# Properties

 Label 2025.2.b.b.649.1 Level $2025$ Weight $2$ Character 2025.649 Analytic conductor $16.170$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.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: $$16.1697064093$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 405) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.b.649.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} -2.00000 q^{4} +O(q^{10})$$ $$q-2.00000i q^{2} -2.00000 q^{4} +5.00000 q^{11} +4.00000i q^{13} -4.00000 q^{16} +4.00000i q^{17} +5.00000 q^{19} -10.0000i q^{22} +6.00000i q^{23} +8.00000 q^{26} +5.00000 q^{29} -9.00000 q^{31} +8.00000i q^{32} +8.00000 q^{34} +10.0000i q^{37} -10.0000i q^{38} +7.00000 q^{41} -2.00000i q^{43} -10.0000 q^{44} +12.0000 q^{46} -2.00000i q^{47} +7.00000 q^{49} -8.00000i q^{52} +8.00000i q^{53} -10.0000i q^{58} +1.00000 q^{59} -2.00000 q^{61} +18.0000i q^{62} +8.00000 q^{64} -6.00000i q^{67} -8.00000i q^{68} +1.00000 q^{71} -8.00000i q^{73} +20.0000 q^{74} -10.0000 q^{76} -12.0000 q^{79} -14.0000i q^{82} +6.00000i q^{83} -4.00000 q^{86} +9.00000 q^{89} -12.0000i q^{92} -4.00000 q^{94} -14.0000i q^{97} -14.0000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} + 10 q^{11} - 8 q^{16} + 10 q^{19} + 16 q^{26} + 10 q^{29} - 18 q^{31} + 16 q^{34} + 14 q^{41} - 20 q^{44} + 24 q^{46} + 14 q^{49} + 2 q^{59} - 4 q^{61} + 16 q^{64} + 2 q^{71} + 40 q^{74} - 20 q^{76} - 24 q^{79} - 8 q^{86} + 18 q^{89} - 8 q^{94}+O(q^{100})$$ 2 * q - 4 * q^4 + 10 * q^11 - 8 * q^16 + 10 * q^19 + 16 * q^26 + 10 * q^29 - 18 * q^31 + 16 * q^34 + 14 * q^41 - 20 * q^44 + 24 * q^46 + 14 * q^49 + 2 * q^59 - 4 * q^61 + 16 * q^64 + 2 * q^71 + 40 * q^74 - 20 * q^76 - 24 * q^79 - 8 * q^86 + 18 * q^89 - 8 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 1.41421i −0.707107 0.707107i $$-0.750000\pi$$
0.707107 0.707107i $$-0.250000\pi$$
$$3$$ 0 0
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ 0 0
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 10.0000i − 2.13201i
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 8.00000 1.56893
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ 8.00000i 1.41421i
$$33$$ 0 0
$$34$$ 8.00000 1.37199
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 10.0000i − 1.62221i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ −10.0000 −1.50756
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ − 2.00000i − 0.291730i −0.989305 0.145865i $$-0.953403\pi$$
0.989305 0.145865i $$-0.0465965\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 8.00000i − 1.10940i
$$53$$ 8.00000i 1.09888i 0.835532 + 0.549442i $$0.185160\pi$$
−0.835532 + 0.549442i $$0.814840\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ − 10.0000i − 1.31306i
$$59$$ 1.00000 0.130189 0.0650945 0.997879i $$-0.479265\pi$$
0.0650945 + 0.997879i $$0.479265\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 18.0000i 2.28600i
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 6.00000i − 0.733017i −0.930415 0.366508i $$-0.880553\pi$$
0.930415 0.366508i $$-0.119447\pi$$
$$68$$ − 8.00000i − 0.970143i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.00000 0.118678 0.0593391 0.998238i $$-0.481101\pi$$
0.0593391 + 0.998238i $$0.481101\pi$$
$$72$$ 0 0
$$73$$ − 8.00000i − 0.936329i −0.883641 0.468165i $$-0.844915\pi$$
0.883641 0.468165i $$-0.155085\pi$$
$$74$$ 20.0000 2.32495
$$75$$ 0 0
$$76$$ −10.0000 −1.14708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 14.0000i − 1.54604i
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.00000 0.953998 0.476999 0.878904i $$-0.341725\pi$$
0.476999 + 0.878904i $$0.341725\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 12.0000i − 1.25109i
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ − 14.0000i − 1.41421i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ 2.00000i 0.197066i 0.995134 + 0.0985329i $$0.0314150\pi$$
−0.995134 + 0.0985329i $$0.968585\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 16.0000 1.55406
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ 0 0
$$109$$ 1.00000 0.0957826 0.0478913 0.998853i $$-0.484750\pi$$
0.0478913 + 0.998853i $$0.484750\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 16.0000i − 1.50515i −0.658505 0.752577i $$-0.728811\pi$$
0.658505 0.752577i $$-0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ 0 0
$$118$$ − 2.00000i − 0.184115i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 4.00000i 0.362143i
$$123$$ 0 0
$$124$$ 18.0000 1.61645
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ 19.0000 1.61156 0.805779 0.592216i $$-0.201747\pi$$
0.805779 + 0.592216i $$0.201747\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 2.00000i − 0.167836i
$$143$$ 20.0000i 1.67248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −16.0000 −1.32417
$$147$$ 0 0
$$148$$ − 20.0000i − 1.64399i
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 24.0000i 1.90934i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ −14.0000 −1.09322
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000i 0.304997i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −20.0000 −1.50756
$$177$$ 0 0
$$178$$ − 18.0000i − 1.34916i
$$179$$ −23.0000 −1.71910 −0.859550 0.511051i $$-0.829256\pi$$
−0.859550 + 0.511051i $$0.829256\pi$$
$$180$$ 0 0
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 20.0000i 1.46254i
$$188$$ 4.00000i 0.291730i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1.00000 0.0723575 0.0361787 0.999345i $$-0.488481\pi$$
0.0361787 + 0.999345i $$0.488481\pi$$
$$192$$ 0 0
$$193$$ 26.0000i 1.87152i 0.352636 + 0.935760i $$0.385285\pi$$
−0.352636 + 0.935760i $$0.614715\pi$$
$$194$$ −28.0000 −2.01028
$$195$$ 0 0
$$196$$ −14.0000 −1.00000
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ − 16.0000i − 1.10940i
$$209$$ 25.0000 1.72929
$$210$$ 0 0
$$211$$ 11.0000 0.757271 0.378636 0.925546i $$-0.376393\pi$$
0.378636 + 0.925546i $$0.376393\pi$$
$$212$$ − 16.0000i − 1.09888i
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ − 2.00000i − 0.135457i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −16.0000 −1.07628
$$222$$ 0 0
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −32.0000 −2.12861
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −2.00000 −0.130189
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −11.0000 −0.708572 −0.354286 0.935137i $$-0.615276\pi$$
−0.354286 + 0.935137i $$0.615276\pi$$
$$242$$ − 28.0000i − 1.79991i
$$243$$ 0 0
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 20.0000i 1.27257i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 30.0000i 1.88608i
$$254$$ −32.0000 −2.00786
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 30.0000i − 1.85341i
$$263$$ 10.0000i 0.616626i 0.951285 + 0.308313i $$0.0997645\pi$$
−0.951285 + 0.308313i $$0.900236\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.0000i 0.733017i
$$269$$ 31.0000 1.89010 0.945052 0.326921i $$-0.106011\pi$$
0.945052 + 0.326921i $$0.106011\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 16.0000i − 0.970143i
$$273$$ 0 0
$$274$$ −24.0000 −1.44989
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.0000i 1.08152i 0.841178 + 0.540758i $$0.181862\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ − 38.0000i − 2.27909i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 6.00000i 0.356663i 0.983970 + 0.178331i $$0.0570699\pi$$
−0.983970 + 0.178331i $$0.942930\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 40.0000 2.36525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 16.0000i 0.936329i
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ − 4.00000i − 0.231714i
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 10.0000i 0.575435i
$$303$$ 0 0
$$304$$ −20.0000 −1.14708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 10.0000i − 0.570730i −0.958419 0.285365i $$-0.907885\pi$$
0.958419 0.285365i $$-0.0921148\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −9.00000 −0.510343 −0.255172 0.966896i $$-0.582132\pi$$
−0.255172 + 0.966896i $$0.582132\pi$$
$$312$$ 0 0
$$313$$ − 4.00000i − 0.226093i −0.993590 0.113047i $$-0.963939\pi$$
0.993590 0.113047i $$-0.0360610\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ 24.0000 1.35011
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ 0 0
$$319$$ 25.0000 1.39973
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 20.0000i 1.11283i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −21.0000 −1.15426 −0.577132 0.816651i $$-0.695828\pi$$
−0.577132 + 0.816651i $$0.695828\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 0 0
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ 6.00000i 0.326357i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −45.0000 −2.43689
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ − 20.0000i − 1.07366i −0.843692 0.536828i $$-0.819622\pi$$
0.843692 0.536828i $$-0.180378\pi$$
$$348$$ 0 0
$$349$$ −13.0000 −0.695874 −0.347937 0.937518i $$-0.613118\pi$$
−0.347937 + 0.937518i $$0.613118\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 40.0000i 2.13201i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ 46.0000i 2.43118i
$$359$$ 27.0000 1.42501 0.712503 0.701669i $$-0.247562\pi$$
0.712503 + 0.701669i $$0.247562\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 50.0000i 2.62794i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 18.0000i − 0.939592i −0.882775 0.469796i $$-0.844327\pi$$
0.882775 0.469796i $$-0.155673\pi$$
$$368$$ − 24.0000i − 1.25109i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 16.0000i 0.828449i 0.910175 + 0.414224i $$0.135947\pi$$
−0.910175 + 0.414224i $$0.864053\pi$$
$$374$$ 40.0000 2.06835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 20.0000i 1.03005i
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 2.00000i − 0.102329i
$$383$$ − 36.0000i − 1.83951i −0.392488 0.919757i $$-0.628386\pi$$
0.392488 0.919757i $$-0.371614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 52.0000 2.64673
$$387$$ 0 0
$$388$$ 28.0000i 1.42148i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −24.0000 −1.20910
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 38.0000i 1.90717i 0.301131 + 0.953583i $$0.402636\pi$$
−0.301131 + 0.953583i $$0.597364\pi$$
$$398$$ 32.0000i 1.60402i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ − 36.0000i − 1.79329i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 50.0000i 2.47841i
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 4.00000i − 0.197066i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −32.0000 −1.56893
$$417$$ 0 0
$$418$$ − 50.0000i − 2.44558i
$$419$$ −16.0000 −0.781651 −0.390826 0.920465i $$-0.627810\pi$$
−0.390826 + 0.920465i $$0.627810\pi$$
$$420$$ 0 0
$$421$$ 13.0000 0.633581 0.316791 0.948495i $$-0.397395\pi$$
0.316791 + 0.948495i $$0.397395\pi$$
$$422$$ − 22.0000i − 1.07094i
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 12.0000i − 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 30.0000i 1.43509i
$$438$$ 0 0
$$439$$ 29.0000 1.38409 0.692047 0.721852i $$-0.256709\pi$$
0.692047 + 0.721852i $$0.256709\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 32.0000i 1.52208i
$$443$$ 6.00000i 0.285069i 0.989790 + 0.142534i $$0.0455251\pi$$
−0.989790 + 0.142534i $$0.954475\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −17.0000 −0.802280 −0.401140 0.916017i $$-0.631386\pi$$
−0.401140 + 0.916017i $$0.631386\pi$$
$$450$$ 0 0
$$451$$ 35.0000 1.64809
$$452$$ 32.0000i 1.50515i
$$453$$ 0 0
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0000i 1.77757i 0.458329 + 0.888783i $$0.348448\pi$$
−0.458329 + 0.888783i $$0.651552\pi$$
$$458$$ 12.0000i 0.560723i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −15.0000 −0.698620 −0.349310 0.937007i $$-0.613584\pi$$
−0.349310 + 0.937007i $$0.613584\pi$$
$$462$$ 0 0
$$463$$ − 6.00000i − 0.278844i −0.990233 0.139422i $$-0.955476\pi$$
0.990233 0.139422i $$-0.0445244\pi$$
$$464$$ −20.0000 −0.928477
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ − 4.00000i − 0.185098i −0.995708 0.0925490i $$-0.970499\pi$$
0.995708 0.0925490i $$-0.0295015\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 10.0000i − 0.459800i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ − 32.0000i − 1.46365i
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ −40.0000 −1.82384
$$482$$ 22.0000i 1.00207i
$$483$$ 0 0
$$484$$ −28.0000 −1.27273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 43.0000 1.94056 0.970281 0.241979i $$-0.0777966\pi$$
0.970281 + 0.241979i $$0.0777966\pi$$
$$492$$ 0 0
$$493$$ 20.0000i 0.900755i
$$494$$ 40.0000 1.79969
$$495$$ 0 0
$$496$$ 36.0000 1.61645
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −7.00000 −0.313363 −0.156682 0.987649i $$-0.550080\pi$$
−0.156682 + 0.987649i $$0.550080\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 24.0000i 1.07117i
$$503$$ − 20.0000i − 0.891756i −0.895094 0.445878i $$-0.852892\pi$$
0.895094 0.445878i $$-0.147108\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 60.0000 2.66733
$$507$$ 0 0
$$508$$ 32.0000i 1.41977i
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 32.0000i − 1.41421i
$$513$$ 0 0
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 10.0000i − 0.439799i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ −30.0000 −1.31056
$$525$$ 0 0
$$526$$ 20.0000 0.872041
$$527$$ − 36.0000i − 1.56818i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 28.0000i 1.21281i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ − 62.0000i − 2.67301i
$$539$$ 35.0000 1.50756
$$540$$ 0 0
$$541$$ 3.00000 0.128980 0.0644900 0.997918i $$-0.479458\pi$$
0.0644900 + 0.997918i $$0.479458\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ 0 0
$$544$$ −32.0000 −1.37199
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ 24.0000i 1.02523i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 25.0000 1.06504
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 36.0000 1.52949
$$555$$ 0 0
$$556$$ −38.0000 −1.61156
$$557$$ 24.0000i 1.01691i 0.861088 + 0.508456i $$0.169784\pi$$
−0.861088 + 0.508456i $$0.830216\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 12.0000i − 0.506189i
$$563$$ − 18.0000i − 0.758610i −0.925272 0.379305i $$-0.876163\pi$$
0.925272 0.379305i $$-0.123837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 3.00000 0.125767 0.0628833 0.998021i $$-0.479970\pi$$
0.0628833 + 0.998021i $$0.479970\pi$$
$$570$$ 0 0
$$571$$ 5.00000 0.209243 0.104622 0.994512i $$-0.466637\pi$$
0.104622 + 0.994512i $$0.466637\pi$$
$$572$$ − 40.0000i − 1.67248i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 16.0000i 0.666089i 0.942911 + 0.333044i $$0.108076\pi$$
−0.942911 + 0.333044i $$0.891924\pi$$
$$578$$ − 2.00000i − 0.0831890i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000i 1.65663i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 36.0000 1.48715
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ 0 0
$$589$$ −45.0000 −1.85419
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 40.0000i − 1.64399i
$$593$$ − 14.0000i − 0.574911i −0.957794 0.287456i $$-0.907191\pi$$
0.957794 0.287456i $$-0.0928094\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ 0 0
$$598$$ 48.0000i 1.96287i
$$599$$ 17.0000 0.694601 0.347301 0.937754i $$-0.387098\pi$$
0.347301 + 0.937754i $$0.387098\pi$$
$$600$$ 0 0
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 34.0000i − 1.38002i −0.723801 0.690009i $$-0.757607\pi$$
0.723801 0.690009i $$-0.242393\pi$$
$$608$$ 40.0000i 1.62221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 4.00000i 0.161558i 0.996732 + 0.0807792i $$0.0257409\pi$$
−0.996732 + 0.0807792i $$0.974259\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 24.0000i 0.966204i 0.875564 + 0.483102i $$0.160490\pi$$
−0.875564 + 0.483102i $$0.839510\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000i 0.721734i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 4.00000i 0.159617i
$$629$$ −40.0000 −1.59490
$$630$$ 0 0
$$631$$ −17.0000 −0.676759 −0.338380 0.941010i $$-0.609879\pi$$
−0.338380 + 0.941010i $$0.609879\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 4.00000 0.158860
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 28.0000i 1.10940i
$$638$$ − 50.0000i − 1.97952i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 0 0
$$643$$ 6.00000i 0.236617i 0.992977 + 0.118308i $$0.0377472\pi$$
−0.992977 + 0.118308i $$0.962253\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 40.0000 1.57378
$$647$$ 10.0000i 0.393141i 0.980490 + 0.196570i $$0.0629804\pi$$
−0.980490 + 0.196570i $$0.937020\pi$$
$$648$$ 0 0
$$649$$ 5.00000 0.196267
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 16.0000i − 0.626608i
$$653$$ − 8.00000i − 0.313064i −0.987673 0.156532i $$-0.949969\pi$$
0.987673 0.156532i $$-0.0500315\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −28.0000 −1.09322
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ 42.0000i 1.63238i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 30.0000i 1.16160i
$$668$$ 24.0000i 0.928588i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ 42.0000i 1.61898i 0.587133 + 0.809491i $$0.300257\pi$$
−0.587133 + 0.809491i $$0.699743\pi$$
$$674$$ 16.0000 0.616297
$$675$$ 0 0
$$676$$ 6.00000 0.230769
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 90.0000i 3.44628i
$$683$$ 48.0000i 1.83667i 0.395805 + 0.918334i $$0.370466\pi$$
−0.395805 + 0.918334i $$0.629534\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ −32.0000 −1.21910
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ 0 0
$$694$$ −40.0000 −1.51838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 28.0000i 1.06058i
$$698$$ 26.0000i 0.984115i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19.0000 0.717620 0.358810 0.933411i $$-0.383183\pi$$
0.358810 + 0.933411i $$0.383183\pi$$
$$702$$ 0 0
$$703$$ 50.0000i 1.88579i
$$704$$ 40.0000 1.50756
$$705$$ 0 0
$$706$$ −36.0000 −1.35488
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 54.0000i − 2.02232i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 46.0000 1.71910
$$717$$ 0 0
$$718$$ − 54.0000i − 2.01526i
$$719$$ −27.0000 −1.00693 −0.503465 0.864016i $$-0.667942\pi$$
−0.503465 + 0.864016i $$0.667942\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 12.0000i − 0.446594i
$$723$$ 0 0
$$724$$ 50.0000 1.85824
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 44.0000i − 1.63187i −0.578144 0.815935i $$-0.696223\pi$$
0.578144 0.815935i $$-0.303777\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ − 46.0000i − 1.69905i −0.527549 0.849524i $$-0.676889\pi$$
0.527549 0.849524i $$-0.323111\pi$$
$$734$$ −36.0000 −1.32878
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ − 30.0000i − 1.10506i
$$738$$ 0 0
$$739$$ 35.0000 1.28750 0.643748 0.765238i $$-0.277379\pi$$
0.643748 + 0.765238i $$0.277379\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 4.00000i − 0.146746i −0.997305 0.0733729i $$-0.976624\pi$$
0.997305 0.0733729i $$-0.0233763\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 32.0000 1.17160
$$747$$ 0 0
$$748$$ − 40.0000i − 1.46254i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ 0 0
$$754$$ 40.0000 1.45671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 28.0000i − 1.01768i −0.860862 0.508839i $$-0.830075\pi$$
0.860862 0.508839i $$-0.169925\pi$$
$$758$$ 8.00000i 0.290573i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2.00000 −0.0723575
$$765$$ 0 0
$$766$$ −72.0000 −2.60147
$$767$$ 4.00000i 0.144432i
$$768$$ 0 0
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 52.0000i − 1.87152i
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 12.0000i 0.430221i
$$779$$ 35.0000 1.25401
$$780$$ 0 0
$$781$$ 5.00000 0.178914
$$782$$ 48.0000i 1.71648i
$$783$$ 0 0
$$784$$ −28.0000 −1.00000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 24.0000i 0.854965i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 8.00000i − 0.284088i
$$794$$ 76.0000 2.69714
$$795$$ 0 0
$$796$$ 32.0000 1.13421
$$797$$ − 4.00000i − 0.141687i −0.997487 0.0708436i $$-0.977431\pi$$
0.997487 0.0708436i $$-0.0225691\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 60.0000i − 2.11867i
$$803$$ − 40.0000i − 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −72.0000 −2.53609
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −7.00000 −0.246107 −0.123053 0.992400i $$-0.539269\pi$$
−0.123053 + 0.992400i $$0.539269\pi$$
$$810$$ 0 0
$$811$$ −21.0000 −0.737410 −0.368705 0.929547i $$-0.620199\pi$$
−0.368705 + 0.929547i $$0.620199\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 100.000 3.50500
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 10.0000i − 0.349856i
$$818$$ 28.0000i 0.978997i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3.00000 0.104701 0.0523504 0.998629i $$-0.483329\pi$$
0.0523504 + 0.998629i $$0.483329\pi$$
$$822$$ 0 0
$$823$$ − 46.0000i − 1.60346i −0.597687 0.801730i $$-0.703913\pi$$
0.597687 0.801730i $$-0.296087\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 44.0000i − 1.53003i −0.644013 0.765015i $$-0.722732\pi$$
0.644013 0.765015i $$-0.277268\pi$$
$$828$$ 0 0
$$829$$ −27.0000 −0.937749 −0.468874 0.883265i $$-0.655340\pi$$
−0.468874 + 0.883265i $$0.655340\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 32.0000i 1.10940i
$$833$$ 28.0000i 0.970143i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −50.0000 −1.72929
$$837$$ 0 0
$$838$$ 32.0000i 1.10542i
$$839$$ −13.0000 −0.448810 −0.224405 0.974496i $$-0.572044\pi$$
−0.224405 + 0.974496i $$0.572044\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ − 26.0000i − 0.896019i
$$843$$ 0 0
$$844$$ −22.0000 −0.757271
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 32.0000i − 1.09888i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −60.0000 −2.05677
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 32.0000i − 1.09310i −0.837427 0.546550i $$-0.815941\pi$$
0.837427 0.546550i $$-0.184059\pi$$
$$858$$ 0 0
$$859$$ −55.0000 −1.87658 −0.938288 0.345855i $$-0.887589\pi$$
−0.938288 + 0.345855i $$0.887589\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 6.00000i − 0.204361i
$$863$$ − 44.0000i − 1.49778i −0.662696 0.748889i $$-0.730588\pi$$
0.662696 0.748889i $$-0.269412\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −52.0000 −1.76703
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −60.0000 −2.03536
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 60.0000 2.02953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 24.0000i 0.810422i 0.914223 + 0.405211i $$0.132802\pi$$
−0.914223 + 0.405211i $$0.867198\pi$$
$$878$$ − 58.0000i − 1.95741i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −25.0000 −0.842271 −0.421136 0.906998i $$-0.638368\pi$$
−0.421136 + 0.906998i $$0.638368\pi$$
$$882$$ 0 0
$$883$$ − 52.0000i − 1.74994i −0.484178 0.874970i $$-0.660881\pi$$
0.484178 0.874970i $$-0.339119\pi$$
$$884$$ 32.0000 1.07628
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ − 6.00000i − 0.201460i −0.994914 0.100730i $$-0.967882\pi$$
0.994914 0.100730i $$-0.0321179\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ − 10.0000i − 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 34.0000i 1.13459i
$$899$$ −45.0000 −1.50083
$$900$$ 0 0
$$901$$ −32.0000 −1.06607
$$902$$ − 70.0000i − 2.33075i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 18.0000i 0.597680i 0.954303 + 0.298840i $$0.0965997\pi$$
−0.954303 + 0.298840i $$0.903400\pi$$
$$908$$ 8.00000i 0.265489i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −49.0000 −1.62344 −0.811721 0.584045i $$-0.801469\pi$$
−0.811721 + 0.584045i $$0.801469\pi$$
$$912$$ 0 0
$$913$$ 30.0000i 0.992855i
$$914$$ 76.0000 2.51386
$$915$$ 0 0
$$916$$ 12.0000 0.396491
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 11.0000 0.362857 0.181428 0.983404i $$-0.441928\pi$$
0.181428 + 0.983404i $$0.441928\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 30.0000i 0.987997i
$$923$$ 4.00000i 0.131662i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −12.0000 −0.394344
$$927$$ 0 0
$$928$$ 40.0000i 1.31306i
$$929$$ −1.00000 −0.0328089 −0.0164045 0.999865i $$-0.505222\pi$$
−0.0164045 + 0.999865i $$0.505222\pi$$
$$930$$ 0 0
$$931$$ 35.0000 1.14708
$$932$$ − 12.0000i − 0.393073i
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 42.0000i − 1.37208i −0.727564 0.686040i $$-0.759347\pi$$
0.727564 0.686040i $$-0.240653\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ 0 0
$$943$$ 42.0000i 1.36771i
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −20.0000 −0.650256
$$947$$ − 18.0000i − 0.584921i −0.956278 0.292461i $$-0.905526\pi$$
0.956278 0.292461i $$-0.0944741\pi$$
$$948$$ 0 0
$$949$$ 32.0000 1.03876
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 16.0000i − 0.518291i −0.965838 0.259145i $$-0.916559\pi$$
0.965838 0.259145i $$-0.0834409\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −32.0000 −1.03495
$$957$$ 0 0
$$958$$ − 30.0000i − 0.969256i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 80.0000i 2.57930i
$$963$$ 0 0
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 40.0000i 1.28631i 0.765735 + 0.643157i $$0.222376\pi$$
−0.765735 + 0.643157i $$0.777624\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 28.0000i 0.895799i 0.894084 + 0.447900i $$0.147828\pi$$
−0.894084 + 0.447900i $$0.852172\pi$$
$$978$$ 0 0
$$979$$ 45.0000 1.43821
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 86.0000i − 2.74437i
$$983$$ 48.0000i 1.53096i 0.643458 + 0.765481i $$0.277499\pi$$
−0.643458 + 0.765481i $$0.722501\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 40.0000 1.27386
$$987$$ 0 0
$$988$$ − 40.0000i − 1.27257i
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ −1.00000 −0.0317660 −0.0158830 0.999874i $$-0.505056\pi$$
−0.0158830 + 0.999874i $$0.505056\pi$$
$$992$$ − 72.0000i − 2.28600i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 48.0000i − 1.52018i −0.649821 0.760088i $$-0.725156\pi$$
0.649821 0.760088i $$-0.274844\pi$$
$$998$$ 14.0000i 0.443162i
$$999$$ 0 0
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 2025.2.b.b.649.1 2
3.2 odd 2 2025.2.b.a.649.2 2
5.2 odd 4 405.2.a.f.1.1 yes 1
5.3 odd 4 2025.2.a.a.1.1 1
5.4 even 2 inner 2025.2.b.b.649.2 2
15.2 even 4 405.2.a.a.1.1 1
15.8 even 4 2025.2.a.f.1.1 1
15.14 odd 2 2025.2.b.a.649.1 2
20.7 even 4 6480.2.a.r.1.1 1
45.2 even 12 405.2.e.g.271.1 2
45.7 odd 12 405.2.e.a.271.1 2
45.22 odd 12 405.2.e.a.136.1 2
45.32 even 12 405.2.e.g.136.1 2
60.47 odd 4 6480.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.a.1.1 1 15.2 even 4
405.2.a.f.1.1 yes 1 5.2 odd 4
405.2.e.a.136.1 2 45.22 odd 12
405.2.e.a.271.1 2 45.7 odd 12
405.2.e.g.136.1 2 45.32 even 12
405.2.e.g.271.1 2 45.2 even 12
2025.2.a.a.1.1 1 5.3 odd 4
2025.2.a.f.1.1 1 15.8 even 4
2025.2.b.a.649.1 2 15.14 odd 2
2025.2.b.a.649.2 2 3.2 odd 2
2025.2.b.b.649.1 2 1.1 even 1 trivial
2025.2.b.b.649.2 2 5.4 even 2 inner
6480.2.a.f.1.1 1 60.47 odd 4
6480.2.a.r.1.1 1 20.7 even 4