0 −2.07706
$$582$$ 0 0
$$583$$ 17576.0i 1.24858i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 14844.0i − 1.04374i −0.853024 0.521872i $$-0.825234\pi$$
0.853024 0.521872i $$-0.174766\pi$$
$$588$$ 0 0
$$589$$ −5760.00 −0.402948
$$590$$ 0 0
$$591$$ −7578.00 −0.527440
$$592$$ 0 0
$$593$$ 20402.0i 1.41283i 0.707797 + 0.706416i $$0.249689\pi$$
−0.707797 + 0.706416i $$0.750311\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3480.00i 0.238571i
$$598$$ 0 0
$$599$$ 10760.0 0.733959 0.366980 0.930229i $$-0.380392\pi$$
0.366980 + 0.930229i $$0.380392\pi$$
$$600$$ 0 0
$$601$$ 14282.0 0.969343 0.484671 0.874696i $$-0.338939\pi$$
0.484671 + 0.874696i $$0.338939\pi$$
$$602$$ 0 0
$$603$$ 756.000i 0.0510559i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 11056.0i 0.739290i 0.929173 + 0.369645i $$0.120521\pi$$
−0.929173 + 0.369645i $$0.879479\pi$$
$$608$$ 0 0
$$609$$ 16560.0 1.10188
$$610$$ 0 0
$$611$$ −5632.00 −0.372907
$$612$$ 0 0
$$613$$ − 16418.0i − 1.08176i −0.841101 0.540878i $$-0.818092\pi$$
0.841101 0.540878i $$-0.181908\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 10374.0i 0.676891i 0.940986 + 0.338445i $$0.109901\pi$$
−0.940986 + 0.338445i $$0.890099\pi$$
$$618$$ 0 0
$$619$$ −5260.00 −0.341546 −0.170773 0.985310i $$-0.554627\pi$$
−0.170773 + 0.985310i $$0.554627\pi$$
$$620$$ 0 0
$$621$$ −4536.00 −0.293113
$$622$$ 0 0
$$623$$ 7920.00i 0.509323i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 3120.00i − 0.198725i
$$628$$ 0 0
$$629$$ −476.000 −0.0301739
$$630$$ 0 0
$$631$$ −21352.0 −1.34708 −0.673542 0.739149i $$-0.735228\pi$$
−0.673542 + 0.739149i $$0.735228\pi$$
$$632$$ 0 0
$$633$$ − 13404.0i − 0.841645i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 5126.00i − 0.318838i
$$638$$ 0 0
$$639$$ −2952.00 −0.182753
$$640$$ 0 0
$$641$$ −29118.0 −1.79422 −0.897108 0.441812i $$-0.854336\pi$$
−0.897108 + 0.441812i $$0.854336\pi$$
$$642$$ 0 0
$$643$$ − 5772.00i − 0.354005i −0.984210 0.177003i $$-0.943360\pi$$
0.984210 0.177003i $$-0.0566401\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 14264.0i − 0.866732i −0.901218 0.433366i $$-0.857326\pi$$
0.901218 0.433366i $$-0.142674\pi$$
$$648$$ 0 0
$$649$$ −5200.00 −0.314511
$$650$$ 0 0
$$651$$ −20736.0 −1.24840
$$652$$ 0 0
$$653$$ 6902.00i 0.413623i 0.978381 + 0.206812i $$0.0663088\pi$$
−0.978381 + 0.206812i $$0.933691\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 342.000i 0.0203085i
$$658$$ 0 0
$$659$$ 20140.0 1.19051 0.595253 0.803539i $$-0.297052\pi$$
0.595253 + 0.803539i $$0.297052\pi$$
$$660$$ 0 0
$$661$$ −3218.00 −0.189358 −0.0946790 0.995508i $$-0.530182\pi$$
−0.0946790 + 0.995508i $$0.530182\pi$$
$$662$$ 0 0
$$663$$ 924.000i 0.0541255i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 38640.0i − 2.24310i
$$668$$ 0 0
$$669$$ −18096.0 −1.04579
$$670$$ 0 0
$$671$$ −38584.0 −2.21985
$$672$$ 0 0
$$673$$ − 7518.00i − 0.430606i −0.976547 0.215303i $$-0.930926\pi$$
0.976547 0.215303i $$-0.0690739\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18114.0i 1.02833i 0.857692 + 0.514164i $$0.171898\pi$$
−0.857692 + 0.514164i $$0.828102\pi$$
$$678$$ 0 0
$$679$$ −20784.0 −1.17469
$$680$$ 0 0
$$681$$ 7908.00 0.444986
$$682$$ 0 0
$$683$$ 23868.0i 1.33716i 0.743638 + 0.668582i $$0.233099\pi$$
−0.743638 + 0.668582i $$0.766901\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14490.0i 0.804699i
$$688$$ 0 0
$$689$$ 7436.00 0.411160
$$690$$ 0 0
$$691$$ −172.000 −0.00946916 −0.00473458 0.999989i $$-0.501507\pi$$
−0.00473458 + 0.999989i $$0.501507\pi$$
$$692$$ 0 0
$$693$$ − 11232.0i − 0.615683i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1708.00i 0.0928194i
$$698$$ 0 0
$$699$$ 8046.00 0.435376
$$700$$ 0 0
$$701$$ −22138.0 −1.19278 −0.596391 0.802694i $$-0.703399\pi$$
−0.596391 + 0.802694i $$0.703399\pi$$
$$702$$ 0 0
$$703$$ − 680.000i − 0.0364818i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 29232.0i 1.55500i
$$708$$ 0 0
$$709$$ −3070.00 −0.162618 −0.0813091 0.996689i $$-0.525910\pi$$
−0.0813091 + 0.996689i $$0.525910\pi$$
$$710$$ 0 0
$$711$$ 2160.00 0.113933
$$712$$ 0 0
$$713$$ 48384.0i 2.54137i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 6960.00i − 0.362519i
$$718$$ 0 0
$$719$$ 15600.0 0.809154 0.404577 0.914504i $$-0.367419\pi$$
0.404577 + 0.914504i $$0.367419\pi$$
$$720$$ 0 0
$$721$$ 2112.00 0.109092
$$722$$ 0 0
$$723$$ − 6006.00i − 0.308943i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 20696.0i 1.05581i 0.849304 + 0.527904i $$0.177022\pi$$
−0.849304 + 0.527904i $$0.822978\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −2632.00 −0.133171
$$732$$ 0 0
$$733$$ − 30778.0i − 1.55090i −0.631408 0.775451i $$-0.717522\pi$$
0.631408 0.775451i $$-0.282478\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4368.00i 0.218314i
$$738$$ 0 0
$$739$$ 11740.0 0.584388 0.292194 0.956359i $$-0.405615\pi$$
0.292194 + 0.956359i $$0.405615\pi$$
$$740$$ 0 0
$$741$$ −1320.00 −0.0654405
$$742$$ 0 0
$$743$$ − 2632.00i − 0.129958i −0.997887 0.0649789i $$-0.979302\pi$$
0.997887 0.0649789i $$-0.0206980\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 10908.0i 0.534274i
$$748$$ 0 0
$$749$$ 864.000 0.0421494
$$750$$ 0 0
$$751$$ 20528.0 0.997440 0.498720 0.866763i $$-0.333804\pi$$
0.498720 + 0.866763i $$0.333804\pi$$
$$752$$ 0 0
$$753$$ 396.000i 0.0191647i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 21646.0i − 1.03928i −0.854384 0.519642i $$-0.826066\pi$$
0.854384 0.519642i $$-0.173934\pi$$
$$758$$ 0 0
$$759$$ −26208.0 −1.25335
$$760$$ 0 0
$$761$$ 18282.0 0.870857 0.435428 0.900223i $$-0.356597\pi$$
0.435428 + 0.900223i $$0.356597\pi$$
$$762$$ 0 0
$$763$$ − 23280.0i − 1.10458i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2200.00i 0.103569i
$$768$$ 0 0
$$769$$ 24190.0 1.13435 0.567174 0.823598i $$-0.308037\pi$$
0.567174 + 0.823598i $$0.308037\pi$$
$$770$$ 0 0
$$771$$ 22842.0 1.06697
$$772$$ 0 0
$$773$$ − 25698.0i − 1.19572i −0.801600 0.597861i $$-0.796018\pi$$
0.801600 0.597861i $$-0.203982\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 2448.00i − 0.113026i
$$778$$ 0 0
$$779$$ −2440.00 −0.112223
$$780$$ 0 0
$$781$$ −17056.0 −0.781449
$$782$$ 0 0
$$783$$ − 6210.00i − 0.283432i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 33436.0i 1.51444i 0.653160 + 0.757220i $$0.273443\pi$$
−0.653160 + 0.757220i $$0.726557\pi$$
$$788$$ 0 0
$$789$$ 14664.0 0.661663
$$790$$ 0 0
$$791$$ 25008.0 1.12412
$$792$$ 0 0
$$793$$ 16324.0i 0.730999i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 37594.0i 1.67083i 0.549623 + 0.835413i $$0.314771\pi$$
−0.549623 + 0.835413i $$0.685229\pi$$
$$798$$ 0 0
$$799$$ −3584.00 −0.158689
$$800$$ 0 0
$$801$$ 2970.00 0.131011
$$802$$ 0 0
$$803$$ 1976.00i 0.0868388i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 3810.00i 0.166194i
$$808$$ 0 0
$$809$$ −4730.00 −0.205560 −0.102780 0.994704i $$-0.532774\pi$$
−0.102780 + 0.994704i $$0.532774\pi$$
$$810$$ 0 0
$$811$$ 8748.00 0.378772 0.189386 0.981903i $$-0.439350\pi$$
0.189386 + 0.981903i $$0.439350\pi$$
$$812$$ 0 0
$$813$$ 3216.00i 0.138733i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 3760.00i − 0.161011i
$$818$$ 0 0
$$819$$ −4752.00 −0.202745
$$820$$ 0 0
$$821$$ 44142.0 1.87645 0.938226 0.346024i $$-0.112468\pi$$
0.938226 + 0.346024i $$0.112468\pi$$
$$822$$ 0 0
$$823$$ − 3992.00i − 0.169079i −0.996420 0.0845397i $$-0.973058\pi$$
0.996420 0.0845397i $$-0.0269420\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 14444.0i − 0.607336i −0.952778 0.303668i $$-0.901789\pi$$
0.952778 0.303668i $$-0.0982114\pi$$
$$828$$ 0 0
$$829$$ −42150.0 −1.76590 −0.882949 0.469468i $$-0.844446\pi$$
−0.882949 + 0.469468i $$0.844446\pi$$
$$830$$ 0 0
$$831$$ 16182.0 0.675508
$$832$$ 0 0
$$833$$ − 3262.00i − 0.135680i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7776.00i 0.321121i
$$838$$ 0 0
$$839$$ 13400.0 0.551394 0.275697 0.961245i $$-0.411091\pi$$
0.275697 + 0.961245i $$0.411091\pi$$
$$840$$ 0 0
$$841$$ 28511.0 1.16901
$$842$$ 0 0
$$843$$ − 7326.00i − 0.299313i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 32952.0i − 1.33677i
$$848$$ 0 0
$$849$$ −8316.00 −0.336165
$$850$$ 0 0
$$851$$ −5712.00 −0.230088
$$852$$ 0 0
$$853$$ − 8658.00i − 0.347531i −0.984787 0.173766i $$-0.944406\pi$$
0.984787 0.173766i $$-0.0555935\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 42826.0i − 1.70701i −0.521084 0.853505i $$-0.674472\pi$$
0.521084 0.853505i $$-0.325528\pi$$
$$858$$ 0 0
$$859$$ −35900.0 −1.42595 −0.712976 0.701189i $$-0.752653\pi$$
−0.712976 + 0.701189i $$0.752653\pi$$
$$860$$ 0 0
$$861$$ −8784.00 −0.347686
$$862$$ 0 0
$$863$$ 3088.00i 0.121804i 0.998144 + 0.0609019i $$0.0193977\pi$$
−0.998144 + 0.0609019i $$0.980602\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 14151.0i − 0.554317i
$$868$$ 0 0
$$869$$ 12480.0 0.487175
$$870$$ 0 0
$$871$$ 1848.00 0.0718910
$$872$$ 0 0
$$873$$ 7794.00i 0.302161i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 35274.0i 1.35817i 0.734058 + 0.679087i $$0.237624\pi$$
−0.734058 + 0.679087i $$0.762376\pi$$
$$878$$ 0 0
$$879$$ 13626.0 0.522860
$$880$$ 0 0
$$881$$ 25042.0 0.957646 0.478823 0.877911i $$-0.341064\pi$$
0.478823 + 0.877911i $$0.341064\pi$$
$$882$$ 0 0
$$883$$ − 12572.0i − 0.479141i −0.970879 0.239570i $$-0.922993\pi$$
0.970879 0.239570i $$-0.0770066\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 21864.0i − 0.827645i −0.910358 0.413823i $$-0.864193\pi$$
0.910358 0.413823i $$-0.135807\pi$$
$$888$$ 0 0
$$889$$ 46464.0 1.75293
$$890$$ 0 0
$$891$$ −4212.00 −0.158370
$$892$$ 0 0
$$893$$ − 5120.00i − 0.191864i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 11088.0i 0.412729i
$$898$$ 0 0
$$899$$ −66240.0 −2.45743
$$900$$ 0 0
$$901$$ 4732.00 0.174968
$$902$$ 0 0
$$903$$ − 13536.0i − 0.498837i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 31236.0i 1.14352i 0.820420 + 0.571761i $$0.193740\pi$$
−0.820420 + 0.571761i $$0.806260\pi$$
$$908$$ 0 0
$$909$$ 10962.0 0.399985
$$910$$ 0 0
$$911$$ −8272.00 −0.300838 −0.150419 0.988622i $$-0.548062\pi$$
−0.150419 + 0.988622i $$0.548062\pi$$
$$912$$ 0 0
$$913$$ 63024.0i 2.28455i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 17568.0i 0.632657i
$$918$$ 0 0
$$919$$ 20200.0 0.725067 0.362533 0.931971i $$-0.381912\pi$$
0.362533 + 0.931971i $$0.381912\pi$$
$$920$$ 0 0
$$921$$ 15348.0 0.549114
$$922$$ 0 0
$$923$$ 7216.00i 0.257332i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 792.000i − 0.0280612i
$$928$$ 0 0
$$929$$ −31010.0 −1.09516 −0.547581 0.836753i $$-0.684451\pi$$
−0.547581 + 0.836753i $$0.684451\pi$$
$$930$$ 0 0
$$931$$ 4660.00 0.164044
$$932$$ 0 0
$$933$$ − 8424.00i − 0.295594i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 39174.0i 1.36580i 0.730510 + 0.682902i $$0.239283\pi$$
−0.730510 + 0.682902i $$0.760717\pi$$
$$938$$ 0 0
$$939$$ −21954.0 −0.762984
$$940$$ 0 0
$$941$$ −4138.00 −0.143353 −0.0716764 0.997428i $$-0.522835\pi$$
−0.0716764 + 0.997428i $$0.522835\pi$$
$$942$$ 0 0
$$943$$ 20496.0i 0.707785i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 23676.0i 0.812425i 0.913779 + 0.406213i $$0.133151\pi$$
−0.913779 + 0.406213i $$0.866849\pi$$
$$948$$ 0 0
$$949$$ 836.000 0.0285961
$$950$$ 0 0
$$951$$ −6738.00 −0.229752
$$952$$ 0 0
$$953$$ 18922.0i 0.643173i 0.946880 + 0.321586i $$0.104216\pi$$
−0.946880 + 0.321586i $$0.895784\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 35880.0i − 1.21195i
$$958$$ 0 0
$$959$$ 53136.0 1.78921
$$960$$ 0 0
$$961$$ 53153.0 1.78420
$$962$$ 0 0
$$963$$ − 324.000i − 0.0108419i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 39656.0i 1.31877i 0.751805 + 0.659385i $$0.229183\pi$$
−0.751805 + 0.659385i $$0.770817\pi$$
$$968$$ 0 0
$$969$$ −840.000 −0.0278480
$$970$$ 0 0
$$971$$ 33228.0 1.09818 0.549092 0.835762i $$-0.314974\pi$$
0.549092 + 0.835762i $$0.314974\pi$$
$$972$$ 0 0
$$973$$ − 480.000i − 0.0158151i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 974.000i 0.0318946i 0.999873 + 0.0159473i $$0.00507640\pi$$
−0.999873 + 0.0159473i $$0.994924\pi$$
$$978$$ 0 0
$$979$$ 17160.0 0.560200
$$980$$ 0 0
$$981$$ −8730.00 −0.284126
$$982$$ 0 0
$$983$$ 13608.0i 0.441534i 0.975327 + 0.220767i $$0.0708560\pi$$
−0.975327 + 0.220767i $$0.929144\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 18432.0i − 0.594425i
$$988$$ 0 0
$$989$$ −31584.0 −1.01548
$$990$$ 0 0
$$991$$ −13472.0 −0.431839 −0.215919 0.976411i $$-0.569275\pi$$
−0.215919 + 0.976411i $$0.569275\pi$$
$$992$$ 0 0
$$993$$ 3996.00i 0.127703i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 3234.00i 0.102730i 0.998680 + 0.0513650i $$0.0163572\pi$$
−0.998680 + 0.0513650i $$0.983643\pi$$
$$998$$ 0 0
$$999$$ −918.000 −0.0290733
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1200.4.f.b.49.1 2
4.3 odd 2 75.4.b.b.49.1 2
5.2 odd 4 240.4.a.e.1.1 1
5.3 odd 4 1200.4.a.t.1.1 1
5.4 even 2 inner 1200.4.f.b.49.2 2
12.11 even 2 225.4.b.e.199.2 2
15.2 even 4 720.4.a.n.1.1 1
20.3 even 4 75.4.a.b.1.1 1
20.7 even 4 15.4.a.a.1.1 1
20.19 odd 2 75.4.b.b.49.2 2
40.27 even 4 960.4.a.b.1.1 1
40.37 odd 4 960.4.a.ba.1.1 1
60.23 odd 4 225.4.a.f.1.1 1
60.47 odd 4 45.4.a.c.1.1 1
60.59 even 2 225.4.b.e.199.1 2
140.27 odd 4 735.4.a.e.1.1 1
180.7 even 12 405.4.e.g.271.1 2
180.47 odd 12 405.4.e.i.271.1 2
180.67 even 12 405.4.e.g.136.1 2
180.167 odd 12 405.4.e.i.136.1 2
220.87 odd 4 1815.4.a.e.1.1 1
420.167 even 4 2205.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 20.7 even 4
45.4.a.c.1.1 1 60.47 odd 4
75.4.a.b.1.1 1 20.3 even 4
75.4.b.b.49.1 2 4.3 odd 2
75.4.b.b.49.2 2 20.19 odd 2
225.4.a.f.1.1 1 60.23 odd 4
225.4.b.e.199.1 2 60.59 even 2
225.4.b.e.199.2 2 12.11 even 2
240.4.a.e.1.1 1 5.2 odd 4
405.4.e.g.136.1 2 180.67 even 12
405.4.e.g.271.1 2 180.7 even 12
405.4.e.i.136.1 2 180.167 odd 12
405.4.e.i.271.1 2 180.47 odd 12
720.4.a.n.1.1 1 15.2 even 4
735.4.a.e.1.1 1 140.27 odd 4
960.4.a.b.1.1 1 40.27 even 4
960.4.a.ba.1.1 1 40.37 odd 4
1200.4.a.t.1.1 1 5.3 odd 4
1200.4.f.b.49.1 2 1.1 even 1 trivial
1200.4.f.b.49.2 2 5.4 even 2 inner
1815.4.a.e.1.1 1 220.87 odd 4
2205.4.a.l.1.1 1 420.167 even 4